Mail Code: 943054042
Email: icmeadmissions@stanford.edu
Web Site: http://icme.stanford.edu/
Courses offered by the Institute for Computational and Mathematical Engineering are listed under the subject code CME on the Stanford Bulletin's ExploreCourses web site.
ICME is a degree granting (M.S./Ph.D.) interdisciplinary institute at the intersection of mathematics, computing, engineering and applied sciences. ICME was founded in 2004, building upon the Scientific Computing and Computational Mathematics Program (est. 1989).
At ICME, we design stateoftheart mathematical and computational models, methods, and algorithms for engineering and science applications. The program collaborates closely with engineers and scientists in academia and industry to develop improved computational approaches and advance disciplinary fields. In particular, it leverages Stanford's strength in engineering applications in the physical, biological, mathematical, and information sciences, and has established connections with nearly 20 departments across five schools at Stanford.
The program identifies research areas that would benefit from a multidisciplinary approach in which computational mathematics plays a role. This multidisciplinary intellectual environment is a core strength of ICME, with interaction among students and faculty with diverse backgrounds and expertise. Students and faculty are active in many research areas: aerodynamics and space applications, fluid dynamics, protein folding, data science including machine learning and recommender systems, ocean dynamics, climate modeling, reservoir engineering, computer graphics, financial mathematics, and many more.
The program trains students and scholars from across Stanford in mathematical modeling, scientific computing, and advanced computational algorithms at the undergraduate and graduate levels. Courses typically provide strong theoretical foundations for the solution of real world problems and numerical computations to facilitate application of mathematical techniques and theories. Training offered includes matrix computations, computational probability and combinatorial optimization, optimization, stochastics, numerical solution of partial differential equations, parallel computer algorithms, and new computing paradigms, amongst others.
ICME offers service courses for undergraduates and graduate students to fulfill departmental requirements, core courses for master's and doctoral students in Computational and Mathematical Engineering, and specialized electives in various application areas.
The ICME master's program offers both specialized and general tracks. Currently, the program is offering specialized tracks in Computational Geosciences, Data Science, Imaging Science, and Mathematical and Computational Finance.
Graduate Programs in Computational and Mathematical Engineering
University regulations governing the M.S. and Ph.D. degrees are described in the “Graduate Degrees” section of this bulletin.
Learning Outcomes (Graduate)
The purpose of the master’s program is to provide students with the knowledge and skills necessary for a professional career or doctoral studies. This is done through coursework in mathematical modeling, scientific computing, advanced computational algorithms, and a set of courses from a specific area of application or field. The latter includes computational geoscience, data sciences, imaging sciences, mathematical and computational finance and other interdisciplinary areas that combine advanced mathematics with the classical physical sciences or with challenging interdisciplinary problems emerging within disciplines such as business, biology, medicine, and information.
The Ph.D. is conferred upon candidates who have demonstrated substantial scholarship and the ability to conduct independent research. Through course work and guided research, the program prepares students to make original contributions in Computational and Mathematical Engineering and related fields.
Master of Science in Computational and Mathematical Engineering
The University’s basic requirements for the M.S. degree are discussed in the “Graduate Degrees” section of this bulletin. The following are specific departmental requirements.
The M.S. degree in Computational and Mathematical Engineering is intended as a terminal professional degree and does not lead to the Ph.D. program. Students interested in the doctoral program should apply directly to the Ph.D. program. Master's students who have maintained a minimum grade point average (GPA) of 3.5 are eligible to take the Ph.D. qualifying exam. Qualifying exams in all six areas must be completed before the start of the second year in the program. Those who pass the qualifying examination and secure a research adviser (three quarters of continuous documented research) may continue into the Ph.D. program upon acceptance by the institute before the end of the second year in the program.
Admission
Prospective applicants should consult the Graduate Admissions and the ICME admissions web pages for complete information on admission requirements and deadlines.
Applications to the M.S. program and all supporting documents must be submitted and received online by January 8, 2019, the deadline published on ICME admissions web page.
Prerequisites
Fundamental courses in mathematics and computing may be needed as prerequisites for other courses in the program. Check the prerequisites of each required course. Recommended preparatory courses include advanced undergraduate level courses in linear algebra, probability, differential equations, stochastics, and numerical methods and proficiency in programming.
Coterminal Master's Program
Stanford undergraduates who want to apply for the coterminal master's degree must submit their application no later than eight weeks before the start of the proposed admit quarter. The application must give evidence that the student possesses a potential for strong academic performance at the graduate level. Graduate Record Examination (GRE) General Test scores are required for application review. A student is eligible to apply for admission once the following conditions have been met:
 completion of six nonSummer quarters at Stanford or two nonSummer quarters at Stanford for transfer students
 completion of 120 units toward graduation (UTG) as shown on the undergraduate transcript, including transfer, Advanced Placement exam, and other external test credit
 declaration of an undergraduate major
University Coterminal Requirements
Coterminal master’s degree candidates are expected to complete all master’s degree requirements as described in this bulletin. University requirements for the coterminal master’s degree are described in the “Coterminal Master’s Program” section. University requirements for the master’s degree are described in the "Graduate Degrees" section of this bulletin.
After accepting admission to this coterminal master’s degree program, students may request transfer of courses from the undergraduate to the graduate career to satisfy requirements for the master’s degree. Transfer of courses to the graduate career requires review and approval of both the undergraduate and graduate programs on a case by case basis.
In this master’s program, courses taken two quarters prior to the first graduate quarter, or later, are eligible for consideration for transfer to the graduate career. No courses taken prior to the first quarter of the sophomore year may be used to meet master’s degree requirements.
Course transfers are not possible after the bachelor’s degree has been conferred.
The University requires that the graduate adviser be assigned in the student’s first graduate quarter even though the undergraduate career may still be open. The University also requires that the Master’s Degree Program Proposal be completed by the student and approved by the department by the end of the student’s first graduate quarter.
Requirements for the Master of Science in Computational and Mathematical Engineering
The master's program consists of 45 units of course work taken at Stanford. No thesis is required; however, students may become involved in research projects during the master's program, particularly to explore an interest in continuing to the doctoral program. Although there is no specific background requirement, significant exposure to mathematics and engineering course work is necessary for successful completion of the program.
There are five tracks in the master's program:
 General CME
 Computational Geosciences
 Data Science
 Imaging Science
 Mathematical and Computational Finance
General CME Track
This track is designed for students interested in studying and developing computational tools in those aspects of applied mathematics central to modeling in the physical and engineering sciences. The curriculum consists of core computational and mathematical engineering courses and programming course work, extensive breadth and depth electives, and seminars. Core courses provide instruction in mathematical and computational tools applicable to a wide range of scientific, industrial and engineering disciplines and augment breadth and depth electives of one’s choosing. Programming requirement ensures proficiency in scientific computing and professional computing skills. Seminars highlight emerging research in engineering and sciences.
Requirements
A candidate is required to complete a program of 45 units of courses numbered 200 or above. Courses below 200 level require special approval from the program office. At least 36 of these must be graded units, passed with a grade point average (GPA) of 3.0 (B) or better.
Requirement 1: Foundational (12 units)
Students must demonstrate foundational knowledge in the field by completing four of the six core courses. Courses in this area must be taken for letter grades.
Units  

CME 302  Numerical Linear Algebra  3 
CME 303  Partial Differential Equations of Applied Mathematics  3 
CME 305  Discrete Mathematics and Algorithms  3 
CME 306  Numerical Solution of Partial Differential Equations  3 
CME 307  Optimization  3 
CME 308  Stochastic Methods in Engineering  3 
or CME 298  Basic Probability and Stochastic Processes with Engineering Applications 
Requirement 2: Programming (3 units)
Three units of programming course work demonstrating programming proficiency. All graduate students in the program are required to complete this programming course for letter grade. Programming proficiency at the level of CME 211 is a hard prerequisite; CME 211 can be applied towards elective requirement.
Units  

CME 211  Software Development for Scientists and Engineers (*can only be counted as an elective)  3 
CME 212  Advanced Software Development for Scientists and Engineers  3 
Requirement 3: Breadth Electives (18 units)
18 units of general electives to demonstrate breadth of knowledge in technical area. The elective course list represents automatically accepted electives within the program. However, electives are not limited to the list below, and the list is expanded on a continuing basis. The elective part of the ICME program is meant to be broad and inclusive of relevant courses of comparable rigor to ICME courses. It is recommended that the selected courses include offerings from (at least) two engineering departments, in addition to CME course work. Courses outside this list can be accepted as electives subject to approval by the student’s program adviser.
Units  

Aeronautics and Astronautics  
AA 214B  Numerical Methods for Compressible Flows  3 
AA 214C  Numerical Computation of Viscous Flow  3 
AA 218  Introduction to Symmetry Analysis  3 
Computational and Mathematical Engineering  
CME 215A/215B  Advanced Computational Fluid Dynamics  3 
CME 263  Introduction to Linear Dynamical Systems  3 
CME 279  Computational Biology: Structure and Organization of Biomolecules and Cells  3 
CME 342  Parallel Methods in Numerical Analysis  3 
CME 364A  Convex Optimization I  3 
CME 371  Computational Biology in Four Dimensions  3 
Computer Science  
CS 205A  3  
CS 221  Artificial Intelligence: Principles and Techniques  34 
CS 228  Probabilistic Graphical Models: Principles and Techniques  34 
CS 229  Machine Learning  34 
CS 255  Introduction to Cryptography  3 
CS 261  Optimization and Algorithmic Paradigms  3 
CS 340  Topics in Computer Systems  34 
CS 348A  Computer Graphics: Geometric Modeling & Processing  34 
Electrical Engineering  
EE 223  Applied Quantum Mechanics II  3 
EE 256  Numerical Electromagnetics  3 
EE 376A  Information Theory  3 
Management Science and Engineering  
MS&E 220  Probabilistic Analysis  34 
MS&E 221  Stochastic Modeling  3 
MS&E 223  Simulation  3 
MS&E 226  "Small" Data: Prediction, Inference, Causality  3 
MS&E 251  Introduction to Stochastic Control with Applications  3 
MS&E 310  Linear Programming  3 
MS&E 316  Discrete Mathematics and Algorithms  3 
MS&E 321  Stochastic Systems  3 
MS&E 322  Stochastic Calculus and Control  3 
Mathematics  
MATH 136  Stochastic Processes  3 
MATH 171  Fundamental Concepts of Analysis  3 
MATH 221B  Mathematical Methods of Imaging  3 
MATH 236  Introduction to Stochastic Differential Equations  3 
MATH 238  Mathematical Finance  3 
Mechanical Engineering  
ME 335A/335B/335C  Finite Element Analysis  3 
ME 346B  Introduction to Molecular Simulations  3 
ME 408  Spectral Methods in Computational Physics  3 
ME 412  Engineering Functional Analysis and Finite Elements  3 
ME 469  Computational Methods in Fluid Mechanics  3 
ME 484  3  
Statistics  
STATS 208  Introduction to the Bootstrap  3 
STATS 217  Introduction to Stochastic Processes I  3 
STATS 219  Stochastic Processes  3 
STATS 250  Mathematical Finance  3 
STATS 305A  Applied Statistics I  23 
STATS 310A/310B/310C  Theory of Probability I  24 
STATS 362  Topic: Monte Carlo  3 
Other  
CEE 281  Mechanics and Finite Elements  3 
CEE 362G  Imaging with Incomplete Information  34 
ENGR 209A  Analysis and Control of Nonlinear Systems  3 
ENERGY 274  Complex Analysis for Practical Engineering  3 
Requirement 4: Specialized Electives (9 units)
Nine units of focused graduate application electives approved by the program adviser, in the areas of engineering, mathematics, physical, biological, information, and other quantitative sciences. These courses should be foundational depth courses relevant to the student's professional development and research interests.
Requirement 5: Seminar (3 units)
One unit of seminar must come from CME 500; two units are up to the student's choice of ICME graduate seminars or other approved seminars. Additional seminar units may not be counted towards the 45unit requirement.
Computational Geosciences Track
The Computational Geosciences (CompGeo) track is designed for students interested in the skills and knowledge required to develop efficient and robust numerical solutions to Earth Science problems using highperformance computing. The CompGeo curriculum is based on four fundamental areas: modern programming methods for Science and Engineering, applied mathematics with an emphasis on numerical methods, algorithms and architectures for highperformance computing and computationally oriented Earth Sciences courses. Earth Sciences/computational project courses give practice in applying methodologies and concepts. CompGeo students are required to complete general and focused application electives (Requirements 3 and 4) from the approved list of courses from the Computational Geosciences program. All other requirements remain the same as set forth above.
Note: Students interested in pursuing the ICME M.S. in the Computational Geosciences (CompGeo) track are encouraged to contact the Computational Geosciences program director before applying.
Students are required to take 45 units of course work, and research credits to earn a master's degree in Computational Geosciences track. The course work follows the requirements of the ICME M.S. degree as above with additional restrictions placed on the general and focused electives.
Requirement 1: Foundational (12 units)
Identical to the general CME master’s track requirement .
Requirement 2: Programming (3 units)
3 units of programming course work demonstrating programming proficiency. All graduate students in the program are required to complete programming course for letter grade. Programming proficiency at the level of CME 211 is a hard prerequisite for CME 212; CME 211 can be applied towards elective requirement.
Units  

