Numerical methods for inverse problems / Michel Kern.

Format:

Book

Author/Creator:

Kern, Michel, author.

Series:

Mathematics and statistics series (ISTE)

Mathematics and Statistics

Language:

English

Subjects (All):

Inverse problems (Differential equations)--Numerical solutions.

Inverse problems (Differential equations).

Physical Description:

1 online resource (234 p.)

Edition:

1st ed.

Place of Publication:

London, England ; Hoboken, New Jersey : iSTE : Wiley, 2016.

Summary:

This book studies methods to concretely address inverse problems. An inverse problem arises when the causes that produced a given effect must be determined or when one seeks to indirectly estimate the parameters of a physical system. The author uses practical examples to illustrate inverse problems in physical sciences. He presents the techniques and specific methods chosen to solve inverse problems in a general domain of application, choosing to focus on a small number of methods that can be used in most applications. This book is aimed at readers with a mathematical and scientific computing background. Despite this, it is a book with a practical perspective. The methods described are applicable, have been applied, and are often illustrated by numerical examples.

Contents:

Cover; Dedication; Title Page; Copyright; Contents; Preface; Book layout; Acknowledgments; PART 1: Introduction and Examples; 1: Overview of Inverse Problems; 1.1. Direct and inverse problems; 1.2. Well-posed and ill-posed problems; 2: Examples of Inverse Problems; 2.1. Inverse problems in heat transfer; 2.2. Inverse problems in hydrogeology; 2.3. Inverse problems in seismic exploration; 2.4. Medical imaging; 2.5. Other examples; PART 2: Linear Inverse Problems; 3: Integral Operators and Integral Equations; 3.1. Definition and first properties; 3.2. Discretization of integral equations

3.2.1. Discretization by quadrature-collocation3.2.2. Discretization by the Galerkin method; 3.3. Exercises; 4: Linear Least Squares Problems - Singular Value Decomposition; 4.1. Mathematical properties of least squares problems; 4.1.1. Finite dimensional case; 4.2. Singular value decomposition for matrices; 4.3. Singular value expansion for compact operators; 4.4. Applications of the SVD to least squares problems; 4.4.1. The matrix case; 4.4.2. The operator case; 4.5. Exercises; 5: Regularization of Linear Inverse Problems; 5.1. Tikhonov's method; 5.1.1. Presentation; 5.1.2. Convergence

5.1.3. The L-curve5.2. Applications of the SVE; 5.2.1. Singular value expansion and Tikhonov's method; 5.2.2. Regularization by truncated SVE; 5.3. Choice of the regularization parameter; 5.3.1. Morozov's discrepancy principle; 5.3.2. The L-curve; 5.3.3. Numerical methods; 5.4. Iterative methods; 5.5. Exercises; PART 3: Nonlinear Inverse Problems; 6: Nonlinear Inverse Problems - Generalities; 6.1. The three fundamental spaces; 6.2. Least squares formulation; 6.2.1. Difficulties of inverse problems; 6.2.2. Optimization, parametrization, discretization

6.3. Methods for computing the gradient - the adjoint state method6.3.1. The finite difference method; 6.3.2. Sensitivity functions; 6.3.3. The adjoint state method; 6.3.4. Computation of the adjoint state by the Lagrangian; 6.3.5. The inner product test; 6.4. Parametrization and general organization; 6.5. Exercises; 7: Some Parameter Estimation Examples; 7.1. Elliptic equation in one dimension; 7.1.1. Computation of the gradient; 7.2. Stationary diffusion: elliptic equation in two dimensions; 7.2.1. Computation of the gradient: application of the general method

7.2.2. Computation of the gradient by the Lagrangian7.2.3. The inner product test; 7.2.4. Multiscale parametrization; 7.2.5. Example; 7.3. Ordinary differential equations; 7.3.1. An application example; 7.4. Transient diffusion: heat equation; 7.5. Exercises; 8: Further Information; 8.1. Regularization in other norms; 8.1.1. Sobolev semi-norms; 8.1.2. Bounded variation regularization norm; 8.2. Statistical approach: Bayesian inversion; 8.2.1. Least squares and statistics; 8.2.2. Bayesian inversion; 8.2.2.1. A priori and a posteriori probabilities; 8.2.2.2. A few estimation techniques

8.2.2.3. References

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781119136965

1119136962

9781119136958

1119136954

OCLC:

945976688