Skip to main content
main-content

Über dieses Buch

Some problems of mathematical physics and analysis can be formulated as the problem of solving the equation f € F, (1) Au = f, where A: DA C U + F is an operator with a non-empty domain of definition D , in a metric space U, with range in a metric space F. The metrics A on U and F will be denoted by P and P ' respectively. Relative u F to the twin spaces U and F, J. Hadamard P-06] gave the following defini­ tion of correctness: the problem (1) is said to be well-posed (correct, properly posed) if the following conditions are satisfied: (1) The range of the value Q of the operator A coincides with A F ("sol vabi li ty" condition); (2) The equality AU = AU for any u ,u € DA implies the I 2 l 2 equality u = u ("uniqueness" condition); l 2 (3) The inverse operator A-I is continuous on F ("stability" condition). Any reasonable mathematical formulation of a physical problem requires that conditions (1)-(3) be satisfied. That is why Hadamard postulated that any "ill-posed" (improperly posed) problem, that is to say, one which does not satisfy conditions (1)-(3), is non-physical. Hadamard also gave the now classical example of an ill-posed problem, namely, the Cauchy problem for the Laplace equation.

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Regularization Method

Abstract
1. Let H, F, and G be Hilbert spaces, and let \( \text{A: H} \to \text{F , L: H} \to \text{G} \) be linear operators, with domains \( \text{D}_{\text{A}} \) and \( \text{D}_{\text{L}} \), respectively. We assume that the set \( \text{D}_{\text{A}} \cap \text{D}_{\text{L}} = :\text{D}_{\text{AL}} \ne \emptyset \) and that a non-empty set \( \text{D} \subseteq \text{D}_{\text{AL}} \) is given a priori.
V. A. Morozov

Chapter 2. Criteria for Selection of Regularization Parameter

Without Abstract
V. A. Morozov

Chapter 3. Regular Methods for Solving Linear and Nonlinear Ill-Posed Problems

Without Abstract
V. A. Morozov

Chapter 4. The Problem of Computation and the General Theory of Splines

Without Abstract
V. A. Morozov

Chapter 5. Regular Methods for Special Cases of the Basic Problem.Algorithms for Choosing the Regularization Parameter

Without Abstract
V. A. Morozov

Backmatter

Weitere Informationen