18.085 - Computational Science and Engineering I (Summer 2014)


Instructors: Rosalie Belanger-Rioux (weeks 1-3, 9-10),
Anand Oza (weeks 4-8)

Office: Rosalie: Room E17-301Q;
Anand: Room E17-401G

Email: robr [at] math . mit . edu;
auoza [at] math . mit . edu

Lectures: M W F 9:30 - 11:00 am, E17-139

Office Hours: Rosalie: Tu and Th 1-2pm in E17-139 (or by appointment); Anand: Tu/Th 1-2pm in E17-139 (or by appointment).

Webpage: http://math.mit.edu/18.085/summer2014/

Stellar (for grades): https://stellar.mit.edu/S/course/18/su14/18.085/
Let us know if you're not on the Stellar membership list for the class!


The goal of this course is to give you:

Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications.

This class is suitable for masters students, advanced undergraduates, or anyone interested in building a foundation in computational science.

Prerequisites: Calculus and some linear algebra

Text Book: Computational Science and Engineering by Gilbert Strang

Grades: 40% problem sets, 60% three in-class quizzes (20-20-20). Lowest problem set score will be dropped. Quizzes' dates will be announced in class.

Problem Sets: Will be due in class on Fridays.

Registration: Registration must be submitted by the end of the first week of the summer session (Friday, June 13). Registration submitted after this deadline is subject to a $50 late fee.
All students attending should be registered either for credit or as listener.

Online lectures: online lectures from Gil Strang are available here. You are encouraged to look at them, either before or after class!


[08.06.2014]  RESCHEDULING: next Monday's class, 8/11, has been rescheduled to Tuesday 8/12, 9:30-11am, same classroom: E17-139. Don't forget!

[06.09.2014]  Welcome to the summer semester! You may find the class webpage from last summer here. Our syllabus will be basically the same, but will appear gradually as the semester unfolds.



Event Date Related Documents
PSET 1 and related Matlab file pset1_1.mDue Friday June 13 noon Matlab codes from the textbook's website.
Matlab tutorials (see especially matrix-vector operations).
PSET 2 and related Matlab file pset2_1.mDue Friday June 20 noon SOLUTIONS.
EXAM 1In class, June 30SOLUTIONS.
See practice exams here (exam #1, 2002-2008). Use common sense if a question looks as if we haven't covered that topic.
PSET 3 and related Matlab file pset3_3.mDue Monday, July 7th at 9:30 AM (beginning of class) SOLUTIONS.
PSET 4 and related Matlab files PSet4Q3_q.m and PSet4Q3_vect_q.m Due Monday, July 14th in class SOLUTIONS.
PSET 5 Due Monday, July 21 in class SOLUTIONS.
PSET 6Due Friday, July 25 in class SOLUTIONS.
EXAM 2In class, Monday, July 28 SOLUTIONS.
See practice exams here (exam #2, 2002-2007). Exam will cover material through Wednesday, July 23.
PSET 7 Due Monday, August 4 at 9:30 AM (beginning of class) SOLUTIONS.
PSET 8 Due on Tuesday August 12 in class. Solutions.
EXAM 3 In class, Friday August 15 Solutions.
See practice exams here (exam #3, 2002-2008). Exam will cover Fourier series and integrals only.
In particular, the 2008 exam is very good (don't think question 2 does not apply to us -- you should be able to do it, although if I were to give it to you in the exam I would use a smaller N, not N=8). SOLUTIONS to Question 2. Question 4a) is phrased a bit strange, I would ask : what is the vector c corresponding to the circulant matrix C such that the convolution of c with x is (2,2,2,2) and the convolution of c with y is (6,-6,6,-6)? Then, take the discrete Fourier transform of each vector, you will need those in b). Question 4c) asks for the eigenvalue first. No need to connect it to vector c hat, we haven't done that.
For the 2007 exam, question 1 might seem a bit unfamiliar, but you can do it too! Question 2 is about filters, but is optional. In 2a), we haven't really talked about X(omega), Y(omega) and A(omega) but you can just use the given formulas for them and see what you get for A. You might find b) confusing as the notation is different from us, but recall how multiplication in some space is the same as convolution in the other space, where the 2 spaces are related by the Fourier transform. In question 3c), write the transform of d in terms of the transforms of b and c. This is the convolution rule!!!
For the 2006 exam, question 1 is VERY similar to question 1 pset 8! Don't do question 2b and c, but 2a is a good one you should know! Question 3, skip c and d.
For the 2005 exam, skip question 1. Question 2: again, it's like question 1 of pset8! Question 3b), the solution has a slight typo, but c) is ok so you can use that to find the solution to b.
For the 2004 exam, question 1a might be hard to do, but you should know what answer to expect even if you can't do the calculation. You should definitely do 1b and 1c! Question 1d is nice but optional since not really the point of the class. If you try it, use the first form of the convolution integral given. Question 2 is a really good one for practice, you should try it! Question 3c: write down u(2) as a function of f and G integratee, but don't compute its actual value.
2003 exam: question 1 is great, but skip e). Do questions 2 and 3.
2002 exam: do questions 1, 2 and 3.


