Michael Holst  
http://ccom.ucsd.edu/~mholst/ 
Professor of Mathematics and Physics UC San Diego 


Math 273A (Advanced Techniques in Computational Mathematics)
Course Topics: Mathematical and Computational Tools for Partial Differential Equations Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu; Regular Office Hours: Mon 1111:50am) Term: Fall 2018 Lecture: 10:00a10:50a MWF, 2402 AP&M TA: None Discussion: None Main Class webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/273a_f18/index.html Textbook(s): I. Stakgold and M. Holst, Green's Functions and Boundary Value Problems. John Wiley & Sons, New York, NY, Third Edition, 855 pages, 2011. This book is free to UCSD students, and a PDF version of the book can be found [ here ]. Printable Syllabus: Can be found [ here ]. CATALOG DESCRIPTION: 273A. ADVANCED TECHNIQUES IN COMPUTATIONAL MATHEMATICS (4) Models of physical systems, calculus of variations, principle of least action. Discretization techniques for variational problems, geometric integrators, advanced techniques in numerical discretization. Projectoriented; projects designed around problems of current interest in science, mathematics, and engineering. Prerequisites: consent of instructor. GENERAL THEME OF THE COURSE: Math 273ABC was designed as part of an NSF CAREER Award to be a new type of projectoriented course on the mathematical foundations of computational mathematics and physics. The main idea of the course is that each participant (including the instructor!) picks a topic of particular interest to his/her own research program, involving the analysis and/or numerical solution of an elliptic, parabolic, or hyperbolic partial differential equation. By attending the lectures and also pursuing their own projects, each participant will gain some mastery of the technical tools for analyzing their particular mathematical model using modern PDE methods (is their model wellposed?), for choosing a "good" discretization method (is the method guaranteed to give the right answer to some known precision?), and for designing algorithms for solving the resulting discrete problem on a workstation or possibly a large cluster (does the algorithm have reasonable complexity, and does it scale on a cluster?). The instructor's particular project this quarter is to analyze and solve a coupled nonlinear elliptic system that arises in mathematical and numerical general relativity; the application is to gravitational wave simulation, and is relevant to NSF's LIGO project. (This problem is the main focus of the instructor's current NSF FRG Project.) GRADES AND IMPORTANT DATES: There are no required homeworks or exams, although I will mention one or two suggested problems nearly every lecture as suggestions for selfstudy after the lecture. Your grade in the course is based (33% each) on (1) attending and participating in class, (2) on your work on your own project, and (3) your 20minute project presentation during the time of the final. NOTE: Your project, and your short presentation at the end of the quarter, can be done individually or in small groups (24 people). By the fourth week of the quarter you need to tell me what you picked for your project; I will discuss the projects in class the first week of the quarter, and can answer any questions that you have. Important dates:
SCHEDULE OF LECTURES: We will cover the three major topics listed below. The three major topics will each take about three weeks of the quarter. The individual subtopics listed within each of the major topics will be covered in varying degree, depending on the available lecture time and the interests of the students. My aim in choosing the topics for the lectures was to provide a general set of mathematical and computational tools for partial differential equations, for mathematicians, scientists, and engineers who want to have a toolkit for doing state of the art computational science. (A secondary aim is to make all of us stronger overall applied mathematicians!) NOTE: My focus will be on tools (mathematical and computational) for elliptic and parabolic PDE; if you choose a project based around hyperbolic PDE, then that is fine, but you will have to do a little selfstudy as well.
