Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method

The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Compu...

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Bibliographic Details
Published in:Mathematics (Basel) 2020-08, Vol.8 (8), p.1297
Main Authors: Yang, Judy P., Lam, Hon Fung Samuel
Format: Article
Language:English
Subjects:
Accuracy
Boundary conditions
Cauchy problems
Collocation methods
gradient approximation
high-order gradient approximation
inverse problem
Inverse problems
Kernels
Mathematical analysis
Methods
multiply connected domain
Operators (mathematics)
Partial differential equations
reproducing kernel approximation
Shape functions
strong form collocation
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Summary:The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically increase, especially when dealing with strong-from equations where high-order derivative operators are required as compared to weak-form approaches for obtaining results with promising levels of accuracy. Under the framework of gradient approximation, the derivatives of reproducing kernel shape functions can be constructed synchronically, thereby alleviating the complexity in computation. In view of this, the present work first introduces the weighted high-order gradient reproducing kernel collocation method in the inverse analysis. The convergence of the method is examined through the weights imposed on the boundary conditions. Then, several configurations of multiply connected domains are provided to numerically investigate the stability and efficiency of the method. To reach the desired accuracy in detecting the outer boundary for two special cases, special treatments including allocation of points and use of ghost points are adopted as the solution strategy. From four benchmark examples, the efficacy of the method in detecting the unknown boundary is demonstrated.
ISSN:2227-7390
2227-7390
DOI:10.3390/math8081297