Topics and dates are tentative
Day Topics (page numbers are approximate and depend on the edition you have)
June 9Four special matrices (1-7)
Intro to springs and masses
June 11Differences vs derivatives, Boundary conditions (13-22)
June 13Solving linear systems (26-31)
The Delta function (36-42)
June 16Discrete Delta function (40-42) -- note we use a delta function with a load of 1/h, not 1 as in the book
Green's function (43)
Eigenvalues and eigenvectors (46-54)
June 18Eigenvalues and eigenvectors continued (54-60)
Positive definiteness (66-70)
June 20Positive definiteness continued (66-70)
Springs and masses (98-102)
June 23Springs and masses continued (102-109), with oscillations and finite differences (111 etc)
June 25Finite differences in time (111-125)
Least squares (128-135)
June 27End of least squares (128-135)
QR, SVD and the condition number (not on the exam, but will be in pset3 -- 78-87)
June 30EXAM 1: book and notes allowed, no electronic devices, no calculators. All topics up to and including June 25.
July 2 Graph models, circuits, and Kirchhoff's laws (142-155).
July 4 No class.
July 7 Matlab refresher
PageRank algorithm
Introduction to trusses (185-199).
July 9 More on trusses (collapse mechanisms: 192-194)
Newton's method in one dimension (171-173).
July 11 Newton's method in N dimensions (171-173).
The hanging bar (231-234).
Weak form of a differential equation (235-236)
July 14 Finite element method in 1D: Galerkin method, hat functions (236-239)
July 16 Finish finite element in 1D: bubble functions (241).
Numerical integration (240-241)
July 18 Plotting piecewise functions in Matlab
Bending equations and cubic finite elements (245-249).
Introduction to the gradient
July 21 Gradient and divergence (255-265)
Laplace's equation
July 23 Laplace's equation (269-275)
July 25 Green's function for Laplace's equation
Fast Poisson solvers (283-288)
July 28 Exam 2 (covers July 2 - 23)
July 30 Fast Poisson solvers
Finite elements in 2D - briefly (293-298)
Fourier series (317-324)
August 1 Fourier series (324-330)
August 4 Discrete Fourier series (346-350)
August 6 Fast Fourier Transform (350-353, not on exam but remember trick of w_N^2=w_(N/2)), Convolution (356-361)
August 8 Convolution part 2 (356-361), filters (361-363)
August 12 We will have class today 9:30-11am in E17-139, as a make-up for Monday august 11. Filters part 2 (361-363), Fourier Integral Transform (367-373).
August 13 2D Fourier, Shannon sampling. Review!
August 15 EXAM 3: covers Fourier material, up to and including august 12 lecture.