Volume 16, Issue 9

The wave equation in the 4-dimension space-time is considered. The concept of wave sources disposed at infinity is introduced. The reasonably wide class of such sources and corresponding wave fields is described.

The inverse problem for a semilinear functional-differential wave equation is studied. The theorems of existence and uniqueness of solution to the inverse problem are proved.

We recover a non-negative time-dependent function, vanishing only at t = 0, in a linear multi-dimensional parabolic equation with a non-integrable degeneration concentrated at t = 0. First we prove a global in time existence and uniqueness result for the direct problem in a general Hilbert space, via Fourier representation of the solution to the direct problem. Then we show a similar result, but local in time only, for the identification problem. Finally, we apply such a result to our specific linear parabolic equation related to a smooth bounded domain in ℝ d , d = 1, 2, 3.

The goal of this paper is to propose and test an iterative coarse-to-fine reconstruction procedure for a certain class of linear inverse problems. This procedure is based on preconditioned iterative regularization through the conjugate gradient (CG) method, through Tikhonov preconditioning, as well as through wavelet low-pass filtering. A quadratic minimization problem associated with a linear inverse problem, can be very problematic if the quadratic form is not diagonal or nearly (block) diagonal. In the present reconstruction strategy, a nearly block diagonal representation of a quadratic form is obtained due to wavelet filtering and preconditioning. In the numerical experiments, the proposed procedure is successfully applied to limited-angle computerized tomography (limited-angle CT). The results of these experiments show that a combined use of wavelet filters and preconditioning can be effective within the present problem class.

We consider an inverse problem of determining a coefficient matrix and an initial value for a first order hyperbolic system. Assuming that the boundary values over a time interval are known, we characterize coefficient matrices and initial values, and prove the uniqueness of some components of the matrix function. The proof is based on a transformation formula and the spectral properties of the corresponding nonsymmetric ordinary differential operator.