Error analysis of collocation method based on reproducing kernel approximation

2009 ◽  
Vol 27 (3) ◽  
pp. 554-580 ◽  
Author(s):  
Hsin-Yun Hu ◽  
Jiun-Shyan Chen ◽  
Wei Hu
Keyword(s):  
Error Analysis ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Kernel Approximation

Solving Inverse Laplace Equation with Singularity by Weighted Reproducing Kernel Collocation Method

2017 ◽  
Vol 09 (05) ◽  
pp. 1750065 ◽  
Author(s):  
Judy P. Yang ◽  
Pai-Chen Guan ◽  
Chia-Ming Fan
Keyword(s):  
Inverse Problems ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Cauchy Problems ◽  
Overdetermined System ◽  
Numerical Examples ◽  
Kernel Approximation ◽  
Laplace Equations ◽  
Laplace Type ◽  
Least Squares Minimization

This work introduces the weighted collocation method with reproducing kernel approximation to solve the inverse Laplace equations. As the inverse problems in consideration are equipped with over-specified boundary conditions, the resulting equations yield an overdetermined system. Following our previous work, the weighted collocation method using a least-squares minimization has shown to solve the inverse Cauchy problems efficiently without using techniques such as iteration and regularization. In this work, we further consider solving the inverse problems of Laplace type and introduce the Shepard functions to deal with singularity. Numerical examples are provided to demonstrate the validity of the method.

Weighted Reproducing Kernel Collocation Method and Error Analysis for Inverse Cauchy Problems

2016 ◽  
Vol 08 (03) ◽  
pp. 1650030 ◽  
Author(s):  
Judy P. Yang ◽  
Pai-Chen Guan ◽  
Chia-Ming Fan
Keyword(s):  
Error Analysis ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Convergence Rates ◽  
Discrete Systems ◽  
Accurate Solution ◽  
Cauchy Problems ◽  
Linear Partial Differential Equations ◽  
Direct Collocation ◽  
Optimal Convergence Rates

In this work, the weighted reproducing kernel collocation method (weighted RKCM) is introduced to solve the inverse Cauchy problems governed by both homogeneous and inhomogeneous second-order linear partial differential equations. As the inverse Cauchy problem is known for the incomplete boundary conditions, how to numerically obtain an accurate solution to the problem is a challenging task. We first show that the weighted RKCM for solving the inverse Cauchy problems considered is formulated in the least-squares sense. Then, we provide the corresponding error analysis to show how the errors in the domain and on the boundary can be balanced with proper weights. The numerical examples demonstrate that the weighted discrete systems improve the accuracy of solutions and exhibit optimal convergence rates in comparison with those obtained by the traditional direct collocation method. It is shown that neither implementation of regularization nor implementation of iteration is needed to reach the desired accuracy. Further, the locality of reproducing kernel approximation gets rid of the ill-conditioned system.

scholarly journals Coefficient-Based Regression with Non-Identical Unbounded Sampling

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jia Cai
Keyword(s):  
Hilbert Space ◽  
Integral Operator ◽  
Error Analysis ◽  
Least Squares ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Least Squares Regression ◽  
Regression Problem ◽  
Novel Technique ◽  
Sample Error

We investigate a coefficient-based least squares regression problem with indefinite kernels from non-identical unbounded sampling processes. Here non-identical unbounded sampling means the samples are drawn independently but not identically from unbounded sampling processes. The kernel is not necessarily symmetric or positive semi-definite. This leads to additional difficulty in the error analysis. By introducing a suitable reproducing kernel Hilbert space (RKHS) and a suitable intermediate integral operator, elaborate analysis is presented by means of a novel technique for the sample error. This leads to satisfactory results.

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