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

An inverse problem of determination of the coefficient $a(x)$ in the equation uₜₜ = Δ u + a(x)u,x ∈ R³ ,t ∈ (0,T) is considered with initial conditions u(x,0) = 0,uₜ (x,0) = δ (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_λ (w) dependent on a large parameter λ is introduced. The main result of this paper is Theorem 1.1. By this theorem, the functional J_λ is uniformly strictly convex on any ball B with the center at the origin with a proper choice of the parameterλ = λ (B). Therefore, by this theorem, a finite-dimensional perturbation of a true solution of the above-mentioned nonlinear Cauchy problem can be found by convex minimization techniques.