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Efficient Fast Method of Fundamental Solution for Large-Scale Problems

This research paper discusses a fast method of fundamental solution for solving large-scale problems efficiently with high accuracy, including motivation, methodology, numerical examples, and conclusion.

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Efficient Fast Method of Fundamental Solution for Large-Scale Problems

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  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 • 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)

  4. Motivation • Dense Matrix • Large-scale problem • Infinite domain … • Speed up ^_^ iterative method • Mainframe T_T • expensive, computer volume large, electricity • PC T_T • Memory, CPU flops

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

  6. Methodology • Fast Multipole Method (FMM) • 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

  8. Methodology • Iterative Method • ill-conditioned

  9. Methodology • Krylov Subspace method::GMRES • Generalized minimal residual method • FMM-BEM • Fail in FMM-MFS • Iterator: open issue

  10. 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

  11. Numerical Examples

  12. Numerical Examples

  13. Numerical Examples

  14. Numerical Examples

  15. Numerical Examples

  16. GPU • Further acceleration

  17. GPU::CUDA • CUDA: Compute Unified Device Architecture

  18. Parallel • Tree Structure • Further Study

  19. Conclusion • FMM-MFS • 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

  20. End Thanks for your attention! Comments? 姜欣榮 Xinrong Jiang hhujiangxr@163.com 08/12/ 2011

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