Uniform Strict Convexity of a Cost Functional for Three-Dimensional Inverse Scattering Problem

Abstract

An inverse problem of determination of the coefficient $a(x)$ in the equation $u_{tt} = \Delta u + a(x)u,x \in \mathbb{R}^3 ,t \in (0,T)$ is considered with initial conditions $u(x,0) = 0,u_t (x,0) = \delta (x)$, and some additional data that can be treated as backscattering information. The goal is to develop a finite-dimensional technique that would be a basis for future computations. We reduce our inverse scattering problem to an equivalent Cauchy problem for a nonlinear hyperbolic integrodifferential equation with the data on the lateral side of a time cylinder. It is assumed that the solution $v(x,t)$ of this equation has the form $v(x,t) = p(x,t) + w(x,t)$, where function $p(x,t)$ is given and function $w(x,t)$ is unknown and has a finite number of nonzero Fourier coefficients. In particular, function $p(x,t)$ can be considered as a first guess. A special cost-functional $J_\lambda (w)$ dependent on a large parameter $\lambda $ is introduced. The main result of this paper is Theorem 1.1. By this the...