Regularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以正規化無網格法求解柯西問題 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 Motivation Numerical Methods Mesh Methods Finite Difference Method Meshless Methods Finite Element Method Boundary Element Method (MFS)(RMM)

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) : 1.Lake of the determination of the domain, its boundary, or an unknown inner boundary. 2.Lake of inference of the governing equation. 3.Lake of identification of boundary conditions and/or initial conditions. 4.Lake of determination of the material properties involved. 5.Lake of determination of the forces or inputs acting in the domain. Cauchy problem

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8 Outlines Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for multiple holes Regularization techniques Numerical example Conclusions

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

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 Formulation for Cauchy problem Regularization techniques Numerical example Conclusions

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

13 In a similar way,

14

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 N Collocation points M 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 Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation with Cauchy problem Regularization techniques Numerical example Conclusions Outlines

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 where is the maximum singular value and is the minimum singular value ill-condition condition number

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

23 Tikhonov techniques (I) (II) Minimize subject to The proposed problem is equivalent to Minimize subject to The Euler-Lagrange equation can be obtained as Where λ is 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 Numerical examples Domain

27 The random error

28 The boundary potential without regularization techniques

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

30 L 2 norm by different regularization techniques TSVD Tikhonov technique Linear regulariztion method

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

32 L 2 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 of kernels 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.