18.085 - Computational Science and Engineering I (Summer 2013)

 

Instructor: Rosalie Belanger-Rioux (week 3 through 7), Roberto Svaldi (week 1-2, 8-9-10)

Office: Rosalie: Room 2-331; Roberto: Room 2-491

Email: robr [at] math . mit .edu, rsvaldi [at] math . mit . edu

Lectures: M W F 9:30 - 11:00 am, E17-129

Office Hours: Rosalie : Th 10am-noon in 2-331; Roberto : W F 3-4 pm (week 1) M 3-4 pm, Th 10-11 am (other weeks), Room 2-491

Webpage: http://math.mit.edu/18.085/summer2013/

Stellar (for grades): https://stellar.mit.edu/S/course/18/su13/18.085/


COURSE DESCRIPTION

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: 30% problem sets, 70% three in-class quizzes (20-20-30). Lowest problem set score will be dropped. Quizzes' dates will be announced in class.

Problem Sets: Will be due in class on Mondays.

Registration: Registration must be submitted by the end of the first week of the summer session (Friday, June 14). Registration submitted after this deadline is subject to a $50 late fee.
The last day of classes, Friday, August 16, is the deadline for adding and dropping subjects.
All students attending should be registered either for credit or as listener.


 

SCHEDULE

Event Date Related Documents
PSET 1Due June 17Solutions, How to construct a sparse matrix
Matlab codes and tutorials
PSET 2Due June 24Solutions
Quiz 1June 28 in classSolutions. See old quiz 1 and related practice problems and solutions here . Also, have a look at question 3 from exam 1 in 2008 and question 1 in exam 1 of 2002 here .
PSET 3Due July 8Solutions
PSET 4Due July 15Matlab file,Solutions, Solution Matlab file.
PSET 5Due July 22Solutions
Quiz 2Friday July 26, in classPractice questions were emailed, see stellar announcement. Solutions
PSET 6Due August 6, by 10am, E18-401U Matlab code, Solutions
PSET 7Due August 12 Solutions
Quiz 3Aug 16, in class Practice questions:
Question 3 on this (Solutions)
Question 3 on this (Solutions)
Question 1 on this (Solutions)
Question 1 on this (Solutions)
Solutions (8/15 quiz, 8/16 quiz)

Syllabus

Topics and dates are tentative
Day Topics (page numbers)
June 10Introduction: examples
Finite differences 2nd order equations (13-21)
Linear algebra basics (685-689)
June 12More linear algebra(685-689)
June 14Fundamental theorem of linear algebra (690)
Resistor network problem (142-151)
Linear equations, Ax=b (686-687)
June 17LU decomposition (78, 26-30)
LU operation count (32-33)
June 19Best fit problems
Least squares
Normal Equations
June 21 Gram-Schmidt (80-81)
QR decomposition (79-81)
June 24QR decomposition (cont.)
Eigenvalues and eigenvectors (46-50)
June 26Using eigenvalues/vectors for dynamic problems such as Markov processes, 1st order systems of DEs (51-54)
Symmetric and positive definite matrices (66-67)
Review
June 28Quiz 1 (topics up to and including June 24)
July 1 Error analysis in solving Ax=b and the condition number (84-87)
Singular value decomposition (81-83)
July 3 SVD (cont.)
Error analysis and condition number revisited
Complex numbers
July 5 Newton's law F=ma and oscillations (98-104)
Pendulum, springs and masses, K=A^TCA (98-104, 111-112)
The wave equation
July 8 Conservation of energy in oscillation problems (112)
Finite differences in time, 1st order systems (113)
Forward and Backward Euler, Trapeziodal method (113)
Leap-frog for 1st and 2nd order equations (114)
July 10Back to leap-frog
Stability (120-121), fixed and free ends (19-21), forcing
July 12Resonance (19)
Curse of dimensionality
First-order equations/systems
Backward Euler for first-order equations (systems) leads to solving an equation (system) at every time step
July 15Structures, trusses (185-194)
Non-linear equilibrium problems (171-175)
July 17Non-linear problems, Newton's method (171-175)
July 19Boundary value problems: the hanging bar (229-232)
Laplace's equation, delta functions (232-234)
integration by parts to get the weak form (235-236)
July 22Galerkin's method using the weak form, F.E.M. steps (236-237)
hat and bubble functions (237-241)
July 24FEM in 2D, Green's formula to get the weak form (293-294)
admissibility conditions on V's, hats become pyramids (295-296)
flexibility of FEM compared to finite differences
July 26Quiz 2 (topics from July 3 (only complex numbers) to July 22.
Jul 29Gradient and divergence (255-265)
Potential flow
Jul 31Laplace equation (269-275)
Computing the flow on a square
Aug 2 Sine and Cosine series
Fourier series (317 -325)
Aug 5 Fourier series (325-331)
Discrete Fourier Transform
Aug 7 Fast Fourier transform
Aug 9 Signals and frequencies
Aug 12 More facts about Fourier Serie
and examples
Aug 14 Review
Aug 16 Quiz 3 (topics form July 29th to August 14th)