The class presents and ties together important notions of computational mathematics for scientists and engineers. It sheds a second light on linear algebra and differential equations. The focus is not on any particular application -- many will be covered from mechanical to electrical systems, graphs, networks, etc. -- but rather on the common mathematical framework that underlies most of them. Both modeling and computation will be covered. The class is suitable for master students, advanced undergraduates, or anyone interested in building a foundation in CSE.
Topics:
- Part 1: Applied Linear Algebra
- Difference matrices and boundary conditions
- Elimination, inverses, eigenvalues
- Positive definite matrices
- Fundamental subspaces and matrix decompositions (QR, SVD)
- Part 2: A Framework for Applied Mathematics
- Stiffness matrices and oscillations
- Least-squares
- Kirchhoff's laws, graphs, etc.
- Part 3: Boundary-Value Problems
- Gradient, divergence, Poisson's equation
- Splines and finite elements
- Finite differences and fast iterative methods
- Part 4: Fourier Series and Integrals
- Periodic functions
- The discrete Fourier transform and the FFT
- Convolution and deconvolution
The class will closely follow the first four chapters of the book Computational Science and Engineering by Gil Strang. The book should be available at the Coop.
- The book's web page, including a few sample sections, codes, and solutions to selected problems.
- Gil Strang's publisher web page, including instructions on how to order the book online with a discount
- The other class page, which will be maintained long after this page is gone, with tons of links to old homework problem sets, quizzes, solutions, and code.
Chapters 5 through 8 of the book are normally covered in 18.086 (not offered this term). It does not hurt to signal your interest to the instructor if you wish to see this class offered next term!
Prerequisites: Calculus, including derivatives,
integrals, linear differential equations, complex numbers, and familiarity
with the noble functions of mathematics (sin, cos, exp, log, etc.).
Vectors and matrices, including elementary operations (mat-vec,
mat-mat multiply, etc.), row reduction, linear (in)dependence of vectors,
linear systems of equations. It helps, but is not necessary, to have taken
a linear algebra class such as 18.06. The homework assignments will
involve basic computer programming in the language of your choice (Matlab
recommended).
Date and Time: Tu-Th, 2:30-4:00, room 4-370.
Instructor: Laurent Demanet. Office hours: W-F, 2:00-4:00,
room 2-392.
Teaching assistant: Dorian Croitoru. Office hours: Tu, 5:00-6:00, room 2-233
40% homework, 60% three in-class quizzes.
The homework problem sets will consist of both theoretical questions (without going into too much proof!) and numerical questions. No late copy will be allowed, but the lowest score will be dropped. Due Thursdays. Please use MATLAB notation to describe algorithms. Use of MATLAB for tedious calculations is encouraged, however you need to know how to do the basic algorithms taught in the course by hand (at least for small matrices) for the quizzes.
You can turn in the homework in a box in room 2-108, before 4:15PM on the day it is due.