CME 211  Software Development for Scientists and Engineers (*can only be used as an elective)  3 
CME 212  Advanced Software Development for Scientists and Engineers  3 
CME 214  Software Design in Modern Fortran for Scientists and Engineers  3 
GEOPHYS 257  Introduction to Computational Earth Sciences  24 
Requirement 3: Breadth Electives in Geosciences (18 units)
18 units of general electives to demonstrate breadth of knowledge in technical area. Courses are currently offered but are not limited to the following specific areas of the School of Earth Sciences:
 Reservoir Simulation
 Geophysical Imaging
 Tectonophysics/Geomechanics
 Climate/Atmosphere/Ocean
 Ecology/Geobiology.
The Earth Science courses, offered in EESS, ERE, GES, and Geophysics is selected based on the area of the student's interest and their research/thesis work, along with the advice and consent of the student's adviser. Students are encouraged to choose a range of courses in order to guarantee breadth of knowledge in Earth Sciences. A maximum of one noncomputationallyoriented course can be counted towards the master’s degree requirements. Following is a list of recommended courses (grouped by area) that can be taken to fulfill the Geosciences course requirement.
Units  

Environmental/Climate/Hydrogeology  
ESS 220  Physical Hydrogeology  4 
ESS 221  Contaminant Hydrogeology and Reactive Transport  4 
ESS 246B  Atmosphere, Ocean, and Climate Dynamics: the Ocean Circulation  3 
CEE 262A  Hydrodynamics  34 
CEE 262B  Transport and Mixing in Surface Water Flows  34 
CEE 262C  Modeling Environmental Flows  3 
CEE 263A  Air Pollution Modeling  34 
CEE 361  Turbulence Modeling for Environmental Fluid Mechanics  24 
Geophysical Imaging  
EE 256  Numerical Electromagnetics  3 
GEOPHYS 210  Basic Earth Imaging  23 
GEOPHYS 211  Environmental Soundings Image Estimation  3 
GEOPHYS 280  3D Seismic Imaging  23 
GEOPHYS 287  Earthquake Seismology  35 
General Computational/Mathematical Geoscineces  
CEE 362G  Imaging with Incomplete Information  34 
CHEM 275  Advanced Physical Chemistry  Single Molecules and Light  3 
CME 372  Applied Fourier Analysis and Elements of Modern Signal Processing  3 
CME 321B  Mathematical Methods of Imaging  3 
ESS 211  Fundamentals of Modeling  35 
ENERGY 291  Optimization of Energy Systems  34 
GS 240  23  
ME 335A  Finite Element Analysis  3 
ME 346B  Introduction to Molecular Simulations  3 
ME 361  Turbulence  3 
ME 469B  Computational Methods in Fluid Mechanics  3 
Reservoir Simulation/Fluid Flow  
ENERGY 223  Reservoir Simulation  34 
ENERGY 224  Advanced Reservoir Simulation  3 
Subsurface/Reservoir Characterization  
ENERGY 241  Seismic Reservoir Characterization  34 
GEOPHYS 202  Reservoir Geomechanics  3 
GEOPHYS 260  Rock Physics for Reservoir Characterization  3 
Structural/Tectonophysics/Geomechanics  
CEE 292  Continuum Mechanics  3 
CEE 294  Computational Poromechanics  3 
GEOPHYS 220  Ice, Water, Fire  35 
GEOPHYS 288A  Crustal Deformation  35 
GEOPHYS 288B  Crustal Deformation  35 
GEOPHYS 290  Tectonophysics  3 
Requirement 4: Practical Component (9 units)
9 units of focused research in computational geosciences. Students are required to either complete a Research Project or an Internship as described below.
Internship and/or Research Project, enrolling in a course such as:  
EARTH 400  Directed Research  3 
EARTH 401  Curricular Practical Training  1 
Research Project
Students who plan to apply to the Ph.D. program need to take 9 units of research. Students will work with the CompGeo program director to find an appropriate adviser and research topic and then enroll in EARTHSCI 400: Directed Research (or a similar SES research course). The successful outcome of a Research Project can be:
 an oral presentation at an international meeting requiring an extended abstract
 a publication submission in a peer reviewed journal.
 a written report
Internship
As an alternative to the Research Project, students have the option of an internship which is recommended for those students interested in a terminal degree. The individual student is responsible for securing and organizing the internship and is required to obtain a faculty adviser and submit a written report on the internship project. Credit for the internship will be obtained through EARTHSCI401: Curricular Practical Training (1 unit) and in this case only 8 units of research are required.
Requirement 5: Seminar (3 units)
3 units of ICME graduate seminars or other approved seminars. Additional seminar units may not be counted towards the 45unit requirement. One of the required seminars for CompGeo must be a seminar course chosen in concert with the student's academic adviser among the seminars offered by the the School of Earth, Energy and Environmental Sciences.
Data Science Track
The Data Science track develops strong mathematical, statistical, computational and programming skills through the foundational and programming requirements. In addition, it provides a fundamental data science education through general and focused electives requirement from courses in data sciences and related areas. Course choices are limited to predefined courses from the data sciences and related courses group. Programming requirement (requirement 2) is extended to 6 units and includes course work in advanced scientific programming and high performance computing. The final requirement is a practical component (requirement 5) for 6 units to be completed through capstone project, data science clinic, or other courses that have strong handson or practical component such as statistical consulting.
Requirement 1: Foundational (12 units)
Students must demonstrate foundational knowledge in the field by completing the following core courses. Courses in this area must be taken for letter grades.
Units  

CME 302  Numerical Linear Algebra  3 
CME 305  Discrete Mathematics and Algorithms  3 
CME 307  Optimization  3 
CME 308  Stochastic Methods in Engineering  3 
or CME 309  Randomized Algorithms and Probabilistic Analysis  
or STATS 310A  Theory of Probability I 
Requirement 2: Programming (6 units)
To ensure that students have a strong foundation in programming, 3 units of advanced scientific programming for letter grade at the level of CME 212 and three units of parallel computing for letter grades are required. Programming proficiency at the level of CME 211 is a hard prerequisite for CME 212; CME 211 can be applied towards elective requirement.
Units  

Advanced Scientific Programming; take 3 units  
CME 211  Software Development for Scientists and Engineers (*can only be used as an elective)  3 
CME 212  Advanced Software Development for Scientists and Engineers  3 
Parallel/HPC Computing; take 3 units  
CME 213  Introduction to parallel computing using MPI, openMP, and CUDA  3 
CME 323  Distributed Algorithms and Optimization  3 
CME 342  Parallel Methods in Numerical Analysis  3 
CS 149  Parallel Computing  34 
CS 315A  3  
CS 316  Advanced MultiCore Systems  3 
Requirement 3: Data Science electives (12 units)
Data Science electives should demonstrate breadth of knowledge in the technical area. The elective course list is defined. Courses outside this list can be accepted as electives subject to approval by the student’s program adviser. Courses that are waived may not be counted towards the master’s degree.
Units  

STATS 200  Introduction to Statistical Inference  3 
or STATS 300A  Theory of Statistics I  
STATS 203  Introduction to Regression Models and Analysis of Variance  3 
or STATS 305A  Applied Statistics I  
STATS 315A  Modern Applied Statistics: Learning  3 
STATS 315B  Modern Applied Statistics: Data Mining  3 
Requirement 4: Specialized electives (9 units)
Choose three courses in specialized areas from the following list. Courses outside this list can be accepted as electives subject to approval by the student’s program adviser.
Units  

BIOE 214  Representations and Algorithms for Computational Molecular Biology  34 
BIOMEDIN 215  Data Driven Medicine  3 
BIOS 221  Modern Statistics for Modern Biology  3 
CS 224W  Analysis of Networks  34 
CS 229  Machine Learning  34 
CS 231N  Convolutional Neural Networks for Visual Recognition  34 
CS 246  Mining Massive Data Sets  34 
CS 448  Topics in Computer Graphics  34 
ECON 293  Machine Learning and Causal Inference  3 
ENERGY 240  Data science for geoscience  3 
OIT 367  Business Intelligence from Big Data  3 
PSYCH 204A  Human Neuroimaging Methods  3 
STATS 290  Computing for Data Science  3 
Requirement 5: Practical component (6 units)
Students are required to take 6 units of practical component that may include any combination of:

Project labs offered by Stanford Data Lab: ENGR 150 Data Challenge Lab, ENGR 350 Data Impact Lab. (limited enrollment; application required.)

Master's Research( CME 291): a research project, supervised by a faculty member and approved by the adviser; should be taken for letter grade only. The research project should be computational in nature. Students should submit a onepage proposal, supported by the faculty member, to ICME student services for approval at least one quarter before.

