Open Access

ARTICLE

Novel Iterative Algorithms Based on Regularization Total Least Squares for Solving the Numerical Solution of Discrete Fredholm Integral Equation

Zichun Yang^1,2,3, Lei Zhang^1,4, Yueyun Cao¹

1 Naval University of Engineering, China 430033.
2 Huazhong University of Science and Technology, Wuhan, China 430074.
3 Center for Aerospace Research & Education, University of California, Irvine Visiting professor.
4 Corresponding Author.

Computer Modeling in Engineering & Sciences 2014, 97(2), 101-130. https://doi.org/10.3970/cmes.2014.097.101

Abstract

Discretization of inverse problems often leads to systems of linear equations with a highly ill-conditioned coefficient matrix. To find meaningful solutions of such systems, one kind of prevailing and representative approaches is the so-called regularized total least squares (TLS) method when both the system matrix and the observation term are contaminated by some noises. We will survey two such regularization methods in the TLS setting. One is the iterative truncated TLS (TTLS) method which can solve a convergent sequence of projected linear systems generated by Lanczos bidiagonalization. The other one is to convert the Tikhonov regularization TLS problem to an unconstrained optimization problem with the properties of a convex function. The optimization problem will be solved with the state-of-the-art conjugate gradient (CG) method, and moreover, the adaptive strategy for selecting regularization parameter is also established. Finally, both the new methods are applied to tackle several Fredholm integral equations of the first kind which are known to be typical ill-posed problems. The results of numerical examples demonstrate that the robustness and effectiveness of the two novel algorithms make a significant improvement in the solution of ill-posed linear problems, i.e., yield more accurate regularized solution than other typical methods.

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