Inverse and ill-posed problems. Theory and applications.
This monograph provides a well-structured and comprehensive survey of various types of inverse problems. In the sequel we summarize some of the key features of this book. The first chapter is devoted to a classification of inverse problems (inverse boundary value problems, inverse source problems, coefficient problems, reverse time problems), and classes of well-posed problems and their ill-posed counterparts are presented. Simple examples are also considered to present first ideas of ill-posedness. Chapter 2 provides a brief introduction to the basic theory of ill-posed problems and their regularization. Topics include but are not restricted to the modulus of continuity, quasi-solutions, Lavrentiev and Tikhonov regularization, general regularization schemes, parameter choice strategies for Tikhonov regularization to solve perturbed equations (including operator perturbations), gradient methods, the conjugate gradient method, the pseudoinverse, and the singular value decomposition. The section on gradient methods contains a comprehensive convergence theory. In Chapter 3, the fundamental concepts for solving ill-conditioned finite-dimensional linear equations are presented. This includes pseudo-solutions, Lavrentiev and Tikhonov regularization, residual principles, the singular value decomposition and Godunov's method. In Chapter 4, integral equations are considered. Lavrentiev regularization for integral equations of the first kind is considered first, both for symmetric Fredholm equations with Hilbert-Schmidt kernels and Volterra equations with smooth kernels. Then for a class of nonlinear Volterra integral equations of the second kind, existence, uniqueness and conditional stability of solutions is considered. Such equations arise, e.g., after some reformulation of a coefficient problem for the wave equation. The chapter concludes with the regularization of related nonlinear Volterra equations of the first kind. Here, Lavrentiev's method and a difference scheme are considered. Chapter 5 deals with the reconstruction of a function from its mean values on a system of manifolds. The inversion of the Radon transform is first considered for radially symmetric functions and then for the general case. Subsequent topics are moment problems and the reconstruction of a function from its spherical means. The final section of this chapter deals with the inverse problem of reconstructing the speed of seismic wave propagation from observations on the surface of the Earth. In Chapter 6, the Sturm-Liouville operator \( l_qy =-y^{\prime\prime} + q(x) y \) is studied. The first section deals with spectral properties of the operator \( l_q \) on finite intervals \( [ 0, \pi ] \) and for separated boundary conditions \( y^\prime(0) - h y(0) = 0, \, y^\prime(\pi) + H y(\pi) = 0 \), with \( h, H \in \mathbb{R} \). For example, the asymptotics of the eigenvalues of \( l_q \) are considered, and some properties of the corresponding eigenfunctions are given. In the next two sections, the inverse Sturm-Liouville problem on a finite interval is considered, i.e., the determination of the potential \( q(x) \) and the coefficients \( h, H \) in the boundary conditions from the spectral data of the operator \( l_q \). Here, uniqueness, existence and reconstruction by the Gelfand-Levitan method is considered. In the next section, stationary inverse scattering problems of the form \( -\Delta u + q(x) u = k^2 u, \, x \in \mathbb{R}^3 \), are considered, with an emphasis on radially symmetric potentials \( q(x) \) which in fact result in inverse Sturm-Liouville problems on the half axis \( 0 \leq x < \infty \). The final section of this chapter is concerned with inverse scattering problems which can be written as initial-boundary value problems for \( u_{tt} = u_{xx} -q(x)u, \, x \in \mathbb{R},\, t \in \mathbb{R} \), with an unknown potential \( q(x) \). Here, a Fourier transform analysis again leads to a spectral Sturm-Liouville problem on a half axis. The focus of Chapter 7 are linear problems for hyperbolic problems. In the first two sections of this chapter, the author considers the problem of reconstructing a function \( q(z,y_1,y_2) \) from the mean values on spheres of arbitrary finite radius \( r \) which in fact can be reformulated as a Cauchy problem for the wave equation. A short section on inverse thermoacoustic problems follows. Then the problem of determining the speed \(c(z,y)\) of wave propagation in the half-space \( (z,y) \in \mathbb{R}_+ \times \mathbb{R}^n \) is considered for the case \( c^2(z,y) = c_0^2(z) + c_1(z,y) \), where \( c_1 \ll c_0^2 \), and \( c_1 \) has compact support. Two related initial-boundary value problems for wave equations are derived from the considered decomposition of \( c^2(z,y) \). For the two corresponding direct problems, results on the existence of solutions are stated, respectively. For the related two inverse problems, existence of solutions, conditional stability and