Thomas R. Lucas

A fast and accurate imaging algorithm in optical/diffusion tomography

An n-dimensional (n = 2,3) inverse problem for the parabolic/diffusion equation , , , is considered. The problem consists of determining the function a(x) inside of a bounded domain given the values of the solution u(x,t) for a single source location on a set of detectors , where is the boundary of . A novel numerical method is derived and tested. Numerical tests are conducted for n = 2 and for ranges of parameters which are realistic for applications to early breast cancer diagnosis and the search for mines in murky shallow water using ultrafast laser pulses. The main innovation of this method lies in a new approach for a novel linearized problem (LP). Such a LP is derived and reduced to a well-posed boundary-value problem for a coupled system of elliptic partial differential equations. A principal advantage of this technique is in its speed and accuracy, since it leads to the factorization of well conditioned, sparse matrices with non-zero entries clustered in a narrow band near the diagonal. The authors call this approach the elliptic systems method (ESM). The ESM can be extended to other imaging modalities.

Some Collocation Methods for Nonlinear Boundary Value Problems

Several projection operators into spaces of spline functions are used to develop new approximation schemes of collocation type for the solutions to nonlinear boundary value problems. Theoretical convergence rates are given along with examples of classes of problems to which the methods are applicable. Numerical results verifying the convergence rates of the methods are presented as well as a discussion of their implementation on an electronic computer.

Imaging the diffusion coefficient in a parabolic inverse problem in optical tomography

The elliptic systems method (ESM), previously developed by the second and third authors, is extended to the reconstruction of the diffusion coefficient of an inverse problem for the parabolic equation in the n-dimensional case (n = 2,3). This inverse problem has applications to optical imaging of small abnormalities hidden in a random media, such as biological tissues, foggy atmospheres, murky water, etc. Results of numerical experiments are presented in the two-dimensional case, for realistic ranges of the parameters.

Numerical Solution of a Subsurface Imaging Inverse Problem

In this paper a new solution method for an inverse problem for the two-dimensional (2D) Helmholtz equation is developed. The underlying application area which motivated this work is the imaging of land mines using ground penetrating radar, formulated as an inverse problem for the Helmholtz equation. A second generation version of the elliptic systems method is developed to numerically solve this problem. Uniqueness and convergence theorems and a variety of numerical results are presented.

Two numerical methods for an inverse problem for the 2-D Helmholtz equation

Two solution methods for the inverse problem for the 2-D Helmholtz equation are developed, tested, and compared. The proposed approaches are based on a marching finite-difference scheme which requires the solution of an overdetermined system at each step. The preconditioned conjugate gradient method is used for rapid solutions of these systems and an efficient preconditioner has been developed for this class of problems. Underlying target applications include the imaging of land mines, unexploded ordinance, and pollutant plumes in environmental cleanup sites, each formulated as an inverse problem for a 2-D Helmholtz equation. The images represent the electromagnetic properties of the respective underground regions. Extensive numerical results are presented.

Asymptotic Expansions for Interpolating Periodic Splines

A detailed study is made of the structure of smooth periodic interpolating splines of odd degree over a uniform mesh. For splines of degree

A high order projection method for nonlinear two point boundary value problems

2r - 1

New imaging algorithm in diffusion tomography

, all terms of the asymptotic expansion of the error at the mesh points of the first

A generalization ofL-splines

2r - 2

Tomographic Images of Land Mines by the Elliptic Systems Method Using GPR: Efficient Solution of the Forward Problem

derivatives are given. This result is used to derive a number of other high order spline approximations.