约束Fractional Tikhonov正则化的模迭代方法

反问题是现在数学物理研究中的一个热点问题,而反问题求解面临的一个本质性困难是不适定性。求解不适定问题的普遍方法是:用与原不适定问题相“邻近”的适定问题的解去逼近原问题的解,这种方法称为正则化方法.如何建立有效的正则化方法是反问题领域中不适定问题研究的重要内容.当前,最为流行的正则化方法有基于变分原理的Tikhonov正则化及其改进方法,此类方法是求解不适定问题的较为有效的方法,在各类反问题的研究中被广泛采用,并得到深入研究.

MODULUS-BASED ITERATIVE METHOD FOR CONSTRAINED FRACTIONAL TIKHONOV REGULARIZATION

Tikhonov regularization is one of the most popular methods for solving linear equations or linear least squares problems with a severely ill-conditioned matrix and an error-contaminated data vector(right-hand side).Also,it is well known that the standard form of Tikhonov regularization may result in an overly smooth approximate solution,for which a fractional Tikhonov regularization method is introduced.In many applications the desired solution is known to lie in the nonnegative cone.It is then natural to require that the approximate solution determined by Tikhonov regularization also lies in this cone.In this paper,we propose an iterative method that employs modulus-based iterative method to compute approximate solution in the nonnegative cone of large-scale fractional.Tikhonov regularization problems.It is called a modulus-based iterative method for constrained fractional Tikhonov regularization.Furthermore,the present paper describes a projection modulus-based iterative method for large-scale constrained fractional Tikhonov regularization.Computed examples illustrate the performances of these methods.