regularization by truncated Fourier series expansion are the main topics. Chapter 8 is devoted to linear problems for parabolic equations. A relation between hyperbolic and parabolic equations is considered first, and then one-dimensional ill-posed initial value problems for heat conduction with reverse time are considered, i.e., \( \tfrac{\partial u}{\partial t} = -\tfrac{\partial^2 u}{\partial x^2},\, 0 < x < l,\, 0 < t < T \) with Dirichlet boundary conditions and the initial condition \( u(x,0) = f(x),\, 0 < x < l \). The approximation properties of some regularization schemes are analyzed in this case, e.g., Lattès--Lions regularization \( \tfrac{\partial u}{\partial t} = -\tfrac{\partial^2 u}{\partial x^2} -\alpha \tfrac{\partial^4 u}{\partial x^4} \), truncated Fourier series expansion employed after separation of variables, and the steepest descent method. For some related one-dimensional initial value problems for heat conduction with reverse time (interior problems, problems of determining sources), uniqueness and existence results are presented. For a related multidimensional direct problem, stability estimates are provided. Chapter 9 deals with linear problems for elliptic equations. The considered inverse problem is of the form \( \Delta u = 0 \) for \( 0 < x, \,y< 1, \, u(0,y) = f(y), \, u_x(0,y) = 0 \) for \( 0 < y < 1 \), and \( u(x,0) = u(x,1) = 0 \) for \( 0 < x < 1 \), with a given function \( f(y) \). Existence and uniqueness of a solution as well as conditional stability are established, and well-posedness of the corresponding direct problem -- here, \( u(0,y) = f(y) \) is replaced by \( u(1,y) = q(y) \) -- and a related adjoint problem are considered. The results are then extended to elliptic problems in several variables, and the gradient method for the approximate solution of the corresponding inverse problem is considered. In Chapter 10, inverse coefficient problems for hyperbolic equations are examined. In the first section, the Cauchy problem \( u_{tt} = u_{xx} - q(x)u + F(x,t), \, x \in \mathbb{R}, \, t > 0, \, u(x,0) = \varphi(x), u_t(x,0) = \psi(x) \) for \( x \in \mathbb{R} \) is considered, where the functions \( \varphi(x),\;\psi(x) \) and \( F(x,t) \) are given, and the function \( q(x) \) is unknown. Existence and uniqueness of the solution \( q(x) \) is shown for smooth data, and then similar questions are addressed for generalized solutions \( q(x) \) in the case \( F(x,t) = 0,\, \varphi(x) = 0 \) and \( \psi(x) = \delta(x) \), the delta function. In the next section, the following one-dimensional inverse acoustic problem is considered: \( (1/c^2(z)) v_{tt} = v_{zz} -(\rho^\prime(z)/\rho(z)) v_z, \;z > 0, \;t > 0, \) with the initial and boundary conditions \( v|_{t<0} \equiv 0, \, v_z|_{z=0} = \delta(t) \) and \( v(+0,t) = g(t) \). An operator version \( Aq=f \) is introduced for this problem, and an analysis of the mapping properties of this operator is presented. For the approximate solution of \( Aq=f\), the method of successive approximation, Landweber iteration, the steepest descent method and Newton's method are considered. The next section introduces an inverse problem related to Maxwell's equations, and then Neumann to Dirichlet maps are introduced for recovering the coefficient \( q(x,y) \) in the equation \( u_{tt} = u_{xx} + \Delta u-q(x,y) u, \, y \in \mathbb{R}^n, \, x \in \mathbb{R}_+,\, 0 \leq t \leq T \). The last two sections of this chapter are devoted to special acoustic equations in two and three dimensions. In Chapter 11, inverse coefficient problems for parabolic and elliptic equations are introduced. First, three inverse problems associated with initial-boundary value problems for the parabolic equation \( u_{t} - (k(x)u_{x})_x + q(x) u = 0, \, 0 < t < T, \, 0 < x < 1 \), are considered, where \( u(x,t) \) and the coefficient \( q(x) \) are unknown, respectively. For each case, uniqueness of the functions \( q(x), u(x,t) \) is shown by reducing the inverse coefficient problem to a Sturm-Liouville problem, respectively. The last two sections of this chapter are devoted to inverse coefficient problems for two special multi-dimensional elliptic equations. The book concludes with two appendices containing some background material from functional analysis as well as supplementary exercises and references. This monograph provides a well-written survey on a wide range of inverse problems, and it is in fact easy to read. It addresses all relevant topics like existence and uniqueness of solutions both of the respective direct and inverse problem, conditional stability, stable approximation of the solution, and regularization in the case of perturbed data. Each chapter contains a collection of exercises. This excellent text will be of interest to experts in the field as well as to graduate students.
Fredholm integral equation of the first kind
Volterra integral equation of the first kind
nonlinear Volterra integral equation of the second kind
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