Michael V. Klibanov

Carleman estimates for coefficient inverse problems and numerical applications

In this monograph, the main subject of the authors considerations is coefficient inverse problems. Arising in many areas of natural sciences and technology, such problems consist of determining the variable coefficients of a certain differential operator defined in a domain from boundary measurements of a solution or its functionals. Although the authors pay strong attention to the rigorous justification of known results, they place the primary emphasis on new concepts and developments.

Inverse problems and Carleman estimates

The author describes a method for proving global uniqueness theorems for one broad class of multidimensional coefficient inverse problems. This method is based on Carleman estimates, and it does not depend essentially on the order or type of differential operator.

A computational quasi-reversiblility method for Cauchy problems for Laplace's equation

This work concerns the use of the method of quasi-reversibility for solving Cauchy problems for Laplace’s equation. The paper begins with a derivation of an error estimate for this method using Carleman’s estimates. Next, a discretization of the method using finite differences is considered. Carleman-type estimates for the discrete scheme are derived and used to establish convergence of the numerical method. Results of numerical experimentations with the method are presented.

Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems

Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems is the first book in which two new concepts of numerical solutions of multidimensional Coefficient Inverse Problems (CIPs) for a hyperbolic Partial Differential Equation (PDE) are presented: Approximate Global Convergence and the Adaptive Finite Element Method (adaptivity for brevity). Two central questions for CIPs are addressed: How to obtain a good approximation for the exact solution without any knowledge of a small neighborhood of this solution, and how to refine it given the approximation. The book also combines analytical convergence results with recipes for various numerical implementations of developed algorithms. The developed technique is applied to two types of blind experimental data, which are collected both in a laboratory and in the field. The result for the blind backscattering experimental data collected in the field addresses a real-world problem of imaging of shallow explosives.

Stability estimates for ill-posed cauchy problems involving hyperbolic equations and inequalities

We consider ill-posed hyperbolic equations and inequalities in Ω × (0,T) with mixed boundary conditions is a partition of . We use Carleman-type estimates to derive Lipschitz stability estimates for such problems.

The phase retrieval problem

In the phase retrieval problem one seeks to recover an unknown function g(t) from the amplitude mod g(k) mod of its Fourier transform. Since phase and amplitude are, in general, independent of each other, it is necessary to make use of other kinds of information which implicitly or explicitly constrain the admissible solutions g(t). In this paper we survey a variety of results explaining circumstances under which g(t) may be uniquely recovered from mod g(k) mod and supplementary information. A number of explicit formulae for the phase are discussed. We pay particular attention to the phase retrieval problem as it arises in certain inverse-scattering applications.

Phaseless inverse scattering and the phase problem in optics

Two related problems are considered: (i) the inverse scattering problem for a potential V(x) supported on the half‐line {x≥0}, when the given data is ‖R−(k)‖, the amplitude of the reflection coefficient and (ii) determination of a function g(t) supported on the half‐line {t≥0} when the given data is ‖g(k)‖, the amplitude of the Fourier transform of g. Under certain conditions on V or g, uniqueness theorems are proved and computational methods are developed. A numerical example of recovery of V(x) from ‖R−(k)‖ is given.

A Globally Convergent Numerical Method for a Coefficient Inverse Problem

A new globally convergent numerical method is developed for a multidimensional coefficient inverse problem for a hyperbolic PDE with applications in acoustics and electromagnetics. On each iterative step the Dirichlet boundary value problem for a second-order elliptic equation is solved. The global convergence is rigorously established, and numerical experiments are presented.

Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data

A special version of the Newton-Kantorovich method is applied to the three-dimensional potential inverse scattering problem in the time domain. The related hyperbolic Cauchy problem with data on the side of the time cylinder is solved by the quasi-reversibility method, and a new stability theorem is established by Carleman-type estimates. The geometrical convergence of the Newton-Kantorovich method, used here, is also established.

Lipschitz stability of an inverse problem for an acoustic equation

An inverse problem of the determination of the coefficient p(x) in the equation is considered. The main difficulty here as compared with the previous results is that the function p(x) is involved together with its derivatives. A Lipschitz stability estimate is obtained using the method of Carleman estimates.