Inverse Problems. Mathematical and analytical techniques with applications to engineering
The book begins with a list of inverse problems ranging from the classical spectral one to those related to scattering theory, tomography, acoustics, electrodynamics, image processing, heat diffusion, and so on. Most of them are discussed throughout the volume. The first part of the book is devoted to the theoretical treatment of inverse problems via variational methods for linear problems -- reading from the celebrated Tikhonov regularization procedure -, to quasisolutions and the Backus and Gilbert method. Then emphasis is laid on possibly noisy problems related to nonlinear -- possibly unbounded -- operators in a Banach space framework and to the discrepancy principle in a Hilbert space setting. A novelty in this treatment consists in the so-called dynamical approach of associating with the nonlinear equation \[ F(u):=B(u)-f=0 \tag{1} \] in a Hilbert space \(H\), a suitable evolution Cauchy problem \[ u'(t)=\Phi(t,u(t)),\qquad u(0)=u_0, \] admitting a unique solution \(u:[0,+\infty)\to H\) for which the limit \(u(+\infty)\) exists and solves equation (1), whenever \(F\) satisfies suitable assumptions. Of course, \(u(t)\) can be chosen, for large \(t\), as an approximation of the searched for solution \(u\) to (1). Such a method, which can be applied to both well posed and ill-posed problems, is endowed with several applications showing how \(\Phi\) can be chosen starting from \(F\). From Chapter 3 on the author begins the treatment of inverse problems related to elliptic equations in one or several dimensions. The starting point is the analysis of the celebrated one-dimensional inverse scattering and inverse spectral problems going back to Gelfand-Levitan and Marchenko. This chapter contains many original contributions by the author such as the so-called C-property. A particular attention is paid to the more difficult inverse scattering problem with fixed energy phase shifts related to the recovery of spherically symmetric potentials. Taking advantage of these one-dimensional results the author can identify an unknown coefficient in one-dimensional hyperbolic or parabolic equations related to a finite space interval, obtaining a uniqueness result. Moreover, the reader will find in this volume a detailed proof of Kreĭn's method along with a proof of the fact that the reconstructed potential actually generates the scattering data it is reconstructed from. It has to be stressed that the author's attention is addressed mainly to inverse problems for scattering theory, involving the identification of obstacles or the reconstruction of potentials. In both cases it means, in general, to show uniqueness of the parameter to be identified, using systematically the C-property. But also the continuous dependence of the parameters on (exact or noisy) data is investigated. In the context of inverse problems for scattering theory particular care is devoted to a detailed analysis of Born's approximation and Born's inversion. The volume also contains the analysis of inverse problems in potential theory and the famous Pompeiu problem as well as many applications of interest to scientists in applied sciences. To conclude, the reviewer emphasizes the author's effort of simplifying both the results in the mathematical literature and his own ones as well as the effort of unifying the theory of the inverse problems illustrated. Moreover, the proofs of the various propositions are almost always inserted in the book so that we can say that the volume is friendly to readers and can help young mathematicians to enter in a not yet so popular -- but highly appealing -- field of mathematics. Finally, the proofs are well written and the misprints are actually a few.
recovering potentials in Schrödinger equations
recovering unknown coefficients in hyperbolic and parabolic equations
scientific article; zbMATH DE number 2126325