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. 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. The 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
		Course List
course	Numerical Linear Algebra	3
course	Partial Differential Equations of Applied Mathematics	3
course	Discrete Mathematics and Algorithms	3
course	Numerical Solution of Partial Differential Equations	3
course or course	Optimization Convex Optimization I	3
course or course	Stochastic Methods in Engineering Basic Probability and Stochastic Processes with Engineering Applications	3

Requirement 2: Programming (3 units)

To ensure that students have a strong foundation in programming, three units of advanced scientific programming for letter grade at the level of course is required. Programming proficiency at the level of course is a hard prerequisite; CME 211 can be applied towards the elective requirement.

		Units
		Course List
course	Software Development for Scientists and Engineers (*can only be counted as an elective)	3
course	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 areas. 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. Six units of independent research can be used to fulfill this requirement with prior approval.

		Units
		Course List
Aeronautics and Astronautics
course	Introduction to Symmetry Analysis	3
Computational and Mathematical Engineering
course	Introduction to Linear Dynamical Systems	3
course	Computational Biology: Structure and Organization of Biomolecules and Cells	3
course	Convex Optimization I	3
course	Computational Biology in Four Dimensions	3
Computer Science
course	Artificial Intelligence: Principles and Techniques	3-4
course	Probabilistic Graphical Models: Principles and Techniques	3-4
course	Machine Learning	3-4
course	Introduction to Cryptography	3
course	Optimization and Algorithmic Paradigms	3
course	Topics in Computer Systems	3-4
course	Computer Graphics: Geometric Modeling & Processing	3-4
Electrical Engineering
course	Applied Quantum Mechanics II	3
Management Science and Engineering
course	Probabilistic Analysis	3-4
course	Stochastic Modeling	3
course	Simulation	3
course	Fundamentals of Data Science: Prediction, Inference, Causality	3
course	Introduction to Stochastic Control with Applications	3
course	Linear Programming	3
course	Discrete Mathematics and Algorithms	3
course	Stochastic Systems	3
course	Stochastic Calculus and Control	3
Mathematics
course	Stochastic Processes	3
course	Fundamental Concepts of Analysis	3

course	Introduction to Stochastic Differential Equations	3
course	Mathematical Finance	3
Mechanical Engineering
course/course/course	Finite Element Analysis	3
course	Introduction to Molecular Simulations	3
course	Spectral Methods in Computational Physics	3
course	Computational Methods in Fluid Mechanics	3
Statistics
course	Bootstrap, Cross-Validation, and Sample Re-use	3
course	Introduction to Stochastic Processes I	3
course	Stochastic Processes	3
course	Mathematical Finance	3
course	Applied Statistics I	3
course/course	Theory of Probability I	3
course	Topic: Monte Carlo	3
Other
course	Mechanics and Finite Elements	3
course	Imaging with Incomplete Information	3-4
course	Machine Learning and Causal Inference	3
course	Analysis and Control of Nonlinear Systems	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 seminar unit 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 45-unit 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 high-performance 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 high-performance 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; see above.

Requirement 2: Programming (3 units)

To ensure that students have a strong foundation in programming, three units of advanced scientific programming for letter grade at the level of course is required. Programming proficiency at the level of is a hard prerequisite; CME 211 can be applied towards the elective requirement.

		Units
		Course List
course	Software Development for Scientists and Engineers (*can only be used as an elective)	3
course	Advanced Software Development for Scientists and Engineers	3
course	Introduction to Computational Earth Sciences	2-4