Other courses that have a strong handson and practical component, such as STATS 390 Consulting Workshop up to 1unit.
Imaging Science Track
The Imaging Science track is designed for students interested in the skills and knowledge required to develop efficient and robust computational tools for imaging science. The curriculum is based on four fundamental areas: mathematical models and analysis for imaging sciences and inverse problems, tools and techniques from modern imaging sciences from medicine, biology, physics/chemistry, and earth science, algorithms in numerical methods and scientific computing and high performance computing skills and architecture oriented towards imaging sciences.
The course work follows the requirements of the general master's degree in the core course requirement. The general and focused elective requirements (requirements 3 and 4 below) are limited to approved courses listed below. Programming requirement (requirement 2) is extended to 6 units and includes course work in advanced scientific programming and high performance computing.
Requirement 1: Foundational (12 units)
Identical to the general ICME master’s program; see above.
Requirement 2: Programming (6 units)
To ensure that students have a strong foundation in programming, 3 units of advanced scientific programming for letter grade at the level of CME 212 and three units of parallel computing for letter grades are required. Programming proficiency at the level of CME 211 is a hard prerequisite for CME 212; CME 211 can be applied towards elective requirement.
Units  

CME 211  Software Development for Scientists and Engineers (*can only be used as an elective)  3 
Advanced Scientific Programming; take 3 units  
CME 212  Advanced Software Development for Scientists and Engineers  3 
CME 214  Software Design in Modern Fortran for Scientists and Engineers  3 
Parallel /HPCComputing; take 3 units  
CME 213  Introduction to parallel computing using MPI, openMP, and CUDA  3 
CME 323  Distributed Algorithms and Optimization  3 
CME 342  Parallel Methods in Numerical Analysis  3 
GEOPHYS 257  Introduction to Computational Earth Sciences  24 
Requirement 3: Imaging Sciences electives (18 units)
Imaging Sciences electives should demonstrate breadth of knowledge in the technical area. The elective course list is defined. Courses outside this list can be accepted as electives subject to approval by the student’s program adviser.
Units  

Take 18 units of the following:  
APPPHYS 232  Advanced Imaging Lab in Biophysics  4 
BIOE 220  Introduction to Imaging and Imagebased Human Anatomy  3 
CEE 362G  Imaging with Incomplete Information  34 
CME 279  Computational Biology: Structure and Organization of Biomolecules and Cells  3 
CME 371  Computational Biology in Four Dimensions  3 
CS 231N  Convolutional Neural Networks for Visual Recognition  34 
EE 236A  Modern Optics  3 
EE 262  TwoDimensional Imaging  3 
EE 355  Imaging Radar and Applications  3 
EE 367  Computational Imaging and Display  3 
EE 368  Digital Image Processing  3 
EE 369A  Medical Imaging Systems I  3 
EE 369B  Medical Imaging Systems II  3 
EE 369C  Medical Image Reconstruction  3 
GEOPHYS 210  Basic Earth Imaging  23 
GEOPHYS 211  Environmental Soundings Image Estimation  3 
GEOPHYS 280  3D Seismic Imaging  23 
MATH 221B  Mathematical Methods of Imaging  3 
MATH 262  Applied Fourier Analysis and Elements of Modern Signal Processing  3 
PSYCH 204A  Human Neuroimaging Methods  3 
Requirement 4: Specialized electives (6 units)
6 units of focused graduate application electives, approved by the ICME graduate adviser, in the areas of engineering, mathematics, physical, biological, information, and other quantitative sciences. These courses should be foundational depth courses relevant to the student's professional development and research interests.
Requirement 5: Seminar (3 units)
One unit of seminar must come from CME 500; two units are up to the student's choice of ICME graduate seminars or other approved seminars. Additional seminar units may not be counted towards the 45unit requirement.
Mathematical and Computational Finance Track
The Mathematical & Computational Finance (MCF) track is an interdisciplinary program that provides education in applied and computational mathematics, statistics, and financial applications for individuals with strong mathematical skills. Upon successful completion of the MCF track in the ICME master's program, students will be prepared to assume positions in the financial industry as data and information scientists, quantitative strategists, risk managers, regulators, financial technologists, or to continue on to their Ph.D. in ICME, MS&E, Mathematics, Statistics, Finance, and other disciplines.
The Institute for Computational and Mathematical Engineering, in close cooperation with Mathematics, Management Science and Engineering and Statistics provides many of the basic courses.
Requirement 1: Foundational (9 units)
Students must demonstrate foundational knowledge in the field by completing the following core courses. Courses in this area must be taken for letter grades.
Units  

CME 302  Numerical Linear Algebra  3 
or CME 303  Partial Differential Equations of Applied Mathematics  
or CME 305  Discrete Mathematics and Algorithms  
CME 307  Optimization  3 
CME 308  Stochastic Methods in Engineering  3 
or MATH 236  Introduction to Stochastic Differential Equations 
Requirement 2: Programming (9 units)
To ensure that students have a strong foundation in programming, six units of advanced programming for letter grade at the level of CME 212 and 3 units of parallel computing for letter grade are required. Programming proficiency at the level of CME 211 is a hard prerequisite for CME 212.
Units  

Advanced Scientific Programming; take 36 units  
CME 211  Software Development for Scientists and Engineers  3 
CME 212  Advanced Software Development for Scientists and Engineers  3 
CME 214  Software Design in Modern Fortran for Scientists and Engineers  3 
Parallel/HPC Computing; take 3 units  
CME 213  Introduction to parallel computing using MPI, openMP, and CUDA  3 
CME 323  Distributed Algorithms and Optimization  3 
CME 342  Parallel Methods in Numerical Analysis  3 
CS 149  Parallel Computing  34 
CS 315A  3  
CS 316  Advanced MultiCore Systems  3 
Requirement 3: Finance electives (9 units)
Choose three courses from the following list. Courses outside this list can be accepted as electives subject to approval by the student’s program adviser.
Units  

Financial Mathematics  
MATH 238  Mathematical Finance  3 
Financial Markets  
FINANCE 320  Debt Markets  3 
FINANCE 620  Financial Markets I  3 
FINANCE 622  Dynamic Asset Pricing Theory  4 
Other  
CS 251  Cryptocurrencies and blockchain technologies  3 
MS&E 347  Credit Risk: Modeling and Management  3 
MS&E 348  Optimization of Uncertainty and Applications in Finance  3 
MS&E 349  Financial Statistics  3 
STATS 240  Statistical Methods in Finance  34 
STATS 241  Datadriven Financial and Risk Econometrics  34 
STATS 244  Quantitative Trading: Algorithms, Data, and Optimization  24 
Requirement 4: Data Science electives (9 units)
Data Science electives should demonstrate breadth of knowledge in the technical area. Courses outside this list can be accepted as electives subject to approval by the student’s program adviser.
Units  

Learning  
CS 229  Machine Learning  34 
CS 230  Deep Learning  34 
CS 246  Mining Massive Data Sets  34 
MS&E 338  Reinforcement Learning  3 
STATS 315A  Modern Applied Statistics: Learning  23 
Mining  
STATS 315B  Modern Applied Statistics: Data Mining  23 
Other  
CS 224N  Natural Language Processing with Deep Learning  34 
MS&E 349  Financial Statistics  3 
STATS 240  Statistical Methods in Finance  34 
STATS 241  Datadriven Financial and Risk Econometrics  34 
Requirement 5: Practical component (9 units)
Students are required to take nine units of practical and project courses ONLY from the courses listed below.
Units  

MS&E 246  Financial Risk Analytics  3 
CME 240  Statistical and Machine Learning Approaches to Problems in Investment Management  3 
CME 241  Reinforcement Learning for Stochastic Control Problems in Finance  3 
CME 291  Master's Research  16 
MS&E 448  Big Financial Data and Algorithmic Trading  3 
Doctor of Philosophy in Computational and Mathematical Engineering
The University’s basic requirements for the Ph.D. degree are outlined in the "Graduate Degrees" section of this bulletin.
Applications to the Ph.D. program and all required supporting documents must be received by December 4, 2018. See Graduate Admissions for information and application materials. See the institute's admissions site for additional details. Applicants should take the Graduate Record Examination by October of the academic year in which the application is submitted.
Admission to the Ph.D. program does not imply that the student is a candidate for the Ph.D. degree. Advancement to candidacy requires superior academic achievement and passing the qualifying examination.
Requirements
 Complete a minimum of 135 units of residency at Stanford, including:
 45 units from the master's program requirements; all six core courses have to be completed for letter grade.
 27 units of electives for letter grade in an area planned with the student's Ph.D. adviser; 12 of these units should come from ICME specialized electives with significant computational content such as the CME 320380 series. The focused and specialized elective component of the ICME program is meant to be broad and inclusive of relevant courses of comparable rigor to ICME courses. The elective course list following represents automatically accepted electives within the program. However, electives are not limited to the list below, and the list is expanded on a continuing basis; courses outside the list can be accepted as electives subject to approval by the student's ICME adviser. Research, directed study, and seminar units are excluded.
 3 units of programming elective demonstrating programming proficiency. Students are required to complete programming course at the level of CME 213 Introduction to parallel computing using MPI, openMP, and CUDA or higher for letter grade.
 60 units of thesis research
 Maintain a grade point average (GPA) of 3.5.
 Pass the ICME qualifying examination before the beginning of the second year.
 Declare candidacy by the end of the second year
 File dissertation reading committee form by the end of third year
 Complete an approved program of original research.
 Complete a written dissertation based on research.
 Pass the oral examination that is a defense of the dissertation research.
Specialized Elective List
See requirement 1b above.
Units  

