MATH 531&532

Advanced Numerical Methods I (Fall 2007)

Monday 3:00-5:50, Room 3584

Instructor: Mo Mu
Rm3443, x7446
mamu@ust.hk
http://www.math.ust.hk/~mamu
Office Hours: to be announced

Course Description

Math531 and its continuation Math 532: Advanced Numerical Methods I & II introduce advanced topics on numerical methods in scientific computing. They will be selected from the following:

 

Discretization (for PDEs, ODEs, Differential-Integral Equations, etc.)

  • Finite difference methods
  • Finite element methods
  • Other topics, including Finite volume methods, Spectral methods, Collocation methods, Boundary integral methods, etc
  • Basic theory of stability, convergence, and error estimates

 

Algebraic Solution

PDE-based algebraic solvers

  • Basic iterative methods: Jacobi, Gauss-Seidel, SOR, CG
  • Advanced iterative methods: Preconditioning, Polynomial Acceleration, Multigrid, Domain Decomposition, Subspace Correction, etc.
  • Direct sparse solvers, Fast algorithms, etc.

Other topics in numerical linear algebra, signal processing, data-mining, etc

  • LU, QR, SVD, Lanczos methods, eigen-problems, data fitting, FFT, etc.

 

Advanced Topics in Scientific Computing

  • Parallel and distributed computing
  • Numerical software
  • Others

 

Topics to be covered in Math531 (Fall 2007)

 

· Review of finite difference methods for elliptic, parabolic and hyperbolic PDEs (teaching pace will depend on students background, whether they took Math331 (UG course on numerical PDEs) or equivalent))

· Finite difference approximation

· Indexing and sparse structures of discrete problems

· Methods for stability and convergence analysis: maximum principle, Fourier analysis, energy method, Lax equivalence theory

· Finite element methods

· Sobolev spaces and weak formulations

· Galerkin and Ritz methods, Finite element methods

· Interpolation theory and error analysis

· Time-dependent problems: semi-discretization, full-discretization, projection operator, stability and convergence analysis

· Other discretization methods (optional, depending on student’s background in Math331, interests, etc)

· Basic PDE-based solution methods (Chap2, Hageman & Young)

· Review of Jacobi method, Gauss-Seidel method, SOR method, Conjugate gradient method

· Basic convergence theory for Jacobi, Gauss-Seidel, and SOR methods (optional, depending on student’s background on Math231: Numerical Analysis or equivalent)

· Rates of convergence and the model problem analysis

· Direct methods, including band solvers, nested dissection, multi-frontal methods, sparse solvers, fast algorithms, etc.

· Computational issues, application and simulation, numerical software

· Parallel and distributed computing

 

Course Material

Lecture notes, handouts, and references

References:
 

·  Mo Mu, From Completing the Squares and Orthogonal Projection to Finite Element Methods, Lecture Notes

·  K.Morton and D.Mayers, Numerical Solution of PDEs, Cambridge University Press, 1994

·  Birkhoff and Lynch, Numerical Solution of Elliptic Problems, SIAM, 1984

·  P. Ciarlet, The Finite Element Method for Elliptic Problems, N. Holland Pub., 1978

·        V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, 2nd Ed., Springer, 2006.

·  J. Thomas, Numerical PDEs--Finite Difference Methods, Springer, 1995

·  Louis Hageman and David Young, Applied Iterative Methods, Academic Press, 1981

·  G. Golub and C. van Loan, Matrix Computations, Johns Hopkins, 1983

·  J. Rice and R. Boisvert, Solving elliptic problems using ELLPACK, Springer, 1984

·  Research papers
 

Course Work and Grading Policy

The course work consists of class attendance, homework and computer projects. Final grades are assigned based on the performance of the course work.

Hope you enjoy this course. Thank you very much!

Last Revised: 15 January 1997