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Welcome to 18.085 / 18.0851!

The class will meet on MWF from 10am to 11am Eastern time, on zoom. The office hours will be on Th at 10am Eastern.

In terms of topics, the class will largely follow the structure used by many instructors in the past, including Prof Strang. The book will be the same as before (CSE by Gil Strang). See below for the syllabus, for more info on how to order the book, and for more reference material.

We recognize that ease of access might be an issue, particularly if you are in a different time zone. As a result, we strongly encourage live participation but do not require it. Recordings of the lectures will be made available. The live component of the class will include ungraded group activities in breakout rooms, so you get to know one another.

There will be no final exam. There will be weekly problem sets, and 2 quizzes. You will have a set amount of time (24h) for completing each quiz after it is handed out. We will operate on an honor system -- no collaboration on the quizzes. The grade breakdown will be 50% psets and 50% quizzes. To further minimize the effect of occasional scheduling conflicts you might have, the lowest pset score and the lowest quiz score will both be dropped.

There will be no recitations, but there will be virtual office hours (time TBA, probably Tuesday mornings). We are running a piazza Links to an external site. site for Q&A, with the ability to post anonymously. The office hours can start with some of the piazza questions. The live Q&A during lecture will likely be through the zoom chat.

18.085 is also a community, we have a discord server Links to an external site.! (be mindful that discord is a bit invasive, it links to spotify, steam, etc. although you can manage that once inside). You can join both piazza and discord even if you're only contemplating enrolling at this point.

If you are running into logistical difficulties before the start of the term, don't hesitate to contact the instructor. Link to loan a Wacom tablet (for graduate students).

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.

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

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).

Reference material

The class will closely 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 Links to an external site., including instructions on how to order the book online with a discount

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!