CEE 362G  Imaging with Incomplete Information  34 
CME 279  Computational Biology: Structure and Organization of Biomolecules and Cells  3 
CME 364B  Convex Optimization II  3 
CME 371  Computational Biology in Four Dimensions  3 
CS 348A  Computer Graphics: Geometric Modeling & Processing  34 
EE 368  Digital Image Processing  3 
MATH 205A  Real Analysis  3 
MATH 215A  Algebraic Topology  3 
MATH 221A  Mathematical Methods of Imaging  3 
MATH 221B  Mathematical Methods of Imaging  3 
MATH 227  Partial Differential Equations and Diffusion Processes  3 
MATH 236  Introduction to Stochastic Differential Equations  3 
MATH 238  Mathematical Finance  3 
ME 335A/335B/335C  Finite Element Analysis  3 
ME 346B  Introduction to Molecular Simulations  3 
ME 351A/351B  Fluid Mechanics  3 
ME 361  Turbulence  3 
ME 408  Spectral Methods in Computational Physics  3 
ME 412  Engineering Functional Analysis and Finite Elements  3 
ME 469  Computational Methods in Fluid Mechanics  3 
MS&E 319  Approximation Algorithms  3 
MS&E 336  
STATS 305A  Applied Statistics I  3 
STATS 305B  Applied Statistics II  3 
STATS 305C  Applied Statistics III  23 
STATS 318  Modern Markov Chains  3 
STATS 366  Modern Statistics for Modern Biology  3 
Note: Students who need to complete 135 units at Stanford, should necessarily complete CME master's requirements. All courses listed under "Requirement 2" under the "Master of Science in Computational and Mathematical Engineering" section can be used for fulfilling the general elective requirement.
Financial Assistance
The department awards a limited number of fellowships, course assistantships, and research assistantships to incoming graduate students. Applying for such assistance is part of submitting the application for admission to the program. Students are appointed for halftime assistantships which provide a tuition scholarship at the 8, 9, 10 unit rate during the academic year and a monthly stipend. Halftime appointments generally require 20 hours of work per week. Most course assistantships and research assistantships are awarded to students in the doctoral program in ICME. If the number of Ph.D. students is not sufficient to staff all course and research assistantship positions available, these positions may be open to master’s students. However, master’s students are not guaranteed financial assistance.
Ph.D. Minor in Computational and Mathematical Engineering
For a minor in Computational and Mathematical Engineering (CME), a doctoral candidate must complete 21 units of approved graduate level courses. These should include three ICME core courses and three ICME graduate electives at the 300 level or above and a programming course at the level of CME212 or higher. All courses must be taken for a letter grade and passed with a grade of ‘B’ or better. Elective courses cannot be cross listed with the primary department. Minor programs should be developed in close discussion between the student and the student's primary Ph.D. adviser.
Master's Student Advising:
The Institute for Computational and Mathematical Engineering (ICME) is committed to providing academic advising in support of our MS students’ education and professional development. When most effective, this advising relationship entails collaborative engagement by both the advisor and the advisee. As a best practice, advising expectations should be discussed and reviewed to ensure mutual understanding. Both the advisor and the advisee are expected to maintain professionalism and integrity.
At the start of graduate study, each student is assigned a master’s program advisor, a member of our faculty who will provide guidance in course selection, exploring academic opportunities, and professional pathways. Typically, the same faculty member serves as program advisor for the duration of a master’s study. Advisors are assigned during New Student Orientation at the beginning of autumn quarter. The first meeting between the program advisor and student should occur once in the first quarter of the first year to discuss the student’s goals and objectives. Students are expected to initiate the meeting with the advisor and complete the master’s program proposal before the final study list deadline of the first quarter in the program. Subsequent meetings with the advisor are strongly encouraged as need arises. Students are required to get approval from advisors for changes to the course plan on file before the changed courses are taken.
In addition, the Director of ICME meets all the master’s students during the ICME New Student Orientation at the start of the first year and is available during the academic year by email and during office hours. The Director also schedules open advising meetings every quarter. In addition to the individual advisors, ICME also provides scholarly and professional development opportunities, including a teaching training program and workshops that address skills relevant to both industrial and academic careers.
ICME also conducts an annual review of all students’ progress. The Director, in conjunction with ICME Student Services, may initiate a meeting with any student deemed to be in academic distress.
The ICME Student Services team is also an important part of the master’s advising team. They inform students and advisors about university and department requirements, procedures, opportunities, and maintain the official records of advisor assignments and course approvals.
Finally, graduate students are active contributors to the advising relationship and we urge them to proactively seek academic and professional guidance and take responsibility for informing themselves about policies and degree requirements for their graduate program. We therefore expect students to read regular communication from the Registrar's office and ICME Student Services regarding upcoming academic deadlines and policy updates, and to be responsible for complying with the university and program requirements.
A statement of University policy on graduate advising is posted under "Graduate Advising" section of this bulletin.
Ph. D. Student Advising:
ICME is committed to providing academic advising in support of our Ph.D. students’ education and professional development. When most effective, this advising relationship entails collaborative engagement by both the advisor and the advisee. As a best practice, advising expectations should be discussed and reviewed to ensure mutual understanding. Both the advisor and the advisee are expected to maintain professionalism and integrity.
The program advisor initially guides students in key areas such as course selection, navigating policies and degree requirements, developing of teaching pedagogy, identifying doctoral research opportunities and exploring academic opportunities and professional pathways. The Director of ICME serves as the program advisor for all incoming Ph.D. students. The program advisor meets with all the doctoral students during the New Student Orientation at the start of the first year and is available during the academic year by email and during office hours. The program advisor or Research advisor/CoAdvisor in conjunction with ICME Student Services may initiate a meeting with any student deemed to be in academic or research distress.
ICME does not require formal research rotations, but students are strongly encouraged to explore research activities in two or three research groups/labs during their first academic year.
In addition to the individual advisors, ICME also provides scholarly and professional development opportunities, including a teaching training program and workshops that address skills relevant to both industrial and academic careers.
Students are required to declare candidacy with a doctoral research advisor by the end of their second year in the program. ICME students can align with faculty across the university who are well versed in supervising research, mentoring doctoral students, and providing funding for the duration of the program. This research supervisor supersedes the program advisor in assuming primary responsibility for advising and mentoring the student. When the research advisor is from outside our department, the student will also identify a doctoral research CoAdvisor from ICME affiliated faculty to provide guidance on departmental requirements, core coursework, and opportunities. We encourage students to decide on their thesis committee within one year after start of candidacy in order to avail themselves of advice from multiple faculty members on the reading committee.
ICME also conducts an annual review of all students’ progress on milestones and research. Research input is solicited and an individual progress report spelling out the forthcoming milestones and any remedial action needed to maintain status is compiled.
The ICME Student Services team is also an important part of the doctoral advising team. They inform students and advisors about university and department requirements, procedures, opportunities, and they maintain the official records of advisor assignments and course approvals. Students are encouraged to talk with the program advisor and the student services office as they consider courses.
Finally, our doctoral students are active contributors to the advising relationship and we urge them to proactively seek academic and professional guidance and take responsibility for informing themselves of policies and degree requirements for their graduate program. We therefore expect students to read regular communication from the Registrar's office and ICME Student Services regarding upcoming academic deadlines and policy updates, and to be responsible for complying with the university and program requirements.
Emeriti: (Professors) Gunnar Carlsson (Mathematics), Antony Jameson (Aeronautics and Astronautics), Walter Murray (Management Science and Engineering), Arogyaswami Paulraj (Electrical Engineering), Michael Saunders (Management Science and Engineering)
Director: Gianluca Iaccarino (Mechanical Engineering)
Professors: Juan Alonso (Aeronautics and Astronautics), Biondo Biondi (Geophysics), Stephen Boyd (Electrical Engineering), Carlos D. Bustamante (Biomedical Data Science, Genetics), Emanuel Candes (Mathematics, Statistics), Persi Diaconis (Mathematics, Statistics), David Donoho (Statistics), Charbel Farhat (Aeronautics and Astronautics, Mechanical Engineering), Ronald Fedkiw (Computer Science), Margot Gerritsen (Energy Resources Engineering), Kay Giesecke (Management Science and Engineering), Peter Glynn (Management Science and Engineering), Ashish Goel (Management Science and Engineering), Leonidas Guibas (Computer Science), Pat Hanrahan (Computer Science, Electrical Engineering), Jerry Harris (Geophysics), Trevor Hastie (Mathematics, Statistics), Gianluca Iaccarino (Mechanical Engineering), Doug James (Computer Science), Peter Kitanidis (Civil and Environmental Engineering), Tze Leung Lai (Statistics), Sanjiva Lele (Mechanical Engineering, Aeronautics and Astronautics), Parviz Moin (Mechanical Engineering), Brad Osgood (Electrical Engineering), George Papanicolaou (Mathematics), Peter Pinsky ( Mechanical Engineering), Lenya Ryzhik (Mathematics), Eric Shaqfeh (Chemical Engineering, Mechanical Engineering), Jonathan Taylor (Statistics), Hamdi Tchelepi (Energy Resources Engineering), Benjamin Van Roy (Management Science and Engineering, Electrical Engineering), Andras Vasy (Mathematics), Lawrence Wein (Graduate School of Business), Wing Wong (Statistics), Yinyu Ye (Management Science and Engineering), Lexing Ying (Mathematics, Institute for Computational and Mathematical Engineering)
Associate Professors: Eric Darve (Mechanical Engineering), Ron Dror (CS, Institute for Computational and Mathematical Engineering), Eric Dunham (Geophysics), Oliver Fringer (Civil and Environmental Engineering), Ramesh Johari (Management Science and Engineering), Adrian Lew (Mechanical Engineering), Ali Mani (Mechanical Engineering), Alison Marsden (Pediatrics, Bioengineering), Amin Saberi (Management Science and Engineering), Andrew Spakowitz (Chemical Engineering)
Assistant Professors: Marco Pavone (Aeronautics and Astronautics), Bala Rajaratnam (Statistics, Enviornmental and Earth System Sciences), Aaron Daniel Sidford (Management Science and Engineering), Jenny Suckale (Geophysics), Johan Ugander (Management Science and Engineering), Mary Wootters (Computer Science)
Senior Lecturer: Vadim Khayms
Lecturer: Hung Le, Ashwin Rao
Academic Staff: William Behrman, Reza BosaghZadeh, Jeremy Evnine, Kari Hanson, Vijay Pande, Hadley Wickham
Courses
CME 10. How to learn Mathematics  New ideas from the science of learning. 1 Unit.
This course will help provide the transition from high school to college learning and encourage the positive ideas and mindsets that shape productive learning. We willnconsider what learning theories have to tell us about mathematics learning, the nature of good teaching and the reasons for ongoing inequities in mathematics learning and participation. This seminar is for those who would like a more positive relationship with mathematics, and are interested in learning about ways to tackle education inequalities. Learning goals: First, it introduces students to theories of learning and in particular the learning of mathematics. Mathematics plays a key role in many students¿ learning identities and is often the cause of low selfesteem and anxiety. Research tells us that this is because mathematics in the US is taught in highly ineffective ways. Indeed there is a large gap between what we know works from research and what happens in most mathematics classrooms. This seminar will give participants an understanding of ways to relate positively to mathematics, to learn mathematics most productively and some of the learning barriers that often deny students the opportunity to engage with mathematics in productive ways.nSecond, the course will teach students about the inequalities that pervade the education system in the United States. We will examine the barriers to the participation of women and students of color and we will consider why social class and race are both strong predictors of mathematics achievement. It is hoped that students will leave the course with greater knowledge of why mathematics is important  to themselves and to the future of society.nCourse participants will be given the opportunity to take part in a mathematics camp, designed to change the pathways of middle school students, similar to this previous camp: https://www.youcubed.org/solvingmath problem/ and to take part in the work of youcubed.org. if they wish.
