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Numerical Solvers for BVPs

By Dong Xu

State Key Lab of CAD&CG, ZJU

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Overview

Introduction Numerical Solvers

– Relaxation Method– Conjugate Gradient– Multigrid Method

Conclusions

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Introduction

What is Boundary Value Problems?

Typical BVPs

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Discretization

Regular Grid Irregular Grid

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Linear System (Matrix)

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Relaxation Methods

0<w<2

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Conjugate Gradient

Steepest Descent Method– Search in the direction of the gradient of given point (lo

cal 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 & ATp (matrix multiplies vector), especi

ally efficient for sparse matrix.– Preconditioning

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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)

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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

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A New Way

“Coarsify” rather than “Simplify” Take H = 2h New residual equation

Approximation Restriction operator Prolongation operator Get new approximation

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Coarse-grid Correction Scheme

Compute the defect on the fine grid. Restrict the defect. Solve exactly on the coarse grid for the cor

rection. Interpolate the correction to the fine grid. Compute the next approximation.

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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.)

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Operators

Smoothing Operator S– Gauss-Seidel, NOT SOR.

Restriction Operator R

Prolongation Operator P

Straight injection, half weighting, full weighting.

Bilinear interpolation

Relationship

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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 –

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2 (named by shape)

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Multigrid Demo

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Full Grid Algorithm

First approximation– Arbitrary, on the finest grid. (Simple Multigrid, uh = 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.

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Full Grid Demo

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Conclusions

One Grid Two Grid Multi-Grid Full Grid

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Reference

Numerical Recipe in C

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Thank you