Numerical Solvers for BVPs

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Numerical Solvers for BVPs

• By Dong Xu
• State Key Lab of CADCG, 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
0ltwlt2
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Conjugate Gradient

• Steepest Descent Method
• Search in the direction of the gradient of given
  point (local approximation).
• The local gradient doesnt 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),
  especially 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
  correction.
• 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.
Relationship
Bilinear interpolation
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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

(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