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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
• 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
• Steepest Descent Method
• Search in the direction of the gradient of given
point (local approximation).
• The local gradient doesnt point to the elliptic
center.
• 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.
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