Welcome to 18.085!

The class will meet on TTh from 11:00am to 12:30pm in 2-190

The instructor office hours will be on TTh at 12:30pm, first in 2-190, and if needed afterwards in 2-247

The TA office hours and pset check-offs will be on M at 1pm and W at 3pm, in 2-390B (might be moved to 2-290 if too popular)

Instructor: Laurent Demanet
TA: Ashu Tripathi, <at943@mit.edu>

Q&A on piazza

Psetpartners

Left: Munich GeoCenter. Right: Dedalus, Keaton Burns

The class will largely follow the structure used by many instructors in the past. Our main reference will be the book on CSE by Gil Strang. See below for the syllabus, for more info on how to order the book, and for more reference material.

There will be no final exam. There will be 10 problem sets, and 3 quizzes. The grade breakdown will be 25% psets and 75% quizzes.

- The quizzes will be on Oct 02, Oct 30, and Dec 04 in class. Quiz makeup requests must be duly justified and must come at least 3 days in advance (except true emergency).

- Your written solutions for each problem set will count for 1% of the grade. In-person check-offs will be offered during the TA office hours, and are each worth 3% of the grade. No more than 25% of the grade can be awarded from problem sets and their check-offs.

There will be no recitations. We are running a piazza site for Q&A, with the ability to post anonymously or just to the instructor. Use psetpartners if you want to join a study team.

Syllabus

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. We will loosely follow this outline:

- 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

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 (like Python, Julia, or Matlab).

If you need a refresher on linear algebra, "Introduction to linear algebra'", also by Gil Strang, is a fantastic book.

Reference material

The class will loosely follow the first four chapters of the book Computational Science and Engineering by Gil Strang.

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

Chapters 5 through 8 of the book are normally covered in 18.086 (not taught this year, in principle). It does not hurt to signal your interest to the instructor if you wish to see 18.086 offered in the future!