Numerical Methods for Inverse Problems

The book "Numerical Methods for Inverse Problems" consists of contemporaneouss articles featuring not only several well-known inverse problems used in the inference of many physical and engineering systems such as that of the partial differential equations but also the statistical and imaging inverse problems. It includes a variety of numerical methods for solving inverse problems such as that of the Tikhonov regularization; finite differences; and orthogonal decomposition; as well as those based on Bayesian inference, artificial neural networks, and quantum annealing. Cover Title Page Copyright DECLARATION ABOUT THE EDITOR TABLE OF CONTENTS List of Contributors List of Abbreviations Preface Chapter 1 A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation Abstract Introduction Mathematical Problem And The Ill-Posedness Analysis Mollification Method And Regularization Solution Parameter Selection And Error Estimates Numerical Examples Conclusions Author Contributions Funding Acknowledgments Conflicts Of Interest References Chapter 2 The Numerical Solution of Forward and Inverse Robin Problems for Laplace’s Equation Abstract Introduction Fast Solution Of The Forward Problem The Robin Inverse Problem And Preconditioner Numerical Examples Acknowledgements Funding Contributions References Chapter 3 Moving Least Squares Method for a One-dimensional Parabolic Inverse Problem Abstract Introduction An Outline Of The Moving Least Squares (Mls) The Inverse Problem And Its Numerical Solution Numerical Experiments And Discussions Conclusion Conflict Of Interests Acknowledgments References Chapter 4 The Inverse Problem of the Heat Equation with Periodic Boundary And Integral Overdetermination Conditions Abstract Introduction Existence And Uniqueness Of The Solution Of The Inverse Problem Continuous Dependence Of (P,U) Upon The Data Numerical Method Numerical Example And Discussions Conclusions Acknowledgements References Chapter 5 An Inverse Problem for a Two-Dimensional Time- Fractional Sideways Heat Equation Abstract Introduction Description Of The 2D Time-Fractional Inverse Diffusion Problem A Modified Method And Convergence Estimates Numerical Aspect Conclusion Data Availability Conflicts Of Interest Authors’ Contributions Acknowledgments References Chapter 6 Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE Abstract Introduction The Forward Problem And The Inverse Problem The Inversion Algorithm And Numerical Inversions Conclusions Conflicts Of Interest Acknowledgments References Chapter 7 Solutions of Direct and Inverse Even-Order Sturm- Liouville Problems Using Magnus Expansion Abstract Introduction Materials And Methods Results Of Direct Sturm–Liouville Problems Results Of Inverse Sturm–Liouville Problems Discussion And Conclusions Author Contributions Funding Acknowledgments Conflicts Of Interest Abbreviations Appendix A References Chapter 8 Numerical Solution of Direct and Inverse Problems for Time-Dependent Volterra Integro-Differential Equation Using Finite Integration Method with Shifted Chebyshev Polynomials Abstract Introduction Preliminaries Numerical Algorithms For Direct And Inverse Problems Of Tvide Numerical Experiments Conclusions And Discussion Author Contributions Acknowledgments Conflicts Of Interest Abbreviations References Chapter 9 Direct and Inverse Problem for Geometric Perturbation of the Laplace Operator in a Strip Abstract Introduction The Direct Problem The Inverse Problem Conclusion Appendices Acknowledgements Contributions References Chapter 10 Solving Large-Scale Inverse Magnetostatic Problems Using the Adjoint Method Abstract Introduction Adjoint Method Numerical Experiments Conclusion Acknowledgements Contributions References Chapter 11 The Problem of the Inverse Lyapunov Exponent and its Applications Abstract Introduction Formulation Of The Problem Results Example Applications Conclusions References Chapter 12 Nonlinear Damping Identification in Nonlinear Dynamic System Based on Stochastic Inverse Approach Abstract Introduction Mathematical Description Nonlinear Damping Identification Stochastic Inverse Formalism Hierarchical Bayesian Formulation Posterior State Exploration Numerical Experiments A Particular Application To Realistic Problem: Ship Roll Motion Conclusions References Chapter 13 Model Order Reduction for Bayesian Approach to Inverse Problems Abstract Background Methods Results And Discussion Conclusion References Chapter 14 Statistical Approaches to the Inverse Problem Introduction Statistics And Inverse Problems Basic Facts About The Meg Inverse Problem Imaging Methods Parametric Methods Highly Automated Dipole Estimation Hades Conclusion Acknowledgements References Index Back Cover