MTH 655-9: Numerical Analysis. Winter 2016
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General information
Assignments
Schedule, modules and assignments

Each module consists of theory, hands-on projects with guidance provided in lab meetings, and HW assignments, and is divided into parts corresponding to different concepts.
Exercises are listed as, e.g., I.A.1 for module I, part A, exercise 1. For your convenience, they are marked as (T): more theoretical, (C): more computational, or as a mixture (T/C) or (C/T) so that students can follow a preferred track. Those marked with (*) require extra effort/advanced background. Only some exercises are assigned as HW, the rest are intended as guided projects. In particular, those marked as BASIC, or PRACTICE, do not need to be turned in, unless specified otherwise.

HW assignments: The core of your work should be your insight. Make sure you provide clearly written out solutions that provide your insight. [A bunch of graphs stapled together without an appropriate guiding narrative, or labels, or some hastily typed code and calculations ... are NOT a solution]. Code is only required if it is substantially different from the templates I provide, or if something went wrong and you need help.
Sending solutions in PDF (only) by email is acceptable but I won't print them. Any of the fabulous graphics you wish to share can be posted in Numerical Art Gallery (forthcoming).
If you want the LaTeX source(s) for the modules, let me know and I can share them.

    Module I: Introductory concepts; interpolation/approximation.

    Background material and definitions: linear and normed spaces [Read 1.1, 1.2]. Interpolation concepts (read 3.2)

  1. 1/4/16: Class cancelled due to inclement weather.
  2. 1/6/16: Overview. Lagrange interpolation. [Read 3.2]
  3. 1/8/16: Class meeting in MLC computer lab (Kidd 108).
    Module I HW minimum: choose at least three parts of B, C, D, E, F, and turn in solutions to at least one exercise from each part you selected.
    Advanced students are welcome to turn in more. Date due: Jan. 22.
  4. 1/11/16: continue details about Lagrange interpolation.
  5. 1/13/16: piecewise interpolation for non-smooth functions. L^p and Sobolev spaces.
  6. 1/15/16: Lab 2: Class meeting in MLC computer lab (Kidd 108). [Background: read 2.4.4]
    LAB 2 HW recommended but not required: turn in at least one of the exercises I.K.2-3. You must be able to solve all BASIC exercises (do not turn in).
  7. 1/20/16: Basics of numerical integration seen as application of polynomial interpolation. Algorithms, error analysis, efficiency. [Read 2.4.4 for overall setup, and class notes]
  8. 1/22/16: Wrap-up simple quadrature rules; adaptive integration and error estimates. Gauss-Legendre (-Lobatto) quadratures: overview.
    Module II: Best approximation.

    Background material: inner product spaces [Read 1.3]. Approximation concepts (read 3.3, 3.4)

  9. 1/25/16: Approximation in normed spaces.
  10. 1/27/16: Approximation in inner product spaces. Variational inequality for convex subsets, and orthogonal projection for subspaces.
  11. 1/29/16: Lab 3: group work.
    Each group (assigned in lab) turns in the solutions to the BASIC part II.A, and II.B. Problem II.C can be done individually or by the group.
    Each individual turns II.D.ii, II.D.iii, and II.E.ii (these problems have easier and harder variants, so you can choose).
    Date due: Fri. Feb 12.
  12. 2/1/16: Orthogonal polynomials [Read Sec.3.5]
  13. 2/3/16: cd orthogonal polynomials; connection to Gauss integration. L^2 and H^1 projections.
  14. 2/5/16: Lab 4: individual work today.
    Module III: Iteration, and applications to solving differential and integral equations.

    Background material on operators in Chapter 2.1-3.

  15. 2/8/16: Operators: range, domain, injectivity, linearity. Examples in finite dimensional and in functional spaces.
  16. 2/10/16: Integral operators as examples. Solving functional equations by iteration. (Read Chapter 5.1-3). Picard (simple/fixed pt) iteration and Banach contraction mapping theorem. Introductory examples of Newton's method.
  17. 2/12/16: Meet in classroom: theory behind solving linear and nonlinear problems by iteration. Estimating order of convergence. Newton's method in R^n.
  18. 2/12/16: [Make up class at 10:00 in MLC Lab]. Lab 5. Practice Newton's method and fixed point iteration.
    Lab 5 solutions do not have to be turned in.
  19. 2/15/16: Applications. Part A: approximating solutions to BVP with Galerkin method. Best approximation in H_0^1. Discrete Laplacian matrix.
  20. 2/17/16: Continue Part A, nonlinear BVP and Newton's method. Comparing FD to FE solutions.
  21. 2/17/16: [Make-up class at 10:00 in STAG 160.] Solving Au=f by (stationary) iteration; [read 5.2]. Overview of solvability of integral equations: Neumann series for the linear operator [More details in Chapter 12].
  22. 2/19/16: Lab 6 in MLC computer lab: large(r) scale applications.
    Lab 6 solutions due Wed 3/2: Choose at least two of parts III.C, III.D, III.E, and from each of these sections solve at least two problems not including the BASIC problems, if any. (Students familiar with any parts should solve the harder problems. More experienced students can solve only the pbms marked as Extra Credit).
  23. 2/22/16: Newton's method wrap-up. Integral equations (more details in Chapter 12)
  24. 2/24/16: Overview of Boundary Integral Method, and moving on to R^d, d>1. (Chapter 13 has details)
    Module IV: Interpolation and approximation in R^d, d>1.

    Read Chapter 14 and Handout from the book by Phillips in Canvas.

  25. 2/26/16: Interpolation/approximation in d>1.
    2/29/16: no class today
  26. 3/2/16: I/A over quadrangulations, triangulations. Quadrature in d>1.
  27. 3/4/16: LAB 7; extended hour.
    Turn in the solutions (at least one substantial problem worked out for each subproject) to at least two IV.A, IV.B, IV.C, IV.D. DUE 3/16 by midnight.
    3/7/16: no class today
    3/9/16: no class today
  28. 3/11/16: Review and overview of reduced order modeling. Handout from the book by Quarteroni.