Ultrasound tomography is modelled by the inverse problem of a 2D Helmholtz equation at fixed frequency with plane-wave irradiation. It is assumed that the field is measured outside the support of the unknown potential f for finitely many incident waves. Starting out from an initial guess f0 for f we propagate the measured field through the object f0 to yield a computed held whose difference to the measurements is in turn backpropagated. The backpropagated field is used to update f0. The propagation as well as the backpropagation are done by a finite difference marching scheme. The whole process is carried out in a single-step fashion, i.e. the updating is done immediately after backpropagating a single wave. It is very similar to the well known ART method in X-ray tomography, with the projection and backprojection step replaced by propagation and backpropagation.

https://iopscience.iop.org/article/10.1088/0266-5611/11/6/007/pdf

A propagation-backpropagation method for ultrasound tomography

As previously mentioned, for problems in mathematical physics Hadamard [95] postulated three requirements: a solution should exist, the solution should be unique, and the solution should depend continuously on the data. The third postulate is motivated by the fact that in all applications the data will be measured quantities. Therefore, one wants to make sure that small errors in the data will cause only small errors in the solution. A problem satisfying all three requirements is called well-posed. Otherwise, it is called ill-posed. As shown in the previous chapter, the direct obstacle scattering problem is well-posed.

Ill-Posed Problems

We examine the scattering of time-harmonic acoustic waves in inhomogeneous media The problem is to recover a spatially varying refractive index in a three-dimensional medium from far-field measurements of scattered waves corresponding to incoming waves from all directions This problem is exponentially ill-posed and of a large scale since a solution of the direct problem corresponds to solving a partial differential equation in 3 for each incident wave We construct a preconditioner for the conjugate gradient method applied to the normal equation to solve the regularized linearized operator equation in each Newton step This reduces the number of operator evaluations dramatically compared to standard regularized Newton methods Our method can also be applied effectively to other exponentially ill-posed problems, for example, in impedance tomography, heat conduction and obstacle scattering To solve the direct problems, we use an improved fast solver for the Lippmann–Schwinger equation suggested by Vainikko

https://iopscience.iop.org/article/10.1088/0266-5611/17/6/314/pdf

On the numerical solution of a three-dimensional inverse medium scattering problem

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.

https://iopscience.iop.org/article/10.1088/0266-5611/10/5/011/pdf

A quasi-Newton method in inverse obstacle scattering

Because leukemia changes the electrical properties of bone marrow, it might be possible to detect the disease via electromagnetic imaging, numerical experiments using inverse problem solving and synthetic data suggest that this approach may work.

S. Gutman

Papers

Regularized Quasi-Newton method for inverse scattering problems