Preface

Mathematical models of the most physical phenomena are governed by initial and boundary value problems for partial differential equations (PDEs). Inverse problems governed by these equations arise naturally in almost all branches of science and engineering. The main objective of this textbook is to introduce students and researchers to inverse problems arising in PDEs. This book presents a systematic exposition of the main ideas and methods in treating inverse problems for PDEs arising in basic mathematical models, though we make no claim to cover all of the topics. More detailed additional information related to each chapter can be found in the following books/lecture notes/monographs of Aster, Borchers, and Thurber [2], Bal [6], Baunmeister [8], Beilina and Klibanov [9], Belishev and Blagovestchenskii [12], Chadan and Sabatier [17], Colton and Kress [19], Engl, Hanke, and Neubauer [23], Groetsch [31], Háo [36], Hofmann [43], Isakov [45, 46], Itou and Jin [48], Kabanikhin [50], Kaipio and E. Somersalo [51], Kirsch [54], Lavrentiev [58], Lavrentiev, Romanov, and Shishatski [59], Lavrentiev, Romanov, and Vasiliev [60], Lebedev, Vorovich, and Gladwell [61], Louis [63], Morozov [68], Nakamura and Potthast [72], Natterer [74], Ramm [84], Romanov [85, 86], Romanov and Kabanikhin [87], Schuster, Kaltenbacher, Hofmann, and Kazimierski [90], Tarantola [92], Tikhonov and Arsenin [97], Tikhonov, Concharsky, Stepanov, and Yagola [98], Vogel [102]. In Introduction, we discuss the nature of ill-posedness in differential and integral equations based on well-known mathematical models. Further, we pursue an in-depth analysis of a reason of ill-posedness of an inverse problem governed by integral operator. We tried to answer the question “why this problem is ill-posed?”, by arriving to the physical meaning of the mathematical model, on one hand, and then explaining this in terms of compact operators, on the other hand. Main notions and tools, including best approximation, Moore–Penrose (generalized) inverse, singular value decomposition, regularization strategy, Tikhonov regularization for linear inverse problems, and Morozov’s discrepancy principle, are given in Chap. 1. In Chap. 2, we tried to illustrate an implementation of all these notions and tools to inverse source problems with final overdetermination for evolution equations and to the backward problem for the parabolic equation, including some numerical