COURSE DESCRIPTION
The goal of this course is to give you:
- practical experience with common computational methods in engineering, and
- enough theoretical understanding to know when those methods can go wrong
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 1 | Due June 17 | Solutions, How to construct a sparse matrix Matlab codes and tutorials |
| PSET 2 | Due June 24 | Solutions |
| Quiz 1 | June 28 in class | Solutions. 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 3 | Due July 8 | Solutions |
| PSET 4 | Due July 15 | Matlab file,Solutions, Solution Matlab file. |
| PSET 5 | Due July 22 | Solutions |
| Quiz 2 | Friday July 26, in class | Practice questions were emailed, see stellar announcement. Solutions |
| PSET 6 | Due August 6, by 10am, E18-401U | Matlab code, Solutions |
| PSET 7 | Due August 12 | Solutions |
| Quiz 3 | Aug 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
| Day | Topics (page numbers) |
|---|---|
| June 10 | Introduction: examples Finite differences 2nd order equations (13-21) Linear algebra basics (685-689) |
| June 12 | More linear algebra(685-689) |
| June 14 | Fundamental theorem of linear algebra (690) Resistor network problem (142-151) Linear equations, Ax=b (686-687) |
| June 17 | LU decomposition (78, 26-30) LU operation count (32-33) |
| June 19 | Best fit problems Least squares Normal Equations |
| June 21 |
Gram-Schmidt (80-81) QR decomposition (79-81) |
| June 24 | QR decomposition (cont.) Eigenvalues and eigenvectors (46-50) |
| June 26 | Using 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 28 | Quiz 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 revisitedComplex 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 10 | Back to leap-frogStability (120-121), fixed and free ends (19-21), forcing |
| July 12 | Resonance (19)Curse of dimensionalityFirst-order equations/systemsBackward Euler for first-order equations (systems) leads to solving an equation (system) at every time step |
| July 15 | Structures, trusses (185-194)Non-linear equilibrium problems (171-175) |
| July 17 | Non-linear problems, Newton's method (171-175) |
| July 19 | Boundary 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 22 | Galerkin's method using the weak form, F.E.M. steps (236-237)hat and bubble functions (237-241) |
| July 24 | FEM 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 26 | Quiz 2 (topics from July 3 (only complex numbers) to July 22. |
| Jul 29 | Gradient and divergence (255-265) Potential flow |
| Jul 31 | Laplace 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) |