1. Fast Method of Fundamental Solution
Xinrong Jiang, PhD candidate
Wen Chen, Prof.
C.S. Chen, Prof.
NTU, Dec. 8, Taipei
Hohai University

2. Outline
Motivation
Methodology
Numerical Example
Conclusion

3. Motivation Radial Basis Functions (RBFs) : Domain Type Boundary Type
Kansa’s Method
Local Method of Particular Solution (LMPS)
Boundary Type
Method of Fundamental Solution (MFS)
Regularized Meshless Method (RMM)
Boundary Knot Method (BKM)
Boundary Particle Method (BPM)
Singular Boundary Method (SBM)
Motivation
Radial Basis Functions Methods: Meshless Method
BKM, MFS, RMM, SBM, …
Dense Matrix -> limitation in solving large-scale problem: infinite domain, plenty scatter
Speed up
mainframe – expensive, volume large, air-condition->electricity…
PC: memory limitation, CPU can not afford the computing with huge flops

4. Motivation Dense Matrix Speed up ^_^ iterative method
Large-scale problem
Infinite domain …
Speed up ^_^ iterative method
Mainframe T_T
expensive, computer volume large, electricity
PC T_T
Memory, CPU flops
Radial Basis Functions Methods: Meshless Method
BKM, MFS, RMM, SBM, …
Dense Matrix -> limitation in solving large-scale problem: infinite domain, plenty scatter
Speed up
mainframe – expensive, volume large, air-condition->electricity…
PC: memory limitation, CPU can not afford the computing with huge flops

5. Motivation Fast algorithm Efficiency  Accuracy Save memory
Fast computing
Efficiency  Accuracy

6. Methodology Fast Multipole Method (FMM) Fast Fourier Transform (FFT)
1987, 1997 new version, Rokhlin and Greengard
Nlog(N)->N
Adaptive
Fast Fourier Transform (FFT)
Precorrected FFT, J White
N*log(N)
Uniform
Hierarchical Matrix, Adaptive Cross-Approximation, etc

7. Methodology collocation, source Points: N Matrix: N*N
Notice that our RBF can be using the summation form for the numerical solution, the coefficient is the unknowns and belongs to the boundary-type method.

9. Methodology
Iterative Method
ill-conditioned

10. Methodology Krylov Subspace method::GMRES Iterator: open issue
Generalized minimal residual method
FMM-BEM
Fail in FMM-MFS
Iterator: open issue
Good iterator will be benefit for the result searching which is an open issue and now the popular iterator in FMM-BEM is Generalized minimal residual method (GMRES), a kind of Krylov subspace method.

11. MFS
easy-to-program, exponential convergence, highly accuracy, geometric flexibility and so on
infinite domain problems, large deformation problems, dynamic crack propagation
etc
FMM-MFS

12. Numerical Examples

13. Numerical Examples

14. Numerical Examples

15. Numerical Examples

16. Numerical Examples

17. GPU
Further acceleration

18. GPU::CUDA CUDA: Compute Unified Device Architecture
clump of piles in ocean

19. Parallel
Tree Structure
Further Study

20. Conclusion FMM-MFS high wave number requires wide band
implement successfully
high precision and speed
high wave number requires wide band
high frequency
large domain with low frequency
ill-conditioned requires suitable iterative method

21. End Thanks for your attention! Comments? 姜欣榮 Xinrong Jiang
08/12/ 2011