1 Numerical Solvers for BVPs By Dong Xu State Key Lab of CAD&CG, ZJU

2 Overview  Introduction  Numerical Solvers –Relaxation Method –Conjugate Gradient –Multigrid Method  Conclusions

3 Introduction  What is Boundary Value Problems?  Typical BVPs

4 Discretization  Regular Grid  Irregular Grid

5 Linear System (Matrix)

6 Relaxation Methods 0<w<2

7 Conjugate Gradient  Steepest Descent Method –Search in the direction of the gradient of given point (local approximation). –The local gradient doesn’t point to the elliptic center.  Conjugate Gradient Method –Search in the direction pointing to the elliptic center. –Iterate at most n steps. (n – the order of the matrix) –Only need Ap & A T p (matrix multiplies vector), especially efficient for sparse matrix. –Preconditioning

8 Multigrid Methods  Multigrid Methods – NOT a single algorithm, BUT a general framework.  Solve elliptic PDEs (BVPs) discretized on N grid points in O(n) operations.  Multigrid means using fine-to-coarse hierarchy to speed up the convergence of a traditional relaxation method.  Another approach is discretizing the same underlying PDE in multiple resolution. (FMG method)

9 Equations  Equation  Discretization  Correction  Residual/Defect  Linear relation between correction and residual  Only knows residual how to get correction? –Approximation –Jacobi iteration: diagonal part –Gauss-Seidel iteration: lower triangle  Get new approximation

10 A New Way  “Coarsify” rather than “Simplify”  Take H = 2h  New residual equation Approximation  Restriction operator  Prolongation operator  Get new approximation

11 Coarse-grid Correction Scheme  Compute the defect on the fine grid.  Restrict the defect.  Solve exactly on the coarse grid for the correction.  Interpolate the correction to the fine grid.  Compute the next approximation.

12 Two-Grid Iteration  Pre-smoothing: Compute by applying steps of a relaxation method to.  Coarse-grid correction: As above, using to give.  Post-smoothing: Compute by applying steps of the relaxation method to. Key Insight: Relaxation methods are good smoothing operators. (High freq. attenuates faster than low freq.)

13 Operators  Smoothing Operator S –Gauss-Seidel, NOT SOR.  Restriction Operator R  Prolongation Operator P Straight injection, half weighting, full weighting. Bilinear interpolation Relationship

14 Multi-Grid  Cycle – One iteration of a multigrid method, from finest grid to coarser grids and back to finest grid again. , the number of two-grid iterations at each intermediate stage (resolution/level).  V-cycle –  W-cycle – (named by shape)

15 Multigrid Demo

16 Full Grid Algorithm  First approximation –Arbitrary, on the finest grid. (Simple Multigrid, u h = 0) –Interpolating from a coarse-grid solution.  Nested Iteration –Get coarse-grid solution from even coarser grids. –At the coarsest grid, start with the exact solution.  Need f at all levels, while simple multigrid only needs f at the finest level.  Produce solutions at all level, while simple multigrid at the finest level.

17 Full Grid Demo

18 Conclusions  One Grid  Two Grid  Multi-Grid  Full Grid

19 Reference  Numerical Recipe in C

20 Thank you