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Author: V.A. Morozov Publisher: Springer Science & Business Media ISBN: 1461252806 Category : Mathematics Languages : en Pages : 275
Book Description
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.
Author: V.A. Morozov Publisher: Springer Science & Business Media ISBN: 1461252806 Category : Mathematics Languages : en Pages : 275
Book Description
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.
Author: A.N. Tikhonov Publisher: Springer Science & Business Media ISBN: 940158480X Category : Mathematics Languages : en Pages : 257
Book Description
Many problems in science, technology and engineering are posed in the form of operator equations of the first kind, with the operator and RHS approximately known. But such problems often turn out to be ill-posed, having no solution, or a non-unique solution, and/or an unstable solution. Non-existence and non-uniqueness can usually be overcome by settling for `generalised' solutions, leading to the need to develop regularising algorithms. The theory of ill-posed problems has advanced greatly since A. N. Tikhonov laid its foundations, the Russian original of this book (1990) rapidly becoming a classical monograph on the topic. The present edition has been completely updated to consider linear ill-posed problems with or without a priori constraints (non-negativity, monotonicity, convexity, etc.). Besides the theoretical material, the book also contains a FORTRAN program library. Audience: Postgraduate students of physics, mathematics, chemistry, economics, engineering. Engineers and scientists interested in data processing and the theory of ill-posed problems.
Author: Anatoly B. Bakushinsky Publisher: Walter de Gruyter ISBN: 3110250659 Category : Mathematics Languages : en Pages : 153
Book Description
Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces.
Author: David Colton Publisher: Springer Science & Business Media ISBN: 3709162963 Category : Mathematics Languages : en Pages : 279
Book Description
Inverse problems are concerned with determining causes for observed or desired effects. Problems of this type appear in many application fields both in science and in engineering. The mathematical modelling of inverse problems usually leads to ill-posed problems, i.e., problems where solutions need not exist, need not be unique or may depend discontinuously on the data. For this reason, numerical methods for solving inverse problems are especially difficult, special methods have to be developed which are known under the term "regularization methods". This volume contains twelve survey papers about solution methods for inverse and ill-posed problems and about their application to specific types of inverse problems, e.g., in scattering theory, in tomography and medical applications, in geophysics and in image processing. The papers have been written by leading experts in the field and provide an up-to-date account of solution methods for inverse problems.
Author: A. A. Samarskii Publisher: Walter de Gruyter ISBN: 3110205793 Category : Mathematics Languages : en Pages : 453
Book Description
The main classes of inverse problems for equations of mathematical physics and their numerical solution methods are considered in this book which is intended for graduate students and experts in applied mathematics, computational mathematics, and mathematical modelling.
Author: Vitalii P. Tanana Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110577216 Category : Mathematics Languages : en Pages : 138
Book Description
The book covers fundamentals of the theory of optimal methods for solving ill-posed problems, as well as ways to obtain accurate and accurate-by-order error estimates for these methods. The methods described in the current book are used to solve a number of inverse problems in mathematical physics. Contents Modulus of continuity of the inverse operator and methods for solving ill-posed problems Lavrent’ev methods for constructing approximate solutions of linear operator equations of the first kind Tikhonov regularization method Projection-regularization method Inverse heat exchange problems
Author: C.R. Cowley Publisher: Springer Science & Business Media ISBN: 9400947143 Category : Science Languages : en Pages : 476
Book Description
This volume contains papers presented at IAU Colloquium No. 90. at the Crimean Astrophysical Observatory in May of 1985. A few additional contributions are included from authors who for various reasons were unable to attend the meeting. Four years have passed since the last major international conference on chemically peculiar stars of the upper main sequence was held in Liege. Belgium in 1981. Previous conferences were held in 1975 (Vienna. Austria) and in 1965 (Greenbelt. Maryland. USA). As the proceedings of this Colloquium show. the recent availability of ultraviolet spectra of large numbers of normal and chemically peculiar A and B stars is having a major impact on the way we study these objects. and has led to many new. exciting and unanticipated results. Simultaneously. the more traditional study of optical spectra has been advanced through the increasing use of very high spectral resolution with high signal-to-noise detectors. The chemically peculiar (CP) stars on the upper main sequence belong in the standard framework within which we understand stellar evolution and the history of matter. Recent work has made it clear that the unusual chemistry and magnetic structure of these objects is of relevance across the broad domain of stellar astronomy. from the upper main sequence to horizontal branch stars and white dwarfs. Metal poor (J>. Boo) as well as metal rich (Ap. Am) stars are an integral part of the picture.
Author: V.A. Morozov
Author: V.A. Morozov
Author: Vladimir Alekseevich Morozov
Author: A.N. Tikhonov
Author: Anatoly B. Bakushinsky
Author: David Colton
Author: Michiel Hazewinkel
Author: A. A. Samarskii
Author: Andrei N. Tikhonov
Author: Vitalii P. Tanana
Author: C.R. Cowley