Same as: EDUC 105, FEMGEN 10
CME 100. Vector Calculus for Engineers. 5 Units.
Computation and visualization using MATLAB. Differential vector calculus: analytic geometry in space, functions of several variables, partial derivatives, gradient, unconstrained maxima and minima, Lagrange multipliers. Introduction to linear algebra: matrix operations, systems of algebraic equations, methods of solution and applications. Integral vector calculus: multiple integrals in Cartesian, cylindrical, and spherical coordinates, line integrals, scalar potential, surface integrals, Green¿s, divergence, and Stokes¿ theorems. Examples and applications drawn from various engineering fields. Prerequisites: 10 units of AP credit (Calc BC with 5, or Calc AB with 5 or placing out of the single variable math placement test: https://exploredegreesnextyear.stanford.edu/undergraduatedegreesandprograms/#aptextt), or MATH 1921.
Same as: ENGR 154
CME 100A. Vector Calculus for Engineers, ACE. 6 Units.
Students attend CME100/ENGR154 lectures with additional recitation sessions; two to four hours per week, emphasizing engineering mathematical applications and collaboration methods. Enrollment by department permission only. Prerequisite: must be enrolled in the regular CME10001 or 02. Application at: https://engineering.stanford.edu/students/programs/engineeringdiversityprograms/additionalcalculusengineers.
CME 102. Ordinary Differential Equations for Engineers. 5 Units.
Analytical and numerical methods for solving ordinary differential equations arising in engineering applications: Solution of initial and boundary value problems, series solutions, Laplace transforms, and nonlinear equations; numerical methods for solving ordinary differential equations, accuracy of numerical methods, linear stability theory, finite differences. Introduction to MATLAB programming as a basic tool kit for computations. Problems from various engineering fields. Prerequisite: 10 units of AP credit (Calc BC with 5, or Calc AB with 5 or placing out of the single variable math placement test: https://exploredegreesnextyear.stanford.edu/undergraduatedegreesandprograms/#aptextt),), or MATH 1921. Recommended: CME100.
Same as: ENGR 155A
CME 102A. Ordinary Differential Equations for Engineers, ACE. 6 Units.
Students attend CME102/ENGR155A lectures with additional recitation sessions; two to four hours per week, emphasizing engineering mathematical applications and collaboration methods. Prerequisite: students must be enrolled in the regular section (CME102) prior to submitting application at:nhttps://engineering.stanford.edu/students/programs/engineeringdiversityprograms/additionalcalculusengineers.
CME 103. Introduction to Matrix Methods. 35 Units.
Introduction to applied linear algebra with emphasis on applications. Vectors, norm, and angle; linear independence and orthonormal sets; applications to document analysis. Clustering and the kmeans algorithm. Matrices, left and right inverses, QR factorization. Leastsquares and model fitting, regularization and crossvalidation. Constrained and nonlinear leastsquares. Applications include timeseries prediction, tomography, optimal control, and portfolio optimization. Undergraduate students should enroll for 5 units, and graduate students should enroll for 3 units. Prerequisites:MATH 51 or CME 100, and basic knowledge of computing (CS 106A is more than enough, and can be taken concurrently). EE103/CME103 and MATH 104 cover complementary topics in applied linear algebra. The focus of EE103 is on a few linear algebra concepts, and many applications; the focus of MATH 104 is on algorithms and concepts.
Same as: EE 103
CME 104. Linear Algebra and Partial Differential Equations for Engineers. 5 Units.
Linear algebra: matrix operations, systems of algebraic equations, Gaussian elimination, undetermined and overdetermined systems, coupled systems of ordinary differential equations, eigensystem analysis, normal modes. Fourier series with applications, partial differential equations arising in science and engineering, analytical solutions of partial differential equations. Numerical methods for solution of partial differential equations: iterative techniques, stability and convergence, time advancement, implicit methods, von Neumann stability analysis. Examples and applications from various engineering fields. Prerequisite: CME 102/ENGR 155A.
Same as: ENGR 155B
CME 104A. Linear Algebra and Partial Differential Equations for Engineers, ACE. 6 Units.
Students attend CME104/ENGR155B lectures with additional recitation sessions; two to four hours per week, emphasizing engineering mathematical applications and collaboration methods. Prerequisite: students must be enrolled in the regular section (CME104) prior to submitting application at: https://engineering.stanford.edu/students/programs/engineeringdiversityprograms/additionalcalculusengineers.
CME 106. Introduction to Probability and Statistics for Engineers. 4 Units.
Probability: random variables, independence, and conditional probability; discrete and continuous distributions, moments, distributions of several random variables. Topics in mathematical statistics: random sampling, point estimation, confidence intervals, hypothesis testing, nonparametric tests, regression and correlation analyses; applications in engineering, industrial manufacturing, medicine, biology, and other fields. Prerequisite: CME 100/ENGR154 or MATH 51 or 52.
Same as: ENGR 155C
CME 108. Introduction to Scientific Computing. 3 Units.
Introduction to Scientific Computing Numerical computation for mathematical, computational, physical sciences and engineering: error analysis, floatingpoint arithmetic, nonlinear equations, numerical solution of systems of algebraic equations, banded matrices, least squares, unconstrained optimization, polynomial interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations, truncation error, numerical stability for time dependent problems and stiffness. Implementation of numerical methods in MATLAB programming assignments. Prerequisites: MATH 51, 52, 53; prior programming experience (MATLAB or other language at level of CS 106A or higher).
Same as: MATH 114
CME 151A. Interactive Data Visualization in D3. 1 Unit.
This fourweek short course introduces D3, a powerful tool for creating interactive data visualizations on the web (d3js.org). The class is geared toward scientists and engineers who want to better communicate their personal projects and research through visualizations on the web. The class will cover the basics of D3: inputting data, creating scales and axes, and adding transitions and interactivity, as well as some of the most used libraries: stack, cluster and force layouts. The class will be based on short workshops and a final project. A background in programming methodology at the level of CS106A is assumed. The course will make use of Javascript, experience is recommended but not necessary.
CME 181. Projects in Applied and Computational Mathematics. 3 Units.
Teams of students use techniques in applied and computational mathematics to tackle problems of their choosing. Students will have the opportunity to pursue openended projects in a variety of areas: economics, physics, political science, operations research, etc. Projects can cover (but are not limited to!) topics such as mathematical modeling of realworld phenomena (population dynamics), datadriven applications (movie recommendations) or complex systems in engineering (optimal control). Each team will be paired with a graduate student mentor working in applied and computational mathematics. Limited enrollment. Prerequisites: CME 100/102/104 or equivalents, or instructor consent. Recommended: CME 106/108 and familiarity with programming at the level of CME 192/193.
CME 192. Introduction to MATLAB. 1 Unit.
This short course runs for the first four weeks/eight lectures of the quarter and is offered each quarter during the academic year. It is highly recommended for students with no prior programming experience who are expected to use MATLAB in math, science, or engineering courses. It will consist of interactive lectures and applicationbased assignments.nThe goal of the short course is to make students fluent in MATLAB and to provide familiarity with its wide array of features. The course covers an introduction of basic programming concepts, data structures, and control/flow; and an introduction to scientific computing in MATLAB, scripts, functions, visualization, simulation, efficient algorithm implementation, toolboxes, and more.
CME 193. Introduction to Scientific Python. 1 Unit.
This short course runs for the first four weeks of the quarter. It is recommended for students who are familiar with programming at least at the level of CS106A and want to translate their programming knowledge to Python with the goal of becoming proficient in the scientific computing and data science stack. Lectures will be interactive with a focus on real world applications of scientific computing. Technologies covered include Numpy, SciPy, Pandas, Scikitlearn, and others. Topics will be chosen from Linear Algebra, Optimization, Machine Learning, and Data Science. Prior knowledge of programming will be assumed, and some familiarity with Python is helpful, but not mandatory.
CME 195. Introduction to R. 1 Unit.
This short course runs for four weeks beginning in the second week of the quarter and is offered in fall and spring. It is recommended for students who want to use R in statistics, science, or engineering courses and for students who want to learn the basics of R programming. The goal of the short course is to familiarize students with R's tools for scientific computing. Lectures will be interactive with a focus on learning by example, and assignments will be applicationdriven. No prior programming experience is needed. Topics covered include basic data structures, File I/O, graphs, control structures, etc, and some useful packages in R.
Same as: STATS 195
CME 200. Linear Algebra with Application to Engineering Computations. 3 Units.
Computer based solution of systems of algebraic equations obtained from engineering problems and eigensystem analysis, Gaussian elimination, effect of roundoff error, operation counts, banded matrices arising from discretization of differential equations, illconditioned matrices, matrix theory, least square solution of unsolvable systems, solution of nonlinear algebraic equations, eigenvalues and eigenvectors, similar matrices, unitary and Hermitian matrices, positive definiteness, CayleyHamilton theory and function of a matrix and iterative methods. Prerequisite: familiarity with computer programming, and MATH51.
Same as: ME 300A
CME 204. Partial Differential Equations in Engineering. 3 Units.
Geometric interpretation of partial differential equation (PDE) characteristics; solution of first order PDEs and classification of secondorder PDEs; selfsimilarity; separation of variables as applied to parabolic, hyperbolic, and elliptic PDEs; special functions; eigenfunction expansions; the method of characteristics. If time permits, Fourier integrals and transforms, Laplace transforms. Prerequisite: CME 200/ME 300A, equivalent, or consent of instructor.
Same as: ME 300B
CME 206. Introduction to Numerical Methods for Engineering. 3 Units.
Numerical methods from a user's point of view. Lagrange interpolation, splines. Integration: trapezoid, Romberg, Gauss, adaptive quadrature; numerical solution of ordinary differential equations: explicit and implicit methods, multistep methods, RungeKutta and predictorcorrector methods, boundary value problems, eigenvalue problems; systems of differential equations, stiffness. Emphasis is on analysis of numerical methods for accuracy, stability, and convergence. Introduction to numerical solutions of partial differential equations; Von Neumann stability analysis; alternating direction implicit methods and nonlinear equations. Prerequisites: CME 200/ME 300A, CME 204/ME 300B.
Same as: ME 300C
CME 207. Numerical Methods in Engineering and Applied Sciences. 3 Units.
Scientific computing and numerical analysis for physical sciences and engineering. Advanced version of CME206 that, apart from CME206 material, includes nonlinear PDEs, multidimensional interpolation and integration and an extended discussion of stability for initial boundary value problems. Recommended for students who have some prior numerical analysis experience. Topics include: 1D and multiD interpolation, numerical integration in 1D and multiD including adaptive quadrature, numerical solutions of ordinary differential equations (ODEs) including stability, numerical solutions of 1D and multiD linear and nonlinear partial differential equations (PDEs) including concepts of stability and accuracy. Prerequisites: linear algebra, introductory numerical analysis (CME 108 or equivalent).
Same as: AA 214A, GEOPHYS 217
CME 211. Software Development for Scientists and Engineers. 3 Units.
Basic usage of the Python and C/C++ programming languages are introduced and used to solve representative computational problems from various science and engineering disciplines. Software design principles including time and space complexity analysis, data structures, objectoriented design, decomposition, encapsulation, and modularity are emphasized. Usage of campus wide Linux compute resources: login, file system navigation, editing files, compiling and linking, file transfer, etc. Versioning and revision control, software build utilities, and the LaTeX typesetting software are introduced and used to help complete programming assignments. Prerequisite: introductory programming course equivalent to CS 106A or instructor consent.
