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

2010/01/03 by A. G. Ramm, Ramm, A. G.
Mathematics · #47A52 #65F22 #65J20 #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1001.0366

openalex publication_date 2010/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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_δ, δ\ are given, ||f-f_δ||≤ δ, and a stable solution u_\d:=R_\d f_\d is defined by the relation lim_{\d→ 0}||R_\d f_\d-y||=0, where y solves the equation Au=f, i.e., Ay=f. In this definition y and f are unknown. Any f∈ B(f_\d,\d) can be the exact data, where B(f_\d,\d):=\f: ||f-f_δ||≤ δ\.The new notion of the stable solution excludes the unknown y and f from the definition of the solution.

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