## 18.086## Mathematical Methods for Engineers II## Spring 2017## Instructor: Homer Reid |

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

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 inmathematics; however, I find it also pretty well describes my own research in physics and engineering.

## Class Schedule

This is aroughschedule 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 day2/14/2017 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 class3/16/2017 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 5/5/2017 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 anindependentproject, 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 paper being presented is not the work of the presenter (there are, of course, other types of talks in which researchers present their own work to their group.) Instead, the paper presented in a journal-club talk is one that the presenter stumbled upon in a journal (or online a preprint archive) and happened to find interesting and potentially relevant to his or her own work---perhaps because it introduces a useful new technique that the research group could adopt.

- The presenter is generally
notexpected to understand every last detail of the paper, nor do the audience members expect to become experts in any new subfield. The goal is simply for the presenter to acquire, and impart to the audience, an executive-level understanding of the research project described in the paper: What problem was tackled, why the problem is interesting, what had been done on it previously, what methods were used in this paper, and what results were obtained.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:

- Peruse a well-known academic journal in your field to find a research article that makes use of some topic that we have covered in 18.305. This may
notbe an article written by you or a close colleague (for example, your roommate); the point is to learn to read a journal paper without having the author around to explain it to you.

- Read the article as thoroughly as you can. As would be the case in an actual journal-club talk, you do
notneed to understand everything in the paper (in fact, if you do, you are kinda missing the point of the assignment). However, youdoneed to have some understanding of the part of the paper that uses the 18.305 technique in question---in particular, you should be able to explain why that technique was chosen, and why it is a better (or worse) choice than other methods that could have been used.

- Write a (roughly)
one-page summaryof the paper at a level roughly commensurate with that of a journal-club talk. That is, assume your audience is familiar with the general field of science or engineering, but not with the specific subject of the paper; briefly explain the problem, its background, and the new contribution made by the paper; and---this is important---discuss the 18.305 technique that was used and explain why it was a good (or bad) choice for the problem.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