CME 212. Advanced Software Development for Scientists and Engineers. 3 Units.
Advanced topics in software development, debugging, and performance optimization are covered. The capabilities and usage of common libraries and frameworks such as BLAS, LAPACK, FFT, PETSc, and MKL/ACML are reviewed. Computer representation of integer and floating point numbers, and interoperability between C/C++ and Fortran is described. More advanced software engineering topics including: representing data in files, signals, unit and regression testing, and build automation. The use of debugging tools including static analysis, gdb, and Valgrind are introduced. An introduction to computer architecture covering processors, memory hierarchy, storage, and networking provides a foundation for understanding software performance. Profiles generated using gprof and perf are used to help guide the performance optimization process. Computational problems from various science and engineering disciplines will be used in assignments. Prerequisites: CME 200 / ME 300A and CME 211.
CME 213. Introduction to parallel computing using MPI, openMP, and CUDA. 3 Units.
This class will give hands on experience with programming multicore processors, graphics processing units (GPU), and parallel computers. Focus will be on the message passing interface (MPI, parallel clusters) and the compute unified device architecture (CUDA, GPU). Topics will include: network topologies, modeling communication times, collective communication operations, parallel efficiency, MPI, dense linear algebra using MPI. Symmetric multiprocessing (SMP), pthreads, openMP. CUDA, combining MPI and CUDA, dense linear algebra using CUDA, sort, reduce and scan using CUDA. Prerequisites include: C programming language and numerical algorithms (solution of differential equations, linear algebra, Fourier transforms).
Same as: ME 339
CME 214. Software Design in Modern Fortran for Scientists and Engineers. 3 Units.
This course introduces software design and development in modern Fortran. Course covers the functional, objectoriented, and parallel programming features introduced in the Fortran 95, 2003, and 2008 standards, respectively, in the context of numerical approximations to ordinary and partial differential equations; introduces objectoriented design and design schematics based on the Unified Modeling Language (UML) structure, behavior, and interaction diagrams; cover the basic use of several opensource tools for software building, testing, documentation generation, and revision control. Recommended: Familiarity with programming in Fortran 90, basic numerical analysis and linear algebra, or instructor approval.
Same as: EARTH 214
CME 215A. Advanced Computational Fluid Dynamics. 3 Units.
High resolution schemes for capturing shock waves and contact discontinuities; upwinding and artificial diffusion; LED and TVD concepts; alternative flow splittings; numerical shock structure. Discretization of Euler and Navier Stokes equations on unstructured meshes; the relationship between finite volume and finite element methods. Time discretization; explicit and implicit schemes; acceleration of steady state calculations; residual averaging; math grid preconditioning. Automatic design; inverse problems and aerodynamic shape optimization via adjoint methods. Pre or corequisite: 214B or equivalent.
Same as: AA 215A
CME 215B. Advanced Computational Fluid Dynamics. 3 Units.
High resolution schemes for capturing shock waves and contact discontinuities; upwinding and artificial diffusion; LED and TVD concepts; alternative flow splittings; numerical shock structure. Discretization of Euler and Navier Stokes equations on unstructured meshes; the relationship between finite volume and finite element methods. Time discretization; explicit and implicit schemes; acceleration of steady state calculations; residual averaging; math grid preconditioning. Automatic design; inverse problems and aerodynamic shape optimization via adjoint methods. Pre or corequisite: 214B or equivalent.
Same as: AA 215B
CME 232. Introduction to Computational Mechanics. 3 Units.
Provides an introductory overview of modern computational methods for problems arising primarily in mechanics of solids and is intended for students from various engineering disciplines. The course reviews the basic theory of linear solid mechanics and introduces students to the important concept of variational forms, including the principle of minimum potential energy and the principles of virtual work. Specific model problems that will be considered include deformation of bars, beams and membranes, plates, and problems in plane elasticity (plane stress, plane strain, axisymmetric elasticity). The variational forms of these problems are used as the starting point for developing the finite element method (FEM) and boundary element method (BEM) approaches providing an important connection between mechanics and computational methods.
Same as: ME 332
CME 237. Networks, Markets, and Crowds. 3 Units.
The course explores the underlying network structure of our social, economic, and technological worlds and uses techniques from graph theory and economics to examine the structure & evolution of information networks, social contagion, the spread of social power and popularity, and information cascades. Prerequisites: basic graph and probability theory.
Same as: MS&E 237
CME 240. Statistical and Machine Learning Approaches to Problems in Investment Management. 3 Units.
This course will approach a variety of problems in investment management, using statistical and machine learning tools to model forecasting problems in the evolution of security prices. Through a combination of lectures and projects, we will investigate pricing and risk models ranging from individual securities up through asset classes. Occasional guest lecturers will present problems they currently face in their day to day work. Prerequisites: Basic background in Probability (e.g.: CME 106) and Mathematical Finance (e.g.: MATH 238), and some facility programming in R and/or Python.
Same as: MS&E 445
CME 241. Reinforcement Learning for Stochastic Control Problems in Finance. 3 Units.
This course will explore a few problems in Mathematical Finance through the lens of Stochastic Control such as Portfolio Management, Optimal Exercise of Derivatives, Order Execution, Personal Finance. For each of these problems, we formulate a suitable Markov Decision Process (MDP), develop Dynamic Programming (DP) solutions, and explore Reinforcement Learning (RL) algorithms. The course emphasizes the theory of DP/RL as well as modeling the practical nuances of these finance problems, and strengthening the understanding through plenty of coding exercises of the methods. Prerequisites: basic background in Probability (eg: CME 106) and Mathematical Finance (eg: MATH 238), and some experience coding in Python; Dynamic Programming or Reinforcement Learning experience not required.
Same as: MS&E 346
CME 242. Mathematical and Computational Finance Seminar. 1 Unit.
May be repeat for credit.
Same as: MS&E 446A, STATS 239
CME 243. Risk Analytics and Management in Finance and Insurance. 24 Units.
Market risk and credit risk, credit markets. Back testing, stress testing and Monte Carlo methods. Logistic regression, generalized linear models and generalized mixed models. Loan prepayment and default as competing risks. Survival and hazard functions, correlated default intensities, frailty and contagion. Risk surveillance, early warning and adaptive control methodologies. Banking and bank regulation, asset and liability management. Prerequisite: STATS 240 or equivalent.
Same as: STATS 243
CME 244. Project Course in Mathematical and Computational Finance. 16 Unit.
For graduate students in the MCF track; students will work individually or in groups on research projects.
CME 245. Topics in Mathematical and Computational Finance. 1 Unit.
Description: Current topics for enrolled students in the MCF program: This course is an introduction to computational, statistical, and optimizations methods and their application to financial markets. Class will consist of lectures and realtime problem solving. Topics: Python & R programming, interest rates, BlackScholes model, financial time series, capital asset pricing model (CAPM), options, optimization methods, and machine learning algorithms. Appropriate for anyone with a technical and solid applied math background interested in honing skills in quantitative finance. Prerequisite: basic statistics and exposure to programming.Can be repeated up to three times.
CME 249. Using Design for Effective Data Analysis. 1 Unit.
Teams of students use techniques in applied and computational mathematics to tackle problems with real world data sets. Application of design methodology adapted for data analysis will be emphasized; leverage design thinking to come up with efficient and effective data driven insights; explore design thinking methodology in small group setting.;apply design thinking to a specific data centric problem and make professional group presentation of the results. Limited enrollment. Prerequisites: CME100/102/104 or equivalents, or instructor consent. Recommended:CME106/108 and familiarity with programming at the level of CME 192/193.
CME 249A. Statistical Arbitrage. 1 Unit.
Course will cover trading strategies that are bottom up, market neutral, with trading driven by statistical or econometric models and strategies such as pair trading and index arbitrage. Models may focus on tendency of short term returns to revert, leads/lags among correlated instruments, volume momentum, or behavioral effects. nTopics include: (a) a taxonomy of market participants and what motivates trading, (b) methods of exploring relationships between instruments, (c) portfolio construction across a large number of instruments, (d) risks inherent in statistical arbitrage (e) nonstationarity of relationships due to changes in market regulations, fluctuations in market volatility and other factors and (f) frictions such as costs of trading and constraints. Students will team to analyze the provided data sets which cover distinct dynamic market regimes.
CME 250. Introduction to Machine Learning. 1 Unit.
A Short course presenting the principles behind when, why, and how to apply modern machine learning algorithms. We will discuss a framework for reasoning about when to apply various machine learning techniques, emphasizing questions of overfitting/underfitting, regularization, interpretability, supervised/unsupervised methods, and handling of missing data. The principles behind various algorithmsthe why and how of using themwill be discussed, while some mathematical detail underlying the algorithmsincluding proofswill not be discussed. Unsupervised machine learning algorithms presented will include kmeans clustering, principal component analysis (PCA), and independent component analysis (ICA). Supervised machine learning algorithms presented will include support vector machines (SVM), classification and regression trees (CART), boosting, bagging, and random forests. Imputation, the lasso, and crossvalidation concepts will also be covered. The R programming language will be used for examples, though students need not have prior exposure to R. Prerequisite: undergraduatelevel linear algebra and statistics; basic programming experience (R/Matlab/Python).
CME 250A. Machine Learning on Big Data. 1 Unit.
A short course presenting the application of machine learning methods to large datasets.Topics include: brief review of the common issues of machine learning, such as, memorizing/overfitting vs learning, test/train splits, feature engineering, domain knowledge, fast/simple/dumb learners vs slow/complex/smart learners; moving your model from your laptop into a production environment using Python (scikit) or R on small data (laptop sized) at first; building math clusters using the open source H2O product to tackle Big Data, and finally to some model building on terabyte sized datasets. Prereqresites: basic knowledge of statistics, matrix algebra, and unixlike operating systems; basic file and text manipulation skills with unix tools: pipes, cut, paste, grep, awk, sed, sort, zip; programming skill at the level of CME211 or CS106A.
CME 251. Geometric and Topological Data Analysis. 3 Units.
Mathematical computational tools for the analysis of data with geometric content, such images, videos, 3D scans, GPS traces  as well as for other data embedded into geometric spaces. Global and local geometry descriptors allowing for various kinds of invariances. The rudiments of computational topology and persistent homology on sampled spaces. Clustering and other unsupervised techniques. Spectral methods for geometric data analysis. Nonlinear dimensionality reduction. Alignment, matching, and map computation between geometric data sets. Function spaces and functional maps.Networks of data sets and joint analysis for segmentation and labeling. The emergence of abstractions or concepts from data. Prerequisites: discrete algorithms at the level of 161; linear algebra at the level of CME103.
Same as: CS 233
CME 253. Introduction to GPU Computing and CUDA. 1 Unit.
Covers the fundamentals of accelerating applications with GPUs (Graphics Processing Units); GPU programming with CUDA and OpenACC, debugging, thrust/CUB, profiling, optimization, debugging, and other CUDA tools. Libraries to easily accelerate compute code will be presented and deployment on larger systems will be addressed, including multiGPU environments. Several practical examples will be detailed, including deep learning. Prerequiste: knowledge of C/C++ at the level of CME211 or CS106b.
