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18.086

Mathematical Methods for Engineers II

Spring 2017

Instructor: Homer Reid

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Course Overview


Course Mechanics

Lectures: 1:00--2:30 pm, Tuesday / Thursdays, 2-131
Grading:
Problem sets: 35%
Journal-club reports: 15%
Independent project(s): 50%
Office hours: 4-6 PM Wednesdays, 2-232C

Course Syllabus

This course is a semester-long transition from the neat, cut-and-dried world of academia---where the task is to imbibe and regurgitate predigested material and reproduce previously-known results on problem sets and exams---to the messy, rough-and-tumble world of research, in which "...we never know what we are talking about, nor whether what we are saying is true."1

The heart of the course is a semester-long independent project that you will complete on your own, working in close consultation with me to identify a topic of interest and you and define a task of appropriate scope.

To facilitate your work on this project and speed you along the way out the door from academia to research, I will provide a steady patter of lectures on a variety of advanced mathematical techniques of general practical usefulness across a wide range of disciplines in engineering and applied science.

To help you master these techniques, I will distribute problem sets including both exercises and more challenging problems covering the content in the lectures.

Meanwhile, to help you see the practical relevance of this material to research in your own area of specialty, you will peruse research journals in your particular field of interest and submit brief journal-club reports summarizing published articles that make use of mathematical techniques discussed in lectures.

1 The quote here is from the great English mathematician and philosopher Bertrand Russell, who was referring specifically to research in mathematics; however, I find it also pretty well describes my own research in physics and engineering.

Class Schedule

This is a rough schedule of the content we will cover and the weeks in which we will cover it. We will cover the material in roughly the order listed below, but we may spend slightly more or less time on certain topics than is budgeted below.
Date Topic Recordings
2/7/2017 Tuesday Course invitation. Lecture video
2/9/2017 Tuesday No class: Snow day
2/14/2017 Tuesday The pantheon of modern mathematical methods for engineers, part 1:
  • Classical deterministic numerical calculus
  • Modern deterministic numerical calculus
  • Numerical linear algebra
Lecture video
2/16/2017 Thursday The pantheon of modern mathematical methods for engineers, part 2:
  • Deterministic numerical optimization
  • Stochastic numerical methods
  • Asymptotic analysis
Lecture video
2/23/2017 Thursday Motivating problems in classical and modern deterministic numerical calculus
2/28/2017 Tuesday Classical numerical quadrature: Newton-Cotes rules
3/02/2017 Thursday Classical timestepping approach to ODE initial-value problems
3/07/2017 Tuesday Finite-difference approach to ODE boundary-value problems
3/09/2017 Thursday
  • Finite-difference solution of Laplace problems: stripline capacitor over ground plane
  • Gauss-Legendre quadrature: Inner products and orthogonal polynomials.
3/14/2017 Tuesday Snow day: no class
3/16/2017 Thursday Gauss-Legendre quadrature: Roots of orthogonal polynomials and weights of Gaussian quadrature schemes.
3/21/2017 Tuesday Spectral methods for PDEs: Surface-integral-equation methods for Laplace problems; application to stripline capacitor problem.
3/23/2017 Thursday Spectral methods for boundary-value problems: Chebyshev polynomials and differentiation matrices.
4/4/2017 Tuesday Numerical linear algebra: Dense-direct methods. LU/Cholesky factorization, Gaussian substitution, QR decomposition, eigendecomposition, SVD.
4/6/2017 Thursday Numerical linear algebra: Sparse-direct methods, iterative methods.
4/11/2017 Tuesday Floating-point arithmetic. Catastrophic loss of floating-point precision. Conditioning.
4/13/2017 Thursday Deterministic optimization I: Newton methods.
4/20/2017 Thursday Deterministic optimization II: Quasi-Newton methods.
4/25/2017 Tuesday Stochastic methods I: Monte-Carlo integration.
4/27/2017 Thursday Stochastic methods II: Markov-chain Monte Carlo.
5/2/2017 Tuesday Stochastic methods III: Floating random-walk solution of stripline capacitance problem.
5/4/2017 Thursday Asymptotic analysis I: Method of multiple scales.
5/9/2017 Tuesday Asymptotic analysis II: Renormalization group.
5/11/2017 Thursday Asymptotic analysis III: Stationary-phase method.
5/16/2017 Tuesday Computational nonlinear algebra: Resultants.
5/18/2017 Thursday Computational nonlinear algebra: Groebner bases.

Problem Sets

Date Due Problem Set Notes
3/10/2017 PSet 1 Notes on finite-difference approach to stripline capacitor problem
3/24/2017 PSet 2
PSet 3 Notes on numerical linear algebra
PSet 4


Collaboration policy: Problem sets

You may (and are encouraged to) work together on PSets, but you must write up and turn in your own solutions. For problems involving computer programs, you must submit a listing of your program and its output together with your PSet. If you work with other students to write your program, each student in your study group must separately type in and execute the program to generate a listing submitted with the PSet.


Collaboration policy: Course project

The course project is an independent project, to be completed by you alone. Of course, you may discuss your work on the project with other students, but this is not a team project. The same is true for the journal-club reports: these are individual assignments, not group work.

About the Journal-Club Reports

An institution common to many fields of science and engineering---in both academic and corporate settings---is the journal club, a regular or irregular sequence of informal talks in which members of a research group take turns describing to their colleagues a research paper (typically from an academic journal) that the presenter happened to find interesting.

In a typical journal club meeting, a group of researchers---which might be a university research group, including students, postdocs, and professors, or which might be a corporate R&D group, including junior and senior engineers, staff scientists, etc.---gather, perhaps over lunch, to look over the paper in question while one of their group members summarizes it and delivers a synopsys of its most interesting points. Journal club presentations are generally characterized by the following features:

The goal of the journal-club assignments in 18.305 is to mimic the experience of an actual journal club by giving you an opportunity to see how the topics and methods covered in our course are used in the actual practice of your favorite field of science or engineering.

For each assignment, you will do the following:

Note: Make sure your submitted writeup includes a link to an online version of the paper you read so that we can peruse it while reading your report. (If you write out your report by hand, you may email us the link.)

Due dates for journal-club reports

Report Due date
JC report 1 During individual meetings, weeks 5/6
JC report 2 4/28/2017
JC report 3 5/12/2017

About the course project


18.086 Spring 2017 Main course page