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

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Presentation transcript:

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.

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

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

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

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)

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

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

2016/1/148 For in 2D

2016/1/149

10

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

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

2016/1/1413 Impose boundary conditions

2016/1/1414

2016/1/1415

2016/1/1416 Numerical Results

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

2016/1/1418

2016/1/1419 c : shape parameter of MQ

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

2016/1/1421

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2016/1/1428

2016/1/1429

2016/1/1430

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2016/1/14 Non-Dirichlet boundary condition

2016/1/1433

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2016/1/1437

2016/1/1438

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

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

2016/1/1441 Local MPS verse Global MPS

2016/1/1442 n: number of neighbor points

2016/1/1443

2016/1/1444

2016/1/1445 LMPS verse LMQ

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,

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48 Profile of exact solution

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

2016/1/1450 Test 1Test 2

2016/1/1451 Normalized Shape parameter where

2016/1/1452

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

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

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

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

2016/1/1457

2016/1/1458 Near Singular Problem II Exact solution

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

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2016/1/1461 Near Singular Problem III

2016/1/1462

2016/1/1463 Adaptive Method First step Second step

2016/1/ rd step 4 th step

2016/1/1465