Thomas S. Angell

An inverse transmission problem for the Helmholtz equation

The problem considered is that of determining the shape of a three-dimensional scattering object, illuminated by an acoustic field, from a knowledge of scattered far-field data. The far-field data are the asymptotic form of the solution of an exterior transmission problem for the Helmholtz equation. The problem is reformulated as an optimisation problem, specifically, finding that surface, in a suitably restricted class, which minimises an appropriate functional of the far field generated by the surface through the solution of the direct problem. Through the use of complete families of solutions, the problem is further reduced to finding a surface which minimises error in satisfying the transmission conditions.

The conductive boundary condition for Maxwell's equations

First, the conductive boundary value problem is derived for the quasi-stationary Maxwell equations that arise in the study of magnetotellurics. Then the boundary integral equation method is used to prove the existence and uniqueness of solutions of the problem. The final section is devoted to a study of the set of far field patterns for scattering problems with plane wave incidence.

The three dimensional inverse scattering problem for acoustic waves

Abstract We consider the inverse scattering problem for an acoustically soft obstacle in R 3 . By assuming a priori that the unknown scattering obstacle is starlike and has its boundary lying in a compact family of Holder continuously differentiable surfaces, it is shown that an optimal solution can be constructed which depends continuously on the measured far field data. Remarks are made on the numerical approximation of the optimal solution.

A constructive method for identification of an impenetrable scatterer

Abstract This paper presents a constructive algorithm for solving the problem of reconstructing the shape of a scattering object from measurements of the scattered far field when the unknown object is illuminated by a known incident wave. The problem is recast as an optimization problem with a penalty term. The cost functional consists of a term which assesses the difference between the measured far field and the far field of the solution of the field equation to a particular surface and the penalty term which measures the error in satisfying the boundary conditions on that surface. A complete family of radiating solutions of the Helmholtz equation is employed to construct approximate solutions by solving finite dimensional minimization problems. Existence of solutions of the original problem as well as the finite dimensional approximations is established. Moreover convergence of the approximate solutions to a solution of the original problem is proven. Some preliminary numerical results are presented to indicate the viability of the method.

The resistive and conductive problems for the Exterior Helmholtz Equation

The existence, uniqueness, and continuous dependence of solutions of the Helmholtz equation, which are subject to either resistive or conductive conditions at the boundary of a closed smooth scatterer, are considered. A boundary integral equation of the second kind for each problem whose unique solution is the trace on the boundary of the unique solution of that problem is devised. Continuous dependence results are proven using the integral equations.

On the optimal control of systems governed by nonlinear volterra equations

Existence theorems are stated and proved for optimal control problems monitored by a nonlinear Volterra-type integral equation. Growth conditions are taken into consideration which ensure the needed compactness on the set of trajectories and entail the existence of the usual optimal solutions for the given problem. Weak or approximate solutions are taken into consideration in a subsequent paper.

A distributed source method for inverse acoustic scattering

We develop a new algorithm for solving inverse acoustic scattering problems. In particular we show that this algorithm can reproduce scattering shapes efficiently, using synthetic data, from only one incident wave in the acoustically hard case and using at most two incident waves for the acoustically soft problem. In order to test the inversion algorithm we generate synthetic data using a technique which combines the distributed source method and the fundamental solution of the Helmholtz equation in order to calculate the scattered field for each of these problems. Numerical results for three-dimensional axially symmetric shapes are compared with those obtained previously by other authors.

A note on approximation of optimal solutions of free problems of the calculus of variations

M. Lavrentiev, and later B. Manià, have given examples of free problems of the Calculus of Variations in which it is impossible to find a piecewiseC1 function, satisfying the same boundary conditions as the absolutely continuous optimal solution, and which simultaneously approximates both the optimal cost and, uniformly, the optimal trajectory. We investigate this question of approximation for free problems inEn.Dispensing with differentiability assumptions on the integrand of the cost functional, we present sufficient conditions under which this approximation problem may be solved thus extending earlier work of Manià, Tonelli, and others for free problems in the plane.

Existence of Optimal Control without Convexity and a Bang-Bang Theorem for Linear Volterra Equations

AbstractWe consider a Mayer problem of optimal control monitored by an integral equation of Volterra type: