# REGULARIZED MULTIQUADRIC METHOD FOR SOLVING INVERSE BOUNDARY VALUE PROBLEMS

Keywords:
Gaussian RBF, Nonlinear PDEs, Variable coefficient PDEs

### Abstract

In this paper, we develop a regularized multiquadric method, which is also a non-iterative numerical method, for solving inverse boundary value problems governed by Laplace equation. The well-known ill-posed Cauchy problem is considered, we assume that the boundary conditions are given only on part of the physical boundary of the solution domain, we have to reconstruct the solution and its normal derivative on the rest un-accessible part of the physical boundary. During the whole solution process, we use the multiquadric and the regularization method to construct a regularized multiquadric method. Numerical experiments are given to demonstrate the effectiveness and efficiency of the proposed method.

### Downloads

Download data is not yet available.

### References

J. Carr, R. Beatson, J. Cherrie, T. Mitchell, W. Fright, B. McCallum and T. R Evans, "Reconstruction and representation of 3D objects with radial basis functions," SIGGRAPH 2001 Proceedings, 2001.

E. Kansa, "Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics ,I. Surface approximations and derivative estimates, in Advances in Partial Deferential Equations," 1990.

J. R. Cannon and Y. Lin, "Determination of parameter p(t) in Holder classes for some semi linear parabolic equations," Inverse Problems 4 595¡ 606, 1988.

J. V. Beck, B. Blackwell, and C. R Clair, "Inverse Heat Conduction. Wiley: New York," 1985.

H. W. Engl, A. K. Louis, and W (Eds). Rundell, "Inverse Problem in Geo logical Applications." SIAM: Philadelphia, 1996.

H. W. Engl, A. K. Louis, and W (Eds), "Inverse Problem in Medical Imaging and Nondestructive Testing," Springer: New York, 1996.

D. Colton, R. Kress. "Inverse Acoustic and Electromagnetic Scattering Theory," Springer: Berlin, 1998.

Y. C. Hon, T. Wei "A fundamental solution method for inverse heat conduction problem." Engineering Analysis with Boundary Elements 28:489¡495, 2004.

N. S. Mera, "The method of fundamental solutions for the backward heat conduction problem: Inverse Problem in Science and Engineering," 13:65-78, 2005.

B. Jin, Y. Zheng, "Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation," Engineering Analysis with Boundary Elements 29:925 ¡ 935, 2005.

Y. C. Hon T. Wei, L. Ling. "Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators," Engineering Analysis with Boundary Elements 31:373 ¡ 385, 2007.

X. Z. Mao, Z. Li, "Least-square-based radial basis collocation method for solving inverse problems of Laplace equation from noisy data, "International Journal for Numerical Methods in Engineering DOI: 10.1002/nme.2880, 2010.

P. A. Ramachandran, Method of fundamental solutions: singular value decomposition analysis. Communications in Numerical Methods in Engineering 789-801, 2002.

P. C .Hansen, "Regularization tools: a Mat Lab package for analysis and solution of discrete ill-posed problems," Numerical Algorithms 6:1-35, 1994.

E. Kansa, "Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics ,I. Surface approximations and derivative estimates, in Advances in Partial Deferential Equations," 1990.

J. R. Cannon and Y. Lin, "Determination of parameter p(t) in Holder classes for some semi linear parabolic equations," Inverse Problems 4 595¡ 606, 1988.

J. V. Beck, B. Blackwell, and C. R Clair, "Inverse Heat Conduction. Wiley: New York," 1985.

H. W. Engl, A. K. Louis, and W (Eds). Rundell, "Inverse Problem in Geo logical Applications." SIAM: Philadelphia, 1996.

H. W. Engl, A. K. Louis, and W (Eds), "Inverse Problem in Medical Imaging and Nondestructive Testing," Springer: New York, 1996.

D. Colton, R. Kress. "Inverse Acoustic and Electromagnetic Scattering Theory," Springer: Berlin, 1998.

Y. C. Hon, T. Wei "A fundamental solution method for inverse heat conduction problem." Engineering Analysis with Boundary Elements 28:489¡495, 2004.

N. S. Mera, "The method of fundamental solutions for the backward heat conduction problem: Inverse Problem in Science and Engineering," 13:65-78, 2005.

B. Jin, Y. Zheng, "Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation," Engineering Analysis with Boundary Elements 29:925 ¡ 935, 2005.

Y. C. Hon T. Wei, L. Ling. "Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators," Engineering Analysis with Boundary Elements 31:373 ¡ 385, 2007.

X. Z. Mao, Z. Li, "Least-square-based radial basis collocation method for solving inverse problems of Laplace equation from noisy data, "International Journal for Numerical Methods in Engineering DOI: 10.1002/nme.2880, 2010.

P. A. Ramachandran, Method of fundamental solutions: singular value decomposition analysis. Communications in Numerical Methods in Engineering 789-801, 2002.

P. C .Hansen, "Regularization tools: a Mat Lab package for analysis and solution of discrete ill-posed problems," Numerical Algorithms 6:1-35, 1994.

Published

2018-10-25

How to Cite

*International Journal of Advanced Computer Technology*,

*7*(5), 01-05. Retrieved from http://ijact.org/index.php/ijact/article/view/2

Section

Articles