Volume 14, Issue 9

An inverse problem of wave propagation into a weakly laterally inhomogeneous medium occupying a half-space is considered in the acoustic approximation. The half-space consists of an upper layer and a semi-infinite bottom separated with an interface. An assumption of a weak lateral inhomogeneity means that the velocity of wave propagation and the shape of the interface depend weakly on the horizontal coordinates, x = ( x 1 , x 2 ), in comparison with the strong dependence on the vertical coordinate, z , giving rise to a small parameter ε ≪ 1. Expanding the velocity in power series with respect to ε , we obtain a recurrent system of 1D inverse problems. We provide algorithms to solve these problems for the zero and first-order approximations. In the zero-order approximation, the corresponding 1D inverse problem is reduced to a system of non-linear Volterra-type integral equations. In the first-order approximation, the corresponding 1D inverse problem is reduced to a system of coupled linear Volterra integral equations. These equations are used for the numerical reconstruction of the velocity in both layers and the interface up to O ( ε 2 ).

In this paper, we consider the inverse problem of estimating simultaneously the five parameters of a jump diffusion process based on return observations of a price trajectory. We show that there occur some ill-posedness phenomena in the parameter estimation problem, because the forward operator fails to be injective and small perturbations in the data may lead to large changes in the solution. We illustrate the instability effect by a numerical case study. To obtain stable approximate solutions of the estimation problem, we use a multi-parameter regularization approach, where a least-squares fitting of empirical densities is superposed by a quadratic penalty term of fitted semi-invariants with weights. A little number of required weights is controlled by a discrepancy principle. For the realization of this control, we propose and justify a fixed point iteration, where an exponent can be chosen arbitrarily positive. A numerical case study completing the paper shows that the approach provides satisfactory results and that the amount of computation can be reduced by an appropriate choice of the free exponent.

In this paper we consider an inverse boundary value problem of determining three dimensional unknown inclusions in an elliptic equation in a bounded domain Ω ⊂ from finite boundary measurements on ∂ Ω. We will show that poly-hedral inclusions in Ω can be uniquely determined up to their convex edges from a single boundary measurement on ∂ Ω.

For the solution to u ( x, t ) – Δ u ( x, t ) + q ( x ) u ( x, t ) = δ ( x 1 ) δ ′´( t ) and u | t <0 = 0, we consider an inverse problem of determining q ( x ), x ∈ Ω from data ƒ = u | S T and g = ( ∂u/∂v )| S T . Here Ω ⊂ {( x 1 , . . . , x n ) ∈ | x 1 > 0}, n ≥ 2, is a bounded domain, S T = {( x, t ) | x ∈ ