Regularized meshless method for solving the Cauchy problem

1. Regularized meshless method for solving the Cauchy problem 以正規化無網格法求解柯西問題 Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 2006/12/16

2. Outlines • Motivation • Statement of problem • Method of fundamental solutions • Desingularized meshless method • Formulation for Cauchy problem • Regularization techniques • Numerical example • Conclusions

3. Outlines • Motivation • Statement of problem • Method of fundamental solutions • Desingularized meshless method • Formulation for Cauchy problem • Regularization techniques • Numerical example • Conclusions

4. Numerical Methods Mesh Methods Meshless Methods Finite Difference Method Finite Element Method Boundary Element Method (MFS) (RMM) Motivation

5. Outlines • Motivation • Statement of problem • Method of fundamental solutions • Desingularized meshless method • Formulation for Cauchy problem • Regularization techniques • Numerical example • Conclusions

6. Statement of problem • Inverse problems (Kubo) : Cauchy problem

7. Cauchy problem

8. Outlines • Motivation • Statement of problem • Method of fundamental solutions • Desingularized meshless method • Formulationfor multiple holes • Regularization techniques • Numerical example • Conclusions

9. d Source point Collocation point — Physical boundary -- Off-set boundary Method of fundamental solutions (MFS) • Method of fundamental solutions (MFS) : Distributed-type Dirichlet problem Neumann problem Single-layer Potential approach d = off-set distance Double-layer potential approach Dirichlet problem Neumann problem

10. The artificial boundary (off-set boundary) distance is debatable. • The diagonal coefficients of influence matrices are singular when the source point coincides the collocation point.

11. Outlines • Motivation • Statement of problem • Method of fundamental solutions • Desingularized meshless method • Formulationfor Cauchy problem • Regularization techniques • Numerical example • Conclusions

12. Source point Collocation point — Physical boundary Desingularized meshless method (DMM) • Desingularized meshless method (DMM) Double-layer potential approach Dirichlet problem Neumann problem where I = Inward normal vector O = Outward normal vector

13. In a similar way,

15. Outlines • Motivation • Statement of problem • Method of fundamental solutions • Desingularized meshless method • Formulation with Cauchy problem • Regularization techniques • Numerical example • Conclusions

16. Formulation with Cauchy problem M Collocation points N Collocation points

17. Derivation of diagonal coefficients of influence matrices. Where

18. where

19. Rearrange the influence matrices together into the linearly algebraic solver system as The linear equations can be generally written as where

20. Outlines • Motivation • Statement of problem • Method of fundamental solutions • Desingularized meshless method • Formulation with Cauchy problem • Regularization techniques • Numerical example • Conclusions

21. (TSVD)Truncated singular value decomposition In the singular value decomposition (SVD), the [A] matrix is decomposed into Where and are column orthonormal matrices, T denotes the matrix transposition, and is a diagonal matrix with nonnegative diagonal elements in nonincreasing order, which are the singular values of . condition number condition number ill-condition is the minimum singular value where is the maximum singular value and

22. truncated number = 1 truncated number = 2 then condition number truncated number

23. Tikhonov techniques Minimize subject to The proposed problem is equivalent to subject to Minimize The Euler-Lagrange equation can be obtained as Where λis the regularization parameter (Lagrange parameter).

24. Linear regularization method The minimization principle in vector notation, where in which

25. Outlines • Motivation • Statement of problem • Method of fundamental solutions • Desingularized meshless method • Formulation for Cauchy problem • Regularization techniques • Numerical example • Conclusions

26. Domain Numerical examples

27. The random error

28. The boundary potential without regularization techniques

29. The boundary potential with different values of λ (or i) Linear regulariztion method Tikhonov technique TSVD

30. L2 norm by different regularization techniques Linear regulariztion method Tikhonov technique TSVD

31. The boundary potential with the optimal value of λ (or i) Linear regulariztion method Tikhonov technique TSVD

32. L2 norm by different regularization techniques

33. The boundary potential with the optimal value of λ (or i)

34. Outlines • Motivation • Statement of problem • Method of fundamental solutions • Desingularized meshless method • Formulation for Cauchy problem • Regularization techniques • Numerical examples • Conclusions

35. Conclusions • Only selection of boundary nodes on the real boundary are required. • Singularity ofkernels is desingularized. • The present results were well compared with exact solutions. • Linear regularization method agreed the analytical solution better than others in this example.

36. The end Thanks for your attentions. Your comment is much appreciated.