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, are 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 non-computationally-oriented 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
		Course List
Environmental/Climate/Hydrogeology
course	Physical Hydrogeology	4
course	Contaminant Hydrogeology and Reactive Transport	3
course	Atmosphere, Ocean, and Climate Dynamics: the Ocean Circulation	3
course	Hydrodynamics	3-4
course	Transport and Mixing in Surface Water Flows	3-4
course	Coastal Ocean Modeling	3
course	Air Pollution Modeling	3-4
Geophysical Imaging
course	Basic Earth Imaging	2-3
course	Environmental Soundings Image Estimation	3
course	3-D Seismic Imaging	2-3
course	Earthquake Seismology	3-5
General Computational/Mathematical Geoscineces
course	Imaging with Incomplete Information	3-4
course	Advanced Physical Chemistry - Single Molecules and Light	3
course	Applied Fourier Analysis and Elements of Modern Signal Processing	3
course	Fundamentals of Modeling	3-5
course	Optimization of Energy Systems	3-4
course	Finite Element Analysis	3
course	Introduction to Molecular Simulations	3
course	Turbulence	3
course	Computational Methods in Fluid Mechanics	3
Reservoir Simulation/Fluid Flow
course	Reservoir Simulation	3-4
course	Advanced Reservoir Simulation	3
Subsurface/Reservoir Characterization
course	Seismic Reservoir Characterization	3-4
course	Reservoir Geomechanics	3
Structural/Tectonophysics/Geomechanics
course	Ice, Water, Fire	3-5
course	Crustal Deformation	3-5
course	Crustal Deformation	3-5

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.

		Units
		Course List
Internship and/or Research Project, enrolling in a course such as:
course	Directed Research	3
course	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 course 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 EARTHSCI 401: Curricular Practical Training (1 unit) and in this case only 8 units of research are required.

Requirement 5: Seminar (3 units)

Three units of ICME graduate seminars or other approved seminars. Additional seminar units may not be counted towards the 45-unit 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. DS track covers both computational data science and machine learning but can be tailored to be more focused on one of the two areas by taking more credits in that concentration (requirement 3 or 4). Course choices are limited to predefined courses from the data sciences and related courses group for requirements 1-5.

Requirement 1: Mathematical and Statistical Foundations (15 units)

Students must demonstrate foundational knowledge in the field by completing the following courses. Courses in this area must be taken for letter grades.

		Units
		Course List
course	Numerical Linear Algebra	3
course	Stochastic Methods in Engineering	3
course or course	Introduction to Statistical Inference Theory of Statistics I	3
course or course	Introduction to Regression Models and Analysis of Variance Applied Statistics I	3
course or course	Modern Applied Statistics: Learning Machine Learning	3

Requirement 2: Experimentation (3 units)

Experimental method and causal considerations are fundamental to data science. The course chosen from this area must be taken for letter grades.

		Course List
course	Design of Experiments	3
course or course	Intermediate Econometrics II (3 units only) Topics in Causal Inference	2-5
course	Introduction to Causal Inference	3

*If both courses are taken, the additional 3 units can count towards the “Machine Learning Methods and Applications” requirement below.

Requirement 3: Scientific Computing (6-12 units)

To ensure that students have a strong foundation in programming, 3 units of scientific software development (CME212) and 3 units from scientific computing foundations and methods for letter grades is required. CME offers a placement test that can be used to directly enroll in CME 212. Students who pass this placement test are not required to take CME 211. Students can choose to take up to six additional units from this group of courses.

*Students must take 6 units from Requirement 3 and 6 units from Requirement 4, with an additional 6 units from either Requirement 3 or 4, for a total of 18 units in the two areas. The additional 6 units may be taken for a non-letter grade.

		Units
		Course List
Software Development; take 3 units
course	Software Development for Scientists and Engineers	3
course	Advanced Software Development for Scientists and Engineers	3
Scientific Computing Foundations and Methods; take 3 units
course	Introduction to parallel computing using MPI, openMP, and CUDA (Scientific Computing Foundations and Methods; take 3 units)	3
course	Discrete Mathematics and Algorithms	3
course	Optimization	3
course	Distributed Algorithms and Optimization	3
course	Convex Optimization I	3
course	Mining Massive Data Sets	3-4

Requirement 4: Machine Learning Methods and Applications (6-12 units)

Six units of coursework from this area is required and should be taken for letter grades. Students can also choose to take up to six additional units from this group of courses.

*Students must take 6 units from Requirement 3 and 6 units from Requirement 4, with an additional 6 units from either Requirement 3 or 4, for a total of 18 units in the two areas. The additional 6 units may be taken for a non-letter grade.

