On explicit expression of the solution to the regularizing by Tikhonov optimization problem in terms of the regularization parameter in the finite-dimensional case

IF 0.3 Q4 MATHEMATICS
A. Chernov
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引用次数: 0

Abstract

It is well known that using the Tikhonov regularization method for solving operator equations of the first kind one has to minimize a regularized residual functional. The minimizer is determined from so called Euler equation which in finite-dimensional case and at its discretization is written as a one-parametric (depending on the regularization parameter) system of linear algebraic equations of special form. Here, there exist various ways of choosing the regularization parameter. In particular, in the frame of principle of generalized residual, it is necessary to solve the corresponding equation of generalized residual with respect to the regularization parameter. And it implies (when solving this equation numerically), in turn, multifold solving a one-parametric system of linear algebraic equations for arbitrary value of the parameter. In this paper we obtain an explicit simple and effective formula of solution to a one-parametric system for an arbitrary value of the parameter. We give an example of computations by above-mentioned formula and also an example of numerical solution of the Fredholm integral equation of the first kind under usage of this formula which substantiates its effectiveness.
有限维正则化参数下Tikhonov优化问题解的显式表达
众所周知,用Tikhonov正则化方法求解第一类算子方程必须最小化正则残差泛函。最小值是由所谓的欧拉方程确定的,在有限维情况下,欧拉方程在其离散化时被写成特殊形式的单参数(取决于正则化参数)线性代数方程组。在这里,存在多种选择正则化参数的方法。特别地,在广义残差原理的框架下,需要求解关于正则化参数的广义残差的相应方程。它意味着(当用数值方法求解这个方程时),反过来,对于任意参数值,多重求解一个单参数线性代数方程组。本文给出了参数为任意值的单参数系统的一个显式的简单有效的解公式。文中给出了用上述公式计算的一个例子和第一类Fredholm积分方程的数值解,证明了该公式的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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