CME 257. Advanced Topics in Scientific Computing with Julia. 1 Unit.
This course will rapidly introduce students to the Julia programming language, with the goal of giving students the knowledge and experience necessary to navigate the language and package ecosystem while using Julia for their own scientific computing needs. The course will begin with learning the basics of Julia, and then introduce students to git version control and package development. Additional topics include: common packages, parallelism, interfacing with shared object libraries, and aspects of Julia's implementation (e.g. core numerical linear algebra). Lectures will be interactive, with an emphasis on collaboration and learning by example. Prerequisites: Data structures at the level of CS106B, experience with one or more scientific computing languages (e.g. Python, Matlab, or R), and some familiarity with the Unix shell. No prior experience with Julia or git is required.
CME 258. Libraries for Numerical Linear Algebra and Optimization. 1 Unit.
This course will cover standard libraries commonly used for numerical linear algebra and optimization, with an emphasis on giving students experience with using the libraries on real examples. The course will cover software for direct methods (BLAS, Atlas, LAPACK, Eigen), iterative methods (ARPACK, Krylov Methods), and linear/nonlinear optimization (MINOS, SNOPT). Prerequisites: at least one course in numerical linear algebra (preferably at the level of CME 200 or CME 302), and one course in numerical optimization, as well as experience with at least one compiled language such as C/C++/Fortran.
CME 262. Imaging with Incomplete Information. 34 Units.
Statistical and computational methods for inferring images from incomplete data. Bayesian inference methods are used to combine data and quantify uncertainty in the estimate. Fast linear algebra tools are used to solve problems with many pixels and many observations. Applications from several fields but mainly in earth sciences. Prerequisites: Linear algebra and probability theory.
Same as: CEE 362G
CME 263. Introduction to Linear Dynamical Systems. 3 Units.
Applied linear algebra and linear dynamical systems with applications to circuits, signal processing, communications, and control systems. Topics: leastsquares approximations of overdetermined equations, and leastnorm solutions of underdetermined equations. Symmetric matrices, matrix norm, and singularvalue decomposition. Eigenvalues, left and right eigenvectors, with dynamical interpretation. Matrix exponential, stability, and asymptotic behavior. Multiinput/multioutput systems, impulse and step matrices; convolution and transfermatrix descriptions. Control, reachability, and state transfer; observability and leastsquares state estimation. Prerequisites: Linear algebra and matrices as in EE 103 or MATH 104; ordinary differential equations and Laplace transforms as in EE 102B or CME 102.
Same as: EE 263
CME 279. Computational Biology: Structure and Organization of Biomolecules and Cells. 3 Units.
Computational techniques for investigating and designing the threedimensional structure and dynamics of biomolecules and cells. These computational methods play an increasingly important role in drug discovery, medicine, bioengineering, and molecular biology. Course topics include protein structure prediction, protein design, drug screening, molecular simulation, cellularlevel simulation, image analysis for microscopy, and methods for solving structures from crystallography and electron microscopy data. Prerequisites: elementary programming background (CS 106A or equivalent) and an introductory course in biology or biochemistry.
Same as: BIOE 279, BIOMEDIN 279, BIOPHYS 279, CS 279
CME 285. Computational Modeling in the Cardiovascular System. 3 Units.
This course introduces computational modeling methods for cardiovascular blood flow and physiology. Topics in this course include analytical and computational methods for solutions of flow in deformable vessels, onedimensional equations of blood flow, cardiovascular anatomy, lumped parameter models, vascular trees, scaling laws, biomechanics of the circulatory system, and 3D patient specific modeling with finite elements; course will provide an overview of the diagnosis and treatment of adult and congenital cardiovascular diseases and review recent research in the literature in a journal club format. Students will use SimVascular software to do clinicallyoriented projects in patient specific blood flow simulations.
Same as: BIOE 285, ME 285
CME 291. Master's Research. 16 Unit.
Students require faculty sponsor. (Staff).
CME 292. Advanced MATLAB for Scientific Computing. 1 Unit.
Short course running first four weeks of the quarter (8 lectures) with interactive online lectures and application based assignment. Students will access the lectures and assignments on https://suclass.stanford.edu. Students will be introduced to advanced MATLAB features, syntaxes, and toolboxes not traditionally found in introductory courses. Material will be reinforced with inclass examples, demos, and homework assignment involving topics from scientific computing. MATLAB topics will be drawn from: advanced graphics (2D/3D plotting, graphics handles, publication quality graphics, animation), MATLAB tools (debugger, profiler), code optimization (vectorization, memory management), objectoriented programming, compiled MATLAB (MEX files and MATLAB coder), interfacing with external programs, toolboxes (optimization, parallel computing, symbolic math, PDEs). Scientific computing topics will include: numerical linear algebra, numerical optimization, ODEs, and PDEs.
CME 298. Basic Probability and Stochastic Processes with Engineering Applications. 3 Units.
Calculus of random variables and their distributions with applications. Review of limit theorems of probability and their application to statistical estimation and basic Monte Carlo methods. Introduction to Markov chains, random walks, Brownian motion and basic stochastic differential equations with emphasis on applications from economics, physics and engineering, such as filtering and control. Prerequisites: exposure to basic probability.
Same as: MATH 158
CME 300. First Year Seminar Series. 1 Unit.
Required for firstyear ICME Ph.D. students; recommended for firstyear ICME M.S. students. Presentations about research at Stanford by faculty and researchers from Engineering, H&S, and organizations external to Stanford. May be repeated for credit.
CME 302. Numerical Linear Algebra. 3 Units.
Solution of linear systems, accuracy, stability, LU, Cholesky, QR, least squares problems, singular value decomposition, eigenvalue computation, iterative methods, Krylov subspace, Lanczos and Arnoldi processes, conjugate gradient, GMRES, direct methods for sparse matrices. Prerequisites: CME 108, MATH 114, MATH 104.
CME 303. Partial Differential Equations of Applied Mathematics. 3 Units.
Firstorder partial differential equations; method of characteristics; weak solutions; elliptic, parabolic, and hyperbolic equations; Fourier transform; Fourier series; and eigenvalue problems. Prerequisite: Basic coursework in multivariable calculus and ordinary differential equations, and some prior experience with a proofbased treatment of the material as in MATH 171 or MATH 61CM (formerly Math 51H).
Same as: MATH 220
CME 305. Discrete Mathematics and Algorithms. 3 Units.
Topics: Basic Algebraic Graph Theory, Matroids and Minimum Spanning Trees, Submodularity and Maximum Flow, NPHardness, Approximation Algorithms, Randomized Algorithms, The Probabilistic Method, and Spectral Sparsification using Effective Resistances. Topics will be illustrated with applications from Distributed Computing, Machine Learning, and largescale Optimization. Prerequisites: CS 261 is highly recommended, although not required.
Same as: MS&E 316
CME 306. Numerical Solution of Partial Differential Equations. 3 Units.
Hyperbolic partial differential equations: stability, convergence and qualitative properties; nonlinear hyperbolic equations and systems; combined solution methods from elliptic, parabolic, and hyperbolic problems. Examples include: Burger's equation, Euler equations for compressible flow, NavierStokes equations for incompressible flow. Prerequisites: MATH 220A or CME 302.
Same as: MATH 226
CME 307. Optimization. 3 Units.
Applications, theories, and algorithms for finitedimensional linear and nonlinear optimization problems with continuous variables. Elements of convex analysis, first and secondorder optimality conditions, sensitivity and duality. Algorithms for unconstrained optimization, and linearly and nonlinearly constrained problems. Modern applications in communication, game theory, auction, and economics. Prerequisites: MATH 113, 115, or equivalent.
Same as: MS&E 311
CME 308. Stochastic Methods in Engineering. 3 Units.
The basic limit theorems of probability theory and their application to maximum likelihood estimation. Basic Monte Carlo methods and importance sampling. Markov chains and processes, random walks, basic ergodic theory and its application to parameter estimation. Discrete time stochastic control and Bayesian filtering. Diffusion approximations, Brownian motion and an introduction to stochastic differential equations. Examples and problems from various applied areas. Prerequisites: exposure to probability and background in analysis.
Same as: MATH 228, MS&E 324
CME 309. Randomized Algorithms and Probabilistic Analysis. 3 Units.
Randomness pervades the natural processes around us, from the formation of networks, to genetic recombination, to quantum physics. Randomness is also a powerful tool that can be leveraged to create algorithms and data structures which, in many cases, are more efficient and simpler than their deterministic counterparts. This course covers the key tools of probabilistic analysis, and application of these tools to understand the behaviors of random processes and algorithms. Emphasis is on theoretical foundations, though we will apply this theory broadly, discussing applications in machine learning and data analysis, networking, and systems. Topics include tail bounds, the probabilistic method, Markov chains, and martingales, with applications to analyzing random graphs, metric embeddings, random walks, and a host of powerful and elegant randomized algorithms. Prerequisites: CS 161 and STAT 116, or equivalents and instructor consent.
Same as: CS 265
CME 321A. Mathematical Methods of Imaging. 3 Units.
Image denoising and deblurring with optimization and partial differential equations methods. Imaging functionals based on total variation and l1 minimization. Fast algorithms and their implementation.
Same as: MATH 221A
CME 321B. Mathematical Methods of Imaging. 3 Units.
Array imaging using Kirchhoff migration and beamforming, resolution theory for broad and narrow band array imaging in homogeneous media, topics in highfrequency, variable background imaging with velocity estimation, interferometric imaging methods, the role of noise and inhomogeneities, and variational problems that arise in optimizing the performance of array imaging algorithms.
Same as: MATH 221B
CME 322. Spectral Methods in Computational Physics. 3 Units.
Data analysis, spectra and correlations, sampling theorem, nonperiodic data, and windowing; spectral methods for numerical solution of partial differential equations; accuracy and computational cost; fast Fourier transform, Galerkin, collocation, and Tau methods; spectral and pseudospectral methods based on Fourier series and eigenfunctions of singular SturmLiouville problems; Chebyshev, Legendre, and Laguerre representations; convergence of eigenfunction expansions; discontinuities and Gibbs phenomenon; aliasing errors and control; efficient implementation of spectral methods; spectral methods for complicated domains; time differencing and numerical stability.
Same as: ME 408
CME 323. Distributed Algorithms and Optimization. 3 Units.
The emergence of clusters of commodity machines with parallel processing units has brought with it a slew of new algorithms and tools. Many fields such as Machine Learning and Optimization have adapted their algorithms to handle such clusters. Topics include distributed and parallel algorithms for: Optimization, Numerical Linear Algebra, Machine Learning, Graph analysis, Streaming algorithms, and other problems that are challenging to scale on a commodity cluster. The class will focus on analyzing parallel and distributed programs, with some implementation using Apache Spark and TensorFlow.
CME 325. Numerical Approximations of Partial Differential Equations in Theory and Practice. 12 Unit.
Finite volume and finite difference methods for initial boundary value problems in multiple space dimensions. Emphasis is on formulation of boundary conditions for the continuous and the discrete problems. Analysis of numerical methods with respect to stability, accuracy, and error behavior. Techniques of treating nonrectangular domains, and effects of nonregular grids.
CME 326. Numerical Methods for Initial Boundary Value Problems. 3 Units.
Initial boundary value problems model many phenomena in engineering and science such as, fluid flow problems, wave propagation, fluidstructure interaction, conjugate heat transfer and financial mathematics. We discuss numerical techniques for such simulations and focus on the underlying principles and theoretical understanding. Emphasis is on stability, convergence and efficiency for methods applied to hyperbolic and parabolic initial boundary value problems.
CME 327. Numerical Methods for Stiff Problems. 3 Units.
Focus is on analysis of numerical techniques for stiff ordinary differential equations, including those resulting from spatial discretization of partial differential equations. Topics include stiffness, convergence, stability, adaptive time stepping, implicit timestepping methods (SDIRK, Rosenbrock), linear and nonlinear system solvers (Fixed Point, Newton, Multigrid, Krylov subspace methods) and preconditioning. Prerequisites: CME200/ME300A or equivalent; or consent of instructor.