Courses in this area must be taken for letter grades.

		Course List
STATS 231
course	Modern Applied Statistics: Data Mining	3
course	Artificial Intelligence: Principles and Techniques	3-4
course	Natural Language Processing with Deep Learning	3-4
course	Deep Learning	3-4
course	Convolutional Neural Networks for Visual Recognition	3-4
course	Reinforcement Learning	3
course	Deep Generative Models	3

Requirement 5: Practical component (3 units)

Students are required to take 3 units of practical component that may include any combination of:

Analytics Accelerator(CME 217)
Master's Research (course): 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 one-page proposal, supported by the faculty member, to ICME student services for approval at least one quarter before.
Other courses that have a strong hands-on and practical component, such as course Consulting Workshop up to 1unit.

Additional Units

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, three units of advanced scientific programming for letter grade at the level of course and three units of parallel computing for letter grade are required. Programming proficiency at the level of course is a hard prerequisite forcourse; CME 211 can be applied towards the elective requirement.

		Units
		Course List
course	Software Development for Scientists and Engineers (*can only be used as an elective)	3
Advanced Scientific Programming; take 3 units
course	Advanced Software Development for Scientists and Engineers	3
course	Software Design in Modern Fortran for Scientists and Engineers	3
Parallel /HPCComputing; take 3 units
course	Introduction to parallel computing using MPI, openMP, and CUDA	3
course	Distributed Algorithms and Optimization	3
course	Introduction to Computational Earth Sciences	2-4

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. Six units of independent research can be used to fulfill this requirement with prior approval.

		Units
		Course List
Take 18 units of the following:
course	Advanced Imaging Lab in Biophysics	4
course	Introduction to Imaging and Image-based Human Anatomy	3
course	Imaging with Incomplete Information	3-4
course	Computational Biology: Structure and Organization of Biomolecules and Cells	3
course	Computational Biology in Four Dimensions	3
course	Convolutional Neural Networks for Visual Recognition	3-4
course	Principles of Robot Autonomy I	3-4
course	Remote Sensing of Land	4
course	Modern Optics	3
course	Three-Dimensional Imaging	3
course	Imaging Radar and Applications	3
course	Computational Imaging and Display	3
course	Digital Image Processing	3
course	Medical Imaging Systems I	3
course	Medical Imaging Systems II	3
course	Medical Image Reconstruction	3
course	Basic Earth Imaging	2-3
course	Environmental Soundings Image Estimation	3
course	3-D Seismic Imaging	2-3
course	Applied Fourier Analysis and Elements of Modern Signal Processing	3
course	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 seminar unit must come from course; 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 45-unit 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 advanced graduate work in applied 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
		Course List
course or course or course	Numerical Linear Algebra Partial Differential Equations of Applied Mathematics Discrete Mathematics and Algorithms	3
course or course	Optimization Convex Optimization I	3
course or course	Stochastic Methods in Engineering Introduction to Stochastic Differential Equations	3

Requirement 2: Programming (6-9 units)

To ensure that students have a strong foundation in programming, six units of advanced programming for letter grade at the level of course and three units of parallel computing for letter grade are required. Programming proficiency at the level of course is a hard prerequisite for course.

		Units
		Course List
Advanced Scientific Programming; take 3-6 units
course	Software Development for Scientists and Engineers	3
course	Advanced Software Development for Scientists and Engineers	3
course	Software Design in Modern Fortran for Scientists and Engineers	3
Parallel/HPC Computing; take 3 units
course	Introduction to parallel computing using MPI, openMP, and CUDA	3
course	Distributed Algorithms and Optimization	3
course	Parallel Computing	3-4
course	Parallel Computing Research Project	3
course	Advanced Multi-Core 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
		Course List
Financial Mathematics
course	Mathematical Finance	3
Financial Markets
course	Debt Markets	3
course	Financial Markets I	3
course	Dynamic Asset Pricing Theory	4
Other
course	Advanced Investment Science	3
course	Credit Risk: Modeling and Management	3
course	Optimization of Uncertainty and Applications in Finance	3
course	Financial Statistics	3
course	Statistical Methods in Finance	3
course	Data-driven Financial Econometrics	3
course	Quantitative Trading: Algorithms, Data, and Optimization	2-4

Requirement 4: Data Science electives (12 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
		Course List
Statistical Modeling
course	Introduction to Statistical Inference	3
course	Introduction to Regression Models and Analysis of Variance	3
course	Introduction to Nonparametric Statistics	3
course	Applied Multivariate Analysis	3
course	Introduction to Time Series Analysis	3
course	A Course in Bayesian Statistics	3
course	Massive Computational Experiments, Painlessly	2
course	Computing for Data Science	3
course	Applied Statistics I	3
course	Applied Statistics II: Generalized Linear Models, Survival Analysis, and Exponential Families	3
course	Applied Statistics III	3
course	Causal Inference	3
Learning
course	Reinforcement Learning for Stochastic Control Problems in Finance	3
course	Natural Language Processing with Deep Learning	3-4
course	Deep Learning	3-4
course	Mining Massive Data Sets	3-4
course	Reinforcement Learning: Behaviors and Applications	3
course	Reinforcement Learning: Frontiers	3
course	Modern Applied Statistics: Learning	3
course	Modern Applied Statistics: Data Mining	3
Other
course	Financial Statistics	3
course	Statistical Methods in Finance	3
course	Data-driven Financial Econometrics	3

Requirement 5: Practical component (6 units)

Students are required to take six units of practical and project courses ONLY from the courses listed below.

		Units
		Course List
course	Technology for Financial Systems	1
course	Reinforcement Learning for Stochastic Control Problems in Finance	3
course	Master's Research	1-6
course	Financial Risk Analytics	3
course	Big Financial Data and Algorithmic Trading	3

Petition Process for Transfer from M.S. to Ph.D. Degree Program

Students admitted to ICME graduate programs are enrolled specifically either into the terminal M.S. or the Ph.D. program. Admission to the Ph.D. program is required for a student to be eligible to work towards the Ph.D. degree.

A student in the terminal M.S. program may petition to be admitted to the Ph.D. program by filing an M.S. to Ph.D. petition form. Petition must include (1) a one-page statement of purpose explaining why the student wishes to transfer to the Ph.D. program, (2) the most recent unofficial transcript, and (3) two letters of recommendation from members of the Stanford faculty, including one from the student’s potential research adviser and at least one from an ICME faculty member belonging to the Academic Council. The M.S. to Ph.D. petition to transfer must be submitted to the student services office before the end of the spring quarter of the first year in the M.S. program to be approved to sit in the June qualifying exams.

Students who wish to submit a petition to add the Ph.D. degree, should plan to complete 18 units of the ICME core classes (CME 302, 303, 305, 306, 307, 308) for letter grade and pass the ICME qualifying exam before the beginning of their second year. If the petition is approved, all the other requirements of the Ph.D. program described in the bulletin apply, i.e. taking all six qualifying exams in June, filing for candidacy, securing a PhD advisor, satisfactory degree progress and degree completion.

Transferring to the Ph.D. program is a competitive process and only highly qualified M.S. students may be admitted. Student’s original application to the graduate program as well as the materials provided for the transfer petition are reviewed. Students must adhere to requirements for the terminal M.S. degree, and plan to confer the M.S. degree in the event that the Ph.D. petition to transfer is not approved.

M.S. to Ph.D. option requirements:

Have a cumulative GPA of 3.5 or higher.
Earn B+ plus or higher grade on all six core courses in the first year.
Pass all six topics of the qualifying exam before the beginning of the second year.
Complete a minimum of six units of research rotations in years one and two. course can be taken for three units at a time for credit/no credit.
Identify and align with a doctoral dissertation adviser before the end of winter quarter of second year who will act as research supervisor and provide funding through the completion of the program.
Submit a petition to the ICME Ph.D. admissions committee before the end of winter quarter of second year; transfer to the Ph.D. program subject to committee approval.

CME-MS - Computational and Mathematical Engineering (MS)

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