Regularized meshless method for solving the Cauchy problem
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
Outlines n n n n Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical example Conclusions 2
Outlines n n n n Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical example Conclusions 3
Motivation Numerical Methods Mesh Methods Finite Difference Method Finite Element Method Meshless Methods Boundary Element Method (MFS) (RMM 4
Outlines n n n n Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical example Conclusions 5
Statement of problem n 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. Cauchy problem 4. Lake of determination of the material properties involved. 5. Lake of determination of the forces or inputs acting in the domain. 6
Cauchy problem 7
Outlines n n n n Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for multiple holes Regularization techniques Numerical example Conclusions 8
Method of fundamental solutions (MFS) d n Method of fundamental solutions (MFS) : Distributed-type Dirichlet problem Single-layer Potential approach Neumann problem d = off-set distance Source point Collocation point — Physical boundary -- Off-set boundary Dirichlet problem Double-layer potential approach Neumann problem 9
n n 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. 10
Outlines n n n n Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical example Conclusions 11
Desingularized meshless method (DMM) n Desingularized meshless method (DMM) Dirichlet problem Double-layer potential approach Neumann problem Source point Collocation point — Physical boundary I = Inward normal vector O = Outward normal vector where 12
In a similar way, 13
14
Outlines n n n n Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation with Cauchy problem Regularization techniques Numerical example Conclusions 15
Formulation with Cauchy problem M Collocation points N Collocation points 16
Derivation of diagonal coefficients of influence matrices. Where 17
where 18
Rearrange the influence matrices together into the linearly algebraic solver system as The linear equations can be generally written as where 19
Outlines n n n n Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation with Cauchy problem Regularization techniques Numerical example Conclusions 20
(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 ill-condition is the maximum singular value and condition number is the minimum singular value 21
truncated number = 1 truncated number = 2 truncated number then condition number 22
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). 23
Linear regularization method The minimization principle in vector notation, where in which 24
Outlines n n n n Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical example Conclusions 25
Numerical examples Domain 26
The random error 27
The boundary potential without regularization techniques 28
The boundary potential with different values of λ (or i) TSVD Tikhonov technique Linear regulariztion method 29
L 2 norm by different regularization techniques TSVD Tikhonov technique Linear regulariztion method 30
The boundary potential with the optimal value of λ (or i) TSVD Tikhonov technique Linear regulariztion method 31
L 2 norm by different regularization techniques 32
The boundary potential with the optimal value of λ (or i) 33
Outlines n n n n Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical examples Conclusions 34
Conclusions n n 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. 35
The end Thanks for your attentions. Your comment is much appreciated. 36
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