1 Adaptive error estimation of the Trefftz method for solving the Cauchy problem Presenter: C.-T. Chen Co-author: K.-H. Chen, J.-F. Lee & J.-T. Chen BEM/MRM 29, 4-6 June 2007, The New Forest, UK

2  Trefftz method for interior problems  Statement of problem  Numerical example  Conclusions  Motivation  Regularization techniques Outlines

3  Trefftz method for interior problems  Statement of problem  Numerical example  Conclusions  Motivation  Regularization techniques Outlines

4 Numerical methods Motivation Mesh methods Finite Difference methods Finite Element methods Boundary Element methods Meshless methods Trefftz methods Method of Fundamental Solution (MFS) (FDM) (FEM) (BEM)

5  Trefftz method for interior problems  Statement of problem  Numerical example  Conclusions  Motivation  Regularization techniques Outlines

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 acting in the domain. Cauchy problem

7 over-specified condition DD DD Lake of identification of boundary conditions and/or initial conditions. case 1 case 2 case 3 case 4

8  Trefftz method for interior problems  Statement of problem  Numerical example  Conclusions  Motivation  Regularization techniques Outlines

9 Field solution : where the number of complete functions the unknown coefficient the T-complete function which satisfies the Laplace equation Trefftz method

10 T-complete set functions : T-complete set Where: Field solution : The unknown coefficient :

11 Normal differential of the boundary solution where

12  Trefftz method for interior problems  Statement of problem  Numerical example  Conclusions  Motivation  Regularization techniques Outlines

13 Tikhonov technique (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).

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

15 over-specified condition D The concept of adaptive error estimation Step 1:

16 D The ill-posed problem Obtain: Step 2:By Trefftz method

17 D Obtain: Step 3: The well-posed problem

18 The optimal parameter Norm The solution is more sensitive The system is distorted The optimal

19 Flow chart of adaptive error estimation Remedied by the Tikhonov techniqueRemedied by the Linear Regularization Method Obtain the left value of the boundary Obtain left the value of the boundary Obtain the right value of the boundary Obtain the right value of the boundary obtain the optimal Lamda value End Trefftz method

20  Trefftz method for interior problems  Statement of problem  Numerical example  Conclusions  Motivation  Regularization techniques Outlines

21 over-specified condition D Numerical example ‧ R Circle case:

22 Analytical field solution :

23 Inverse Problem with artificial Contamination over-specified condition D ‧ R

24 1% random errors contaminating the input data

25 Numerical solution without regularization techniques

26 Numerical field solution without regularization techniques

27 Numerical solutions remedied by 3 different (200 nodes)

28 Numerical field solutions remedied by the Tikhonov technique with 3 different (200 nodes)

29 Numerical field solutions remedied by the linear regularization method with 3 different (200 nodes)

30 Obtain the optimal parameters by computing the Norm deriving from comparing numerical solution with analytic solution

31 The norm error of the Tikhonov technique is lower than the linear regularization method

32 The Tikhonov technique and the Linear Regularization Method Numerical solutions with optimal (200 nodes)

33 Numerical field solutions with optimal (200 nodes) The Tikhonov technique The Linear Regularization Method

34 Under no exact solution, the optimal results obtained by using the adaptive error estimation The Tikhonov techniqueThe Linear Regularization Method

35  Trefftz method for interior problems  Statement of problem  Numerical example  Conclusions  Motivation  Regularization techniques Outlines

36 1. The optimal parameters make the system insensitive to contaminating noise. 2. The present results were well compared with exact solutions. 3. The Tikhonov technique agreed the analytical solution better than another in this example. 4. Under no exact solution, the optimal results are obtained by employing the adaptive error estimation. Conclusions

37 Thanks for your attentions. Your comment is much appreciated.

38 × The Norm deriving from adaptive error estimation

39 Figure 9(a) The numerical field solution remedied by the Tikhonov technique with 3 different lambdas (200nodes)

40 Numerical solution being remedied by the Linear Regularization Method with 3 different lambdaes(200 nodes) Figure 9(b) The numerical field solution remedied by the linear regularization method with 3 different lambdas (200nodes)

41 × Figure 14 (a), 14(b) Numerical field solution being remedied by the Tikhonov technique and the linear regularization method with optimal lambda (200 nodes) The Tikhonov technique: The Linear Regularization Method:

42 Numerical solution being remedied by the Tikhonov technique Of 40 nodes and 200 nodes with optimal lambda

43 Numerical solution being remedied by the Linear Regularization Method Of 40 nodes and 200 nodes with optimal lambda