On a new notion of the solution to an ill-posed problem

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:234

Article

Ramm, A. G.

关键词: III-posed problems;    Regularizer;    Stable solution of ill-posed problems;   

DOI  :  10.1016/j.cam.2010.04.032

来源: Elsevier

【 摘 要 】

A new understanding of the notion of the stable solution to ill-posed problems is proposed. The new notion is more realistic than the old one and better fits the practical computational needs. A method for constructing stable solutions in the new sense is proposed and justified. The basic point is: in the traditional definition of the stable solution to an ill-posed problem Au = f. where A is a linear or nonlinear operator in a Hilbert space H, it is assumed that the noisy data (f(delta) , delta) are given, vertical bar vertical bar f - f(delta) vertical bar vertical bar <= delta, and a stable solution u(delta) := R(delta)f(delta) is defined by the relation lim(delta -> 0) vertical bar vertical bar R(delta)f(delta) - y vertical bar vertical bar = 0, where y solves the equation Au = f, i.e., Ay = f. In this definition y and f are unknown. Any f is an element of B(f(delta), delta) can be the exact data, where B(f(delta) . delta) :-= {f : vertical bar vertical bar f - f(delta) vertical bar vertical bar <= delta}. The new notion of the stable solution excludes the unknown y and f from the definition of the solution. The solution is defined only in terms of the noisy data, noise level, and an a priori information about a compactum to which the solution belongs. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free