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This archived information is dated to the 2011-12 academic year only and may no longer be current.

For currently applicable policies and information, see the current Stanford Bulletin.

Doctor of Philosophy in Computational and Mathematical Engineering

Applications to the Ph.D. program and all required supporting documents must be received by December 6, 2011. See /admissions for full information. Prospective graduate students should see for information and application materials. 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.


  1. Complete a minimum of 135 units of residency at Stanford, including:
    1. 45 units from the master's program
    2. 27 units of focused electives 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 320-380 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.
    3. 60 units of thesis research
    4. 3 units of free electives
  2. Maintain a grade point average (GPA) of 3.5.
  3. Pass the qualifying examination administered by ICME.
  4. Complete an approved program of original research.
  5. Complete a written dissertation based on research.
  6. Pass the oral examination that is a defense of the dissertation research.

Specialized Elective List—See requirement 1b above.

CEE 362G. Stochastic Inverse Modeling and Data Assimilation Methods

CME 364A,B. Convex Optimization I,II

CS 348A. Computer Graphics: Geometric Modeling

EE 363. Linear Dynamical Systems

EE 368. Digital Image Processing

MATH 205A. Real Analysis

MATH 215A. Complex Analysis, Geometry and Topology

MATH 217A. Differential Geometry

MATH 221. Mathematical Methods of Imaging

MATH 227. Partial Differential Equations and Diffusion Processes

MATH 236. Introduction to Stochastic Differential Equations

MATH 237. Stochastic Equations and Random Media

MATH 238. Mathematical Finance

ME 335A,B,C. Finite Element Analysis

ME 346B. Introduction to Molecular Simulations

ME 351A,B. Fluid Mechanics

ME 361. Turbulence

ME 408. Spectral Methods in Computational Physics

ME 412. Engineering Functional Analysis and Finite Elements

ME 469A,B. Computational Methods in Fluid Mechanics

MS&E 319. Approximation Algorithms

MS&E 336. Topics in Game Theory with Engineering Applications

STATS 305. Introduction to Statistical Modeling

STATS 306A,B. Methods for Applied Statistics

STATS 318. Modern Markov Chains

STATS 366. Computational Biology

Note: All courses listed under "Requirement 3" under the "Master of Science in Computational and Mathematical Engineering" section can be used for fulfilling the general elective requirement.


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 half-time assistantships which provide a tuition scholarship at the 8, 9, 10 unit rate during the academic year and a monthly stipend. Half-time 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.

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