CME 328. Advanced Topics in Partial Differential Equations. 3 Units.
Contents change each time and is taught as a topics course, most likely by a faculty member visiting from another institution. May be repeated for credit. Topic in 201213: numerical solution of timedependent partial differential equations is a fundamental tool for modeling and prediction in many areas of science and engineering. In this course we explore the stability, accuracy, efficiency, and appropriateness of specialized temporal integration strategies for different classes of partial differential equations including stiff problems and fully implicit methods, operator splitting and semiimplicit methods, extrapolation methods, multirate time integration, multiphysics problems, symplectic integration, and temporal parallelism. Prerequisites: recommended CME303 and 306 or with instructor's consent.
CME 330. Applied Mathematics in the Chemical and Biological Sciences. 3 Units.
Mathematical solution methods via applied problems including chemical reaction sequences, mass and heat transfer in chemical reactors, quantum mechanics, fluid mechanics of reacting systems, and chromatography. Topics include generalized vector space theory, linear operator theory with eigenvalue methods, phase plane methods, perturbation theory (regular and singular), solution of parabolic and elliptic partial differential equations, and transform methods (Laplace and Fourier). Prerequisites: CME 102/ENGR 155A and CME 104/ENGR 155B, or equivalents.
Same as: CHEMENG 300
CME 334. Advanced Methods in Numerical Optimization. 3 Units.
Topics include interiorpoint methods, relaxation methods for nonlinear discrete optimization, sequential quadratic programming methods, optimal control and decomposition methods. Topic chosen in first class; different topics for individuals or groups possible. Individual or team projects. May be repeated for credit.
Same as: MS&E 312
CME 335. Advanced Topics in Numerical Linear Algebra. 3 Units.
Possible topics: Classical and modern (e.g., focused on provable communication minimization) algorithms for executing dense and sparsedirect factorizations in highperformance, distributedmemory environments; distributed dense eigensolvers, dense and sparsedirect triangular solvers, and sparse matrixvector multiplication; unified analysis of distributed Interior Point Methods for symmetric cones via algorithms for distributing Jordan algebras over products of secondorder cones and Hermitian matrices. May be repeated for credit. Prerequisites: CME 302 and CME 304 (or equivalents).
CME 336. Linear and Conic Optimization with Applications. 3 Units.
Linear, semidefinite, conic, and convex nonlinear optimization problems as generalizations of classical linear programming. Algorithms include the interiorpoint, barrier function, and cutting plane methods. Related convex analysis, including the separating hyperplane theorem, Farkas lemma, dual cones, optimality conditions, and conic inequalities. Complexity and/or computation efficiency analysis. Applications to combinatorial optimization, sensor network localization, support vector machine, and graph realization. Prerequisite: MS&E 211 or equivalent.
Same as: MS&E 314
CME 338. LargeScale Numerical Optimization. 3 Units.
The main algorithms and software for constrained optimization emphasizing the sparsematrix methods needed for their implementation. Iterative methods for linear equations and least squares. The simplex method. Basis factorization and updates. Interior methods. The reducedgradient method, augmented Lagrangian methods, and SQP methods. Prerequisites: Basic numerical linear algebra, including LU, QR, and SVD factorizations, and an interest in MATLAB, sparsematrix methods, and gradientbased algorithms for constrained optimization. Recommended: MS&E 310, 311, 312, 314, or 315; CME 108, 200, 302, 304, 334, or 335.
CME 342. Parallel Methods in Numerical Analysis. 3 Units.
Emphasis is on techniques for obtaining maximum parallelism in numerical algorithms, especially those occurring when solving matrix problems, partial differential equations, and the subsequent mapping onto the computer. Implementation issues on parallel computers. Topics: parallel architecture, programming models (MPI, GPU Computing with CUDA ¿ quick review), matrix computations, FFT, fast multiple methods, domain decomposition, graph partitioning, discrete algorithms. Prerequisites: 302 or 200 (ME 300A), 213 or equivalent, or consent of instructor. Recommended: differential equations and knowledge of a highlevel programming language such as C or C++ (F90/95 also allowable).
CME 345. Model Reduction. 3 Units.
Model reduction is an indispensable tool for computationalbased design and optimization, statistical analysis, embedded computing, and realtime optimal control. This course presents the basic mathematical theory for projectionbased model reduction. Topics include: notions of linear dynamical systems and projection; projectionbased model reduction; error analysis; proper orthogonal decomposition; Hankel operator and balancing of a linear dynamical system; balanced truncation method: modal truncation and other reduction methods for linear oscillators; model reduction via moment matching methods based on Krylov subspaces; introduction to model reduction of parametric systems and notions of nonlinear model reduction. Course material is complemented by a balanced set of theoretical, algorithmic and Matlab computer programming assignments. Prerequisites: CME 200 or equivalent, CME 263 or equivalent and basic numerical methods for ODEs.
CME 356. Engineering Functional Analysis and Finite Elements. 3 Units.
Concepts in functional analysis to understand models and methods used in simulation and design. Topology, measure, and integration theory to introduce Sobolev spaces. Convergence analysis of finite elements for the generalized Poisson problem. Extensions to convectiondiffusionreaction equations and elasticity. Upwinding. Mixed methods and LBB conditions. Analysis of nonlinear and evolution problems. Prerequisites: 335A,B, CME 200, CME 204, or consent of instructor. Recommended: 333, MATH 171.
Same as: ME 412
CME 358. Finite Element Method for Fluid Mechanics. 3 Units.
Mathematical theory of the finite element method for incompressible flows; related computational algorithms and implementation details. Poisson equation; finite element method for simple elliptic problems; notions of mathematical analysis of noncoercive partial differential equations; the infsup or BabushkaBrezzi condition and its applications to the Stokes and Darcy problems; presentation of stable mixed finite element methods and corresponding algebraic solvers; stabilization approaches in the context of advectiondiffusion equation; numerical solution of the incompressible NavierStokes equations by finite element method. Theoretical, computational, and MATLAB computer programming assignments. Prerequisites: foundation in multivariate calculus and ME 335A or equivalent.
CME 362. An Introduction to Compressed Sensing. 3 Units.
Compressed sensing is a new data acquisition theory asserting that one can design nonadaptive sampling techniques that condense the information in a compressible signal into a small amount of data. This revelation may change the way engineers think about signal acquisition. Course covers fundamental theoretical ideas, numerical methods in largescale convex optimization, hardware implementations, connections with statistical estimation in high dimensions, and extensions such as recovery of data matrices from few entries (famous Netflix Prize).
Same as: STATS 330
CME 364A. Convex Optimization I. 3 Units.
Convex sets, functions, and optimization problems. The basics of convex analysis and theory of convex programming: optimality conditions, duality theory, theorems of alternative, and applications. Leastsquares, linear and quadratic programs, semidefinite programming, and geometric programming. Numerical algorithms for smooth and equality constrained problems; interiorpoint methods for inequality constrained problems. Applications to signal processing, communications, control, analog and digital circuit design, computational geometry, statistics, machine learning, and mechanical engineering. Prerequisite: linear algebra such as EE263, basic probability.
Same as: CS 334A, EE 364A
CME 364B. Convex Optimization II. 3 Units.
Continuation of 364A. Subgradient, cuttingplane, and ellipsoid methods. Decentralized convex optimization via primal and dual decomposition. Monotone operators and proximal methods; alternating direction method of multipliers. Exploiting problem structure in implementation. Convex relaxations of hard problems. Global optimization via branch and bound. Robust and stochastic optimization. Applications in areas such as control, circuit design, signal processing, and communications. Course requirements include project. Prerequisite: 364A.
Same as: EE 364B
CME 371. Computational Biology in Four Dimensions. 3 Units.
Cuttingedge research on computational techniques for investigating and designing the threedimensional structure and dynamics of biomolecules, cells, and everything in between. These techniques, which draw on approaches ranging from physicsbased simulation to machine learning, play an increasingly important role in drug discovery, medicine, bioengineering, and molecular biology. Course is devoted primarily to reading, presentation, discussion, and critique of papers describing important recent research developments. Prerequisite: CS 106A or equivalent, and an introductory course in biology or biochemistry. Recommended: some experience in mathematical modeling (does not need to be a formal course).
Same as: BIOMEDIN 371, BIOPHYS 371, CS 371
CME 372. Applied Fourier Analysis and Elements of Modern Signal Processing. 3 Units.
Introduction to the mathematics of the Fourier transform and how it arises in a number of imaging problems. Mathematical topics include the Fourier transform, the Plancherel theorem, Fourier series, the Shannon sampling theorem, the discrete Fourier transform, and the spectral representation of stationary stochastic processes. Computational topics include fast Fourier transforms (FFT) and nonuniform FFTs. Applications include Fourier imaging (the theory of diffraction, computed tomography, and magnetic resonance imaging) and the theory of compressive sensing.
Same as: MATH 262
CME 375. Advanced Topics in Convex Optimization. 3 Units.
Modern developments in convex optimization: semidefinite programming; novel and efficient firstorder algorithms for smooth and nonsmooth convex optimization. Emphasis on numerical methods suitable for large scale problems arising in science and engineering. Prerequisites: convex optimization (EE 364), linear algebra (MATH 104), numerical linear algebra (CME 302); background in probability, statistics, real analysis and numerical optimization.
Same as: MATH 301
CME 390. Curricular Practical Training. 1 Unit.
Educational opportunities in high technology research and development labs in applied mathematics. Qualified ICME students engage in internship work and integrate that work into their academic program. Students register during the quarter they are employed and complete a research report outlining their work activity, problems investigated, results, and followon projects they expect to perform. May be repeated three times for credit.
CME 399. Special Research Topics in Computational and Mathematical Engineering. 115 Unit.
Graduatelevel research work not related to report, thesis, or dissertation. May be repeated for credit.
CME 400. Ph.D. Research. 115 Unit.
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CME 444. Computational Consulting. 13 Unit.
Advice by graduate students under supervision of ICME faculty. Weekly briefings with faculty adviser and associated faculty to discuss ongoing consultancy projects and evaluate solutions. May be repeated for credit.
CME 500. Departmental Seminar. 1 Unit.
Special topic  Artificial Intelligence (AI) in Real Life: Senior industry technical leaders will share insights about how they are actively using artificial intelligence, applications of machine learning and deep learning in their organizations; will raise awareness about the challenges and unintended consequences of AI and address the interdisciplinary skills required to successfully implement AI solutions in their organizations. Speakers will share their vision for an AI future, and address student questions. Weekly speakers will come from many different fields and AI application areas.
CME 510. Linear Algebra and Optimization Seminar. 1 Unit.
Recent developments in numerical linear algebra and numerical optimization. Guest speakers from other institutions and local industry. Goal is to bring together scientists from different theoretical and application fields to solve complex scientific computing problems. May be repeated for credit.
CME 520. Topics in Simulation of Human Physiology & Anatomical Systems. 1 Unit.
Biweekly interdisciplinary lecture series on the development of computational tools for modeling and simulation of human physiological and anatomical systems. Lectures by instructors and guest speakers on topics such as surgical simulation, anatomical & surgical Modeling, neurological Systems, and biomedical models of human movement. Group discussions, team based assignments, and project work.nPrerequisite: Medical students, residents or fellows from school of medicine, and computationally oriented students with a strong interest to explore computational and mathematical methods related to the health sciences.
Same as: SURG 253
CME 801. TGR Project. 0 Units.
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CME 802. TGR Dissertation. 0 Units.
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