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Numerical Solvers for BVPs

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Search in the direction of the gradient of given point (local approximation) ... Multi-Grid. Full Grid. 19. Reference. Numerical Recipe in C. 20. Thank you ... – PowerPoint PPT presentation

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Title: Numerical Solvers for BVPs


1
Numerical Solvers for BVPs
  • By Dong Xu
  • State Key Lab of CADCG, 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
0ltwlt2
7
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

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.
Relationship
Bilinear interpolation
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,
    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.

17
Full Grid Demo
18
Conclusions
  • One Grid
  • Two Grid
  • Multi-Grid
  • Full Grid

19
Reference
  • Numerical Recipe in C

20
Thank you
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