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

✍ Scribed by A.G. Ramm

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 251 KB
• Volume: 234
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2010.04.032

✦ Synopsis

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 illposed 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, ff δ ≤ δ, and a stable solution u δ := R δ f δ is defined by the relation lim δ→0 R δ f δy = 0, where y solves the equation Au = f , i.e., Ay = f . In this definition y and f are unknown. Any f ∈ B(f δ , δ) can be the exact data, where

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.