Methods for Solving Incorrectly Posed ProblemsSome 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. |
Contents
INTRODUCTION | 1 |
7 | 24 |
Section 7 | 41 |
Section 9 | 49 |
CHAPTER 3 | 65 |
The Computation of the Estimation | 76 |
Examples of Regular Methods | 88 |
The Principle of Residual Optimality | 98 |
The Regularization Method for Nonlinear | 108 |
CHAPTER 4 | 123 |
Properties of Smoothing Families | 132 |
Section 19 | 141 |
Section 21 | 159 |
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A. N. Tikhonov accuracy algorithm approximate solutions approximation condition approximation method assertion assume Au-Au Au-f Banach space basic problem Cauchy-Schwarz inequality choosing completion condition computation conditions of Theorem convex Corollary criterion D₁ defined denote element û Euler Euler equation exists following theorem formula Furthermore given H₁ Hilbert space holds ill-posed ill-posed problems implies inequality interpolating K-regular L₂ Lemma Let H lim sup linear operator lower semi-continuity Lu-g matrix minimizing Newton's method norm obtain obviously optimal priori Proof proved pseudosolutions quadratic form rate of convergence regularization method regularization parameter regularized solutions relation lim Remark satisfied Section sequence solution û splines sufficiently small tion u₁ unique solution v₁ values vector weakly compact yields α α α α α