William Rundell

An introduction to inverse scattering and inverse spectral problems

Inverse problems attempt to obtain information about structures by non-destructive measurements. This introduction to inverse problems covers three central areas: inverse problems in electromagnetic scattering theory; inverse spectral theory; and inverse problems in quantum scattering theory.

Reconstruction techniques for classical inverse Sturm-Liouville problems

This paper gives constructive algorithms for the classical inverse Sturm-Liouville problem. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. Two methods of solving this latter problem are then provided, and numerical examples are presented.

Nonlinear integral equations and the iterative solution for an inverse boundary value problem

Determining the shape of a perfectly conducting inclusion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modelled as an inverse boundary value problem for harmonic functions. We present a novel solution method for such inverse boundary value problems via a pair of nonlinear and ill-posed integral equations for the unknown boundary that can be solved by linearization, i.e., by regularized Newton iterations. We present a mathematical foundation of the method and illustrate its feasibility by numerical examples.

The determination of a discontinuity in a conductivity from a single boundary measurement

We consider the determination of the interior domain where D is characterized by a different conductivity from the surrounding medium. This amounts to solving the inverse problem of recovering the piecewise constant conductivity in from boundary data consisting of Cauchy data on the boundary of the exterior domain . We will compute the derivative of the map from the domain D to this data and use this to obtain both qualitative and quantitative measures of the solution of the inverse problem.

Inverse scattering for shape and impedance

We consider the inverse problem of determining both the shape and the impedance of a two-dimensional scatterer from a knowledge of the far-field pattern of the scattering of time-harmonic acoustic or electromagnetic waves by solving the ill posed nonlinear equation for the operator that maps the boundary and the boundary impedance of the scatterer onto the far-field pattern. We establish results on the injectivity of the linearized map and obtain satisfactory reconstructions by a regularized Newton iteration.

ITERATIVE METHODS FOR THE RECONSTRUCTION OF AN INVERSE POTENTIAL PROBLEM

This paper considers an inverse potential problem which seeks to recover the shape of an obstacle separating two different densities by measurements of the potential. A representation for the domain derivative of the corresponding operator is established and this allows the investigation of several iterative methods for the solution of this ill-posed problem.

An inverse problem for a one-dimensional time-fractional diffusion problem

We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented.

Surveys on solution methods for inverse problems

Convergence Rates Results for Iterative Methods for Solving Nonlinear III-Posed Problems.- Iterative Regularization Techniques in Image Reconstruction.- A Survey of Regularization Methods for First-Kind Volterra Equations.- Layer Stripping.- The Linear Sampling Method in Inverse Scattering Theory.- Carleman Estimates and Inverse Problems in the Last Two Decades.- Local Tomographic Methods in Sonar.- Efficient Methods in Hyperthermia Treatment Planning.- Solving Inverse Problems with Spectral Data.- Low Frequency Electromagnetic Fields in High Contrast Media.- Inverse Scattering in Anisotropic Media.- Inverse Problems as Statistics.

A quasi-Newton method in inverse obstacle scattering

A quasi-Newton method is presented for the approximate solution of the inverse problem of determining the shape of a sound-soft or perfectly conducting infinitely long cylindrical obstacle from a knowledge of the far-field pattern for the scattering of time-harmonic plane waves. The numerical implementation of the method is described and it is illustrated by numerical examples that the method yields satisfactory reconstructions.

Variational formulation of problems involving fractional order differential operators

In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order α ∈ (1, 2) on the unit interval (0, 1). These fractional derivatives lead to nonsymmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space Hα/2 0 (0, 1) but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. The analytical theory is then applied to the Sturm-Liouville problem involving a fractional derivative in the leading term. Finally, extensive numerical results are presented to illustrate the error estimates for the source problem and eigenvalue problem.