| Description |
1 online resource (xv, 475 pages) : illustrations |
| Series |
Inverse and ill-posed problems series ; 55 |
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Inverse and ill-posed problems series ; v. 55.
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| Contents |
Preface; Denotations; 1 Basic concepts and examples; 1.1 On the definition of inverse and ill-posed problems; 1.2 Examples of inverse and ill-posed problems; 2 Ill-posed problems; 2.1 Well-posed and ill-posed problems; 2.2 On stability in different spaces; 2.3 Quasi-solution. The Ivanov theorems; 2.4 The Lavrentiev method; 2.5 The Tikhonov regularization method; 2.6 Gradient methods; 2.7 An estimate of the convergence rate with respect to the objective functional; 2.8 Conditional stability estimate and strong convergence of gradient methods applied to ill-posed problems |
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2.9 The pseudoinverse and the singular value decomposition of an operator3 Ill-posed problems of linear algebra; 3.1 Generalization of the concept of a solution. Pseudo-solutions; 3.2 Regularization method; 3.3 Criteria for choosing the regularization parameter; 3.4 Iterative regularization algorithms; 3.5 Singular value decomposition; 3.6 The singular value decomposition algorithm and the Godunov method; 3.7 The square root method; 3.8 Exercises; 4 Integral equations; 4.1 Fredholm integral equations of the first kind; 4.2 Regularization of linear Volterra integral equations of the first kind |
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4.3 Volterra operator equations with boundedly Lipschitz-continuous kernel4.4 Local well-posedness and uniqueness on the whole; 4.5 Well-posedness in a neighborhood of the exact solution; 4.6 Regularization of nonlinear operator equations of the first kind; 5 Integral geometry; 5.1 The Radon problem; 5.2 Reconstructing a function from its spherical means; 5.3 Determining a function of a single variable from the values of its integrals. The problem of moments; 5.4 Inverse kinematic problem of seismology; 6 Inverse spectral and scattering problems |
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6.1 Direct Sturm-Liouville problem on a finite interval6.2 Inverse Sturm-Liouville problems on a finite interval; 6.3 The Gelfand-Levitan method on a finite interval; 6.4 Inverse scattering problems; 6.5 Inverse scattering problems in the time domain; 7 Linear problems for hyperbolic equations; 7.1 Reconstruction of a function from its spherical means; 7.2 The Cauchy problem for a hyperbolic equation with data on a time-like surface; 7.3 The inverse thermoacoustic problem; 7.4 Linearized multidimensional inverse problem for the wave equation; 8 Linear problems for parabolic equations |
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8.1 On the formulation of inverse problems for parabolic equations and their relationship with the corresponding inverse problems for hyperbolic equations8.2 Inverse problem of heat conduction with reverse time (retrospective inverse problem); 8.3 Inverse boundary-value problems and extension problems; 8.4 Interior problems and problems of determining sources; 9 Linear problems for elliptic equations; 9.1 The uniqueness theorem and a conditional stability estimate on a plane |
| Summary |
The text demonstrates the methods for proving the existence (if et all) and finding of inverse and ill-posed problems solutions in linear algebra, integral and operator equations, integral geometry, spectral inverse problems, and inverse scattering problems. It is given comprehensive background material for linear ill-posed problems and for coefficient inverse problems for hyperbolic, parabolic, and elliptic equations. A lot of examples for inverse problems from physics, geophysics, biology, medicine, and other areas of application of mathematics are included |
| Bibliography |
Includes bibliographical references and index |
| Notes |
9.2 Formulation of the initial boundary value problem for the Laplace equation in the form of an inverse problem. Reduction to an operator equation |
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Print version record |
| Subject |
Inverse problems (Differential equations)
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Boundary value problems -- Improperly posed problems.
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MATHEMATICS -- Differential Equations -- General.
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Boundary value problems -- Improperly posed problems
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Inverse problems (Differential equations)
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| Form |
Electronic book
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| ISBN |
9783110224016 |
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3110224011 |
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3110224003 |
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9783110224009 |
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