## 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:** 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!

## RECENT UPDATE

**[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.

# SCHEDULE

Event | Date | Related Documents |
---|---|---|

PSET 1 and related Matlab file pset1_1.m | Due Friday June 13 noon | Matlab codes from the textbook's website. Matlab tutorials (see especially matrix-vector operations). SOLUTIONS. |

PSET 2 and related Matlab file pset2_1.m | Due Friday June 20 noon | SOLUTIONS. |

EXAM 1 | In class, June 30 | SOLUTIONS. 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.m | Due 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 6 | Due Friday, July 25 in class | SOLUTIONS. |

EXAM 2 | In 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. |

# Syllabus

Day | Topics (page numbers are approximate and depend on the edition you have) |
---|---|

June 9 | Four special matrices (1-7) Intro to springs and masses |

June 11 | Differences vs derivatives, Boundary conditions (13-22) |

June 13 | Solving linear systems (26-31) The Delta function (36-42) |

June 16 | Discrete 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 18 | Eigenvalues and eigenvectors continued (54-60) Positive definiteness (66-70) |

June 20 | Positive definiteness continued (66-70) Springs and masses (98-102) |

June 23 | Springs and masses continued (102-109), with oscillations and finite differences (111 etc) |

June 25 | Finite differences in time (111-125) Least squares (128-135) |

June 27 | End of least squares (128-135) QR, SVD and the condition number (not on the exam, but will be in pset3 -- 78-87) |

June 30 | EXAM 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 11 | NO CLASS!!! RESCHEDULED FOR AUGUST 12 |

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. |