Course 18.086: Mathematical Methods for Engineers II
(Spring 2007)

Department of Mathematics
Massachusetts Institute of Technology



The grading will be based on homework exercises and a project. There will be no exams.


This course consists of three main topics: initial value problems, solving large systems, and optimization. The goal of the course is to provide a good start into each of these fields, focusing more on fundamental ideas than on involved details. Focus will be given on the mathematical understanding as well as on applying the presented concepts. Homework will include programming exercises, using Matlab. Outline of the topics:
  1. Initial value problems
    Linear initial value problems such as the wave equation and the heat equation admit closed form solutions in simple geometries. In a more complex setup they have to be solved numerically. Many important aspects seen for the linear problems regarding stability and covergence transfer to nonlinear initial value problems. Important examples for these are flow problems (fluid flow, traffic flow, etc.). Solutions to nonlinear problems may become non-smooth, and numerical methods have to consider this fact.
    Topics: Stiff problems, wave equation, heat equation, Airy equation, convection-diffusion, conservations laws, front propagation, Navier-Stokes equations, Fourier methods, finite differences, consistency, stability, covergence order, Lax equivalence theorem, CFL-condition, leapfrog method, staggered grids, shocks, upwind, Lax-Wendroff, finite volume methods, level set method
  2. Solving large systems
    The discretization of partial differential equations by finite difference or finite element methods leads to large sparse linear systems, either directly for linear problems or as an auxiliary subproblem for many nonlinear problems. Gaussian elimination destroys the sparse structure, so solvers are required which make use of the specific sparse matrix structure.
    Topics: Applications yielding sparse matrices, elimination with reordering, iterative methods, preconditioning, incomplete LU, multigrid, Krylov subspaces, conjugate gradient method
  3. Optimization and minimum principles
    Optimization problems search for the minimizer of some quantity (cost function), possibly given constraints. Quadratic cost functions lead to linear systems using Lagrange multipliers and Kuhn-Tucker conditions. Saddle point problems, regularization and calculus of variations will be presented as fundamental concepts. A different world in encountered in the case of linear cost functions. Applications are operations research and network problems. Solution algorithms are the simplex method or interior point methods. The underlying principle in all approaches is the concept of duality.
    Topics: Weighted least squares, duality, constrained minimization, inverse problems, calculus of variations, saddle point problems, linear programming, network problems, simplex method, interior point methods


1 Wed02/07 1Introduction
Fri02/09 2Ordinary differential equations6.2
Mon02/12 3Some scalar linear initial value problems6.1
Wed02/14 4Von Neumann stability analysis6.6 & 6.3
Fri02/16 5Finite difference methods for the one-way wave equation6.3
Tue02/20 6Modified equationLeVeque 11.1
21Wed02/21 7Lax equivalence theorem6.3
Fri02/23 8Numerical error analysis & convection-diffusion problems6.3 & 6.6
Mon02/26 9Wave equation, leapfrog, staggered grids6.4
Wed02/2810Nonlinear conservation laws: examples, method of characteristics6.8
Fri03/0211Nonlinear conservation laws: weak solutions6.8
Mon03/0512Nonlinear conservation laws: high resolution shock capturing6.8
Wed03/0713Nonlinear conservation laws: finite volume & particle methods6.8
32Fri03/0914Higher space dimensions
Mon03/1215Systems of initial value problems
Wed03/1416Level set method6.9
Fri03/1617Navier-Stokes equationGriebel
Mon03/1918Navier-Stokes equationGriebel
Wed03/2119Sparse matrices in applications7.1
3Fri03/2320Elimination with reordering7.1
03/26-03/30Spring break
4 Mon04/0221ILU, preconditioning, iterative methods7.2
Wed04/0422Iterative methods7.2
Fri04/0623Multigrid methods7.3
Mon04/0924Multigrid methods7.3
Wed04/1125Krylov subspace methods7.4
Fri04/1326Krylov subspace methods7.4
04/16-04/17Patriot's vacation
4Wed04/1827Least squares problems8.1
5 Fri04/2028Saddle point formulation & weighted least squares8.1
Mon04/2329Duality & minimization with constraints8.1
Fri04/2731Inverse problems8.2
Mon04/3032Tychonov regularization & ill posed operators in function spaces8.2
Wed05/0233Calculus of variations8.3
5Fri05/0434Calculus of variations8.3
Mon05/0735Saddle point Stokes problem8.5
Wed05/0936Linear programming8.6
Fri05/1137Linear programming8.6
Sat05/12 Project presentations
Mon05/1438Project presentations
Wed05/1639Project presentations

Course Materials

Homework Exercises

How to submit your solutions:

Matlab Programs

Course Projects

The following project are worked on: Project presentations on Saturday, 05/12/2007, 10:00am-3:00pm, in 2-132.

Video Lectures

There is information about the previous year's courses available on the MIT OpenCourseWare. In particular the video lectures by Gilbert Strang are strongly recommendable.
The course 18.085 is not a prerequisite for 18.086. Still, if you have not heard 18.085, you can watch some of the video lectures to get into the mood for numerical methods.


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