Math Modeling II (Fall, 2009)
Instructor: Xin Li (Office Hours: MWF 10:30-11:30)
Classroom: HEC 103
Time: 9:30 - 10:20 MWF
Objectives: We will discuss selected topics from numerical approximation, numerical optimization, and numerical solution of differential equations (both ordinary and partial differential equations) and their applications in solving real world problems in machine learning, image analysis, and computer vision.
Homework assignments and tests: Regular homework will be assigned and graded. Four mid size projects are planned for the semester (for both theory and computing/programming). There will be two mid term tests and a comprehensive final examination.
Grading:
| Homework (8) | 20 points each |
| Project (4) | 40 points each |
| Mid term tests (2) | 100 points each |
| Final exam | 200 points |
| TOTAL | 720 points |
Grade scale: A: over 620 points; B: 520 - 619 points; C: 420 - 519 points; D: 320 - 419; F: less than 320 points
List of topics:
| 8/24 | Splines of degree 1 | HW:P.381: 1,2,3,6,10,11 | |
| 8/26,8/28 | Quadratic and cubic splines | HW: P.382:15,16,17 | Computer Problem: P.384: 3 |
| 8/31 | Algorithm for natural cubic spline interpolation | HW: P.399-: 1,3,4,5,12,21,27 | |
| 9/2 | Smoothness Theorem, space curves, B-splines, Bezier curves | HW: P.420-: 3,11,13,19,20,23,34,37,38 | Computer Problem: P.403: 4 |
| 9/4 | B-splines and wavelets (paper1,paper2,Professor Wang's slides) | Sample codes for interpolating natural cubic splines: Spline3Coef.m, Spine3Eval.m,TestSpline3.m | Implement and reproduce figure 5 in paper2 |
| 9/9 | Multiresolution Analysis | First home work collection (only HW assignments, not Computer Problems) due HW: Study paper 1 and paper2 |
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| 9/11 | Applications of splines and wavelets | HW: Study appendix of paper 1 | |
| 9/14 | Image compression | ||
| 9/16 | 2D wavelets, intro to MRA | Project: Implement the algorithm in paper 1 and use it for 2D image compression (reproduce figures 5,6&9 with your choice of image) with a brief report. | |
| 9/18 | Numerical solution of DE | HW: 436-: 1(b),(f), 2(c), 3(c), 5, 8,9,15 | P. 438: 5 |
| 9/21 | Error Analysis for Euler's Method | Verify the final inequality for |en|. | |
| 9/23 | Understanding Runge-Kutta methods | See the end of the notes | Computer Problems and project are due on Friday! |
| 9/25 | Variations of RK: RKF,ADM | HW: P455: 2,3,8 | P.461: 2,3,4 |
| 9/28 | System of ODEs | See the end of notes | Due on Monday 10/5 |
| 10/2 | BVP for ODE: shooting method | Verify equation (2) on p. 566. P. 568: 5(a), 6 | Due on |
| 10/5 | BVP: FD method | P. 578: 2, 7 | Implement program BVP1 and reproduce the table on p. 573. (Due: 10/19) |
| 10/9 | BVP: FEM | See the end of the notes | Due on |
| 10/12 | Parabolic problem | P.595: 7,9,11,12 | Due on 10/19 |
| 10/16 | Hyperbolic problem | P.604: 1,2,4, Computer problem 2 | Due on 10/19 |
| 10/19 | Review | Topics, sample hint/solution | |
| 10/23 | Midterm Examination I | ||
| 10/26 | Elliptic problem: FD method | P.620: 1(a), 3,4,6 | Due on 11/4 |
| 10/30 | Elliptic problem: FEM | P.621: 7, Computer problem 5. | Due on 11/4 |
| 11/2 | Project 2, Project 3 | ROF paper, CS paper for Project 3. Sample image. Sample code for ROF | |
| 11/6 | Deriving Euler's equation | p.34: 1,2,3 | Due: 11/13 |
| 11/9 | Golden Ratio Search | 16.1.6, 16.1.10,16.1.15 | Due: 11/13 |
| 11/13 | Fibonacci Search | 16.1.3-6 | Due: 11/18 |
| 11/16 | Art and Craft of Math Modeling | Distinguished Guest Lecturer: Professor K. R. Rajagopal | |
Textbooks and references: We will follow some materials from the second half of Cheney&Kincaid (Numerical Mathematics and Computing, 6th edition). Additional topics will be supplemented by papers and lecture notes.
