Numerical methods for inverse problems (Q2803797)

This book, consisting of 8 chapters and being divided into 3 parts, is devoted to the numerical methods for solving ill-posed inverse problems. The efficient solution of inverse problems has received much attention in the last few decades. A wide variety of techniques have been developed.NEWLINENEWLINEThe present book presents a number of useful numerical methods for solving inverse problems, especially the least squares method, the Tikhonov regularization, the singular value decomposition, and the adjoint state method. Throughout, the methods are illustrated by numerical examples.NEWLINENEWLINEThe book is divided into three independent parts: Part I (consisting of Chapters 1--2) presents an overview of basic concepts of ill-posed inverse problems and gives a few examples. In Chapter 1, the basic concepts are introduced, and then a few exemplary inverse problems, e.g., in heat transfer, hydrology and seismic exploration, medical imaging, are described. Part II (consisting of Chapters 3--5) is devoted to the numerical solution of linear inverse problems. Chapter 3 focuses on integral operators and their discretization by quadrature-collocation and the Galerkin method. Then in Chapter 4, the singular value decomposition for matrix and compact operators in Hilbert spaces is given. Finally, in Chapter 5, various useful regularization techniques are discussed, e.g., the Tikhonov regularization, the truncated singular value decomposition, the iterative regularization method and the crucial issue of parameter choice rules. Part III (consisting of Chapters 6--8) is devoted to nonlinear inverse problems. In Chapter 6, the general setting of nonlinear inverse problems is described, and the important issue of computing the gradient by the adjoint method is emphasized. In Chapter 7, several examples with the elliptic partial differential equation and the heat equations are given. Finally, in Chapter 8, several more recent topics are briefly touched, including bounded variation regularization, the Bayesian inversion, and a few practical issues. The book concludes with three appendices and about 120 bibliography entries. At the end of each chapter, a number of exercises ise provided. The book is oriented towards an applied audience, with basic knowledge of differential equations and linear algebra. It may be used as a textbook for a course in applied inverse problems.