1. A Localized Method of Particular Solutions for Solving Near Singular Problems C.S. Chen, Guangming Yao, D.L. Young Department of Mathematics University of Southern Mississippi U.S.A.

2. 2016/1/142 OutlineOutline Radial Basis Functions The global approaches of the method of particular solutions Numerical examples of global method Local approach of the method of particular solutions Numerical examples of local method Near Singular Problems

3. 2016/1/143 Radial Basis Functions Linear: Cubic: Multiquadrics: Polyharmonic Spines: Gaussian:

4. 2016/1/144 Assume that To approximate f bywe usually require fitting the given data set of pairwise distinct centres with the imposed conditions The linear system is well-posed if the interpolation matrix is non-singular Surface Reconstruction Scheme

5. 2016/1/145 The Splitting Method Consider the following equation Whereis a bounded open nonempty domain with sufficiently regular boundary Letwheresatisfying but does not necessary satisfy the boundary condition in (11). (10) (11) v satisfies (12) (13) (14)

6. 2016/1/146 Assume that and that we can obtain an analytical solutionto Then To approximate f bywe usually require fitting the given data set of pairwise distinct centres with the imposed conditions Particular Solutions

7. 2016/1/147 The linear system is well-posed if the interpolation matrix is non-singular where and (*)

8. 2016/1/148 For in 2D

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11. 2016/1/1411 Where G(r) is the fundamental solution of L Boundary Method is required.

12. 2016/1/1412 The Method of Particular Solutions (MPS) where

13. 2016/1/1413 Impose boundary conditions

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16. 2016/1/1416 Numerical Results

17. 2016/1/1417 Example I Analytical solution: Computational Domain:

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19. 2016/1/1419 c : shape parameter of MQ

20. 2016/1/1420 Consider the Poisson’s equation Given a large data set where

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32. 2016/1/14 Non-Dirichlet boundary condition

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39. 2016/1/1439 The absolute errors of LMAPS with L=1, n=5, S n =100, c=8.9

40. 2016/1/14 L=1, S n = 100, N=225.

41. 2016/1/1441 Local MPS verse Global MPS

42. 2016/1/1442 n: number of neighbor points

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45. 2016/1/1445 LMPS verse LMQ

46. 2016/1/14 Near Singular Problem I C.S. Chen, G. Kuhn, J. Li, G. Mishuris, Radial basis functions for solving near singular Poisson’s problems, Communication in Numerical Methods in Engineering, 2003, 19, 333-347.

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48. 48 Profile of exact solution

49. 2016/1/1449 CS-RBF 400 quasi-random points

50. 2016/1/1450 Test 1Test 2

51. 2016/1/1451 Normalized Shape parameter where

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53. 2016/1/1453 Sobel quasi-random nodes Von-Del Corput quasi-random nodes Random nodes

54. 2016/1/1454 Speed up N=10,000 CPU = 0.5/3.42 s N=40,000 CPU = 3.31/14.06 s N=62,500 CPU = 7.01/25.28 s

55. 2016/1/1455 RMSE error verse shape parameter for a=1.6 and various mesh sizes LMPS

56. 2016/1/1456 RMSE error verse shape parameter for h=1/200, and various value of a.

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58. 2016/1/1458 Near Singular Problem II Exact solution

59. 2016/1/14 Profile of f(x,y) f(1,1,) = -15,861, f(0,0)=237

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61. 2016/1/1461 Near Singular Problem III

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63. 2016/1/1463 Adaptive Method First step Second step

64. 2016/1/1464 3 rd step 4 th step

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