Slitex Foilset CPS615 Gauss Seidel Finite Element Methods and Conjugate Gradient
This discusses sequential and parallel Gauss Seidel and Jacobi Iteration
aThis is followed by an elementary discussion of Finite Element Methods applied
to Laplace's equation in two dimensions. This motivates the following detailed
account of Conjugate Gradient method including parallelism and analogies with
optimization
1:Finite and Infinite Dimensional Matrices as Operators
2:d/dx as an Operator and Its Scalar Products
3:d/dx as a Hermitean Operator
4:The Laplacian as a (Matrix) Operator
5:Mapping of Function Spaces to a Finite Number of Dimensions
6:Mathematical and Pseudo Code Form of Gauss Seidel Iteration Method
7:Mathematical (Matrix) Form of Gauss Seidel
8:Parallelism in Gauss-Seidel Iteration
9:Matrix Example Stencil
10:Matrix---Wavefront Parallelism for Gauss Seidel
11:The Red-Black Two Phase Parallel Gauss Seidel Iteration
12:Analysis of Parallel Red Black Gauss Seidel
13:Eigenvalues of Gauss Seidel Iteration Matrix
14:Comparison of Convergence of Gauss-Seidel and Jacobi Iteration
15:Successive Overrelaxation Iteration Method (SOR)
16:Convergence of SOR Compared to Jacobi and Gauss Seidel
17:Estimate of Over Relaxation Parameter
18:Pseudo Code for SOR---Successive Over Relaxation
19:Integral Formulation of Finite Element Method
20:Variation in Integral
21:Equivalence of Integral and Differential Formulation of Laplace's Equation
22:Discretization of Integral
23:Triangular Elements in Two Dimensions
24:Example for Two-Dimensional Triangular Elements
25:Bilinear Form of Integral with Triangular Elements
26:Formula for Stiffness Matrix Element
27:Finite Element Equations
28:Structure of Stiffness Matrix and Its Assembly
29:Conditions on Triangulation
30:Introduction to Poor Person's Conjugate Gradient
31:Conjugate Gradient Iteration for Quadratic Form
32:Conjugate Gradient and Method of Steepest Descent
33:Conjugate Gradient for Finite Element Problems
34:Poor Person's Conjugate Gradient and Eigenvalues of Matrix
35:Diagonalization of Quadratic Form
36:Diagonalization of Conjugate Gradient Equations
37:Convergence of Conjugate Gradient in Diagonalized Form
38:Clarification of Eigenvalue Analysis for Conjugate Gradient and Jacobi Iteration
39:Intuitive Description of Poor Person's Conjugate Gradient Algorithm
40:Improvement of Poor Person's Conjugate Gradient with Orthonormal Iteration
41:Full Conjugate Gradient Algorithm
42:Overview of Parallelism in Conjugate Gradient
43:Parallel Issues in Calculation of Matrix Elements
44:Scalar Products in Parallel Conjugate Gradient
45:Preconditioning in Conjugate Gradient
46:Convergence of Conjugate Gradient