F11_CSE316_Numeric_and_symbolic_computation

=CSE316: Numeric and Symbolic Computation.=

Program:
BS(CS)

Semester:
Fall 2011

Instructor:
Dr. H. bin Zubair

**Course Lead:**
Dr. H. bin Zubair

**Credit Hours:**
3 credit hours.

Prerequisite(s):
MTS101 (Calculus-1), MTS201 (Logic and discrete structures), CSE202 (Numerical analysis and algorithms).

Course Description:
This course is based chiefly on scientific computing, and has requires a specific mathematical skill-set for its smooth flow. The course mainly teaches how to solve computational problems from real applications, based on linear partial differential equations. A big part is on the emphasis of choosing a suitable discretization for the PDE, the rest is largely devoted to solving the resulting linear systems iteratively. The principal method dealt in the course is multigrid, with in-depth study of components and their Fourier analysis.

Course Objectives:
On culmination of the course, successful students are expected to be able to:
 * Model beginners to mid level mathematical problems using PDEs.
 * Decide which discretization method should be used for formulating a corresponding numerical problem.
 * Discretize the problem into suitable domains.
 * Construct a suitable multigrid solution method for its iterative solution.
 * Solve the problem in real time on a multi-core architecture.

Books:
The course would follow many different printed resources, some of which appear below:
 * Applied Numerical Analysis by Gerald & Wheatley, 5th Edition or later.
 * Multigrid Tutorial by Briggs, Henson, and McCormick 2nd Edition (available gratis on the Internet).
 * Multigrid by Trottenberg et al, 2001. Academic Press.
 * Templates for the solution of sparse linear systems (available gratis on the Internet).

Web Resources:
The following is a very useful website, which contains lots of free multigrid software. http://www.mgnet.org

Grading Policy:
The following grading policy would be followed in this course.
 * **Assessment** ||= **Points** ||
 * First Term Exam (closed book exam) ||= 20 ||
 * Second Term Exam (closed book exam) ||= 20 ||
 * Group Programming Projects (small scale) ||= 20 ||
 * Terminal Exam (closed book) ||= 40 ||

**Topics Covered (Salient Outlines) **
The following is a rough dissemination of topics into an approximate number of lectures. Matrix inversion using row operations. Crout's LU decomposition. Matlab methods and routines for direct solution ||= 6 || Review: Jacobi, Gauss-Seidel, SOR, SSOR CG, Bi-CGSTAB, GMRES, IDR(//s//) (only algorithms) Preconditioning for Krylov-subspace solvers. ||= 3 || Review of Kroneckor-tensor products. Equidistant Grid based averaging (restriction) in 2D. Full-Weighting (FW), Half-weighting (HW), and injection ops. Equidistant Grid based interpolation (prolongation) in 2D. Linear, bilinear, and cubic ops. Matlab implementation of restriction and prolongation ||= 4 || Basic categories of elliptic PDEs. Vertex-centered and cell-centered FDM discretization. Multigrid components. Smoothers and transfer operators. ||= 10 || The 2-grid algorithm. Migraton to V-cycle multigrid. Other cycle types such as W and F. Complexity analysis of V-cycle multigrid. Calibrating performance. ||= 3 ||
 * __**Topics**__ ||= __**Approximate # of lectures**__ ||
 * __**Review of basic numerical methods**__ ||= 2 ||
 * __**Direct methods for Linear System****s (Intro)**__
 * **__Iterative methods for linear systems (Intro)__**
 * **__Averaging and Interpolation (Intro)__**
 * __**Multigrid based solution of elliptic PDEs (Intro)**__
 * __**Local Fourier Smoothing Analysis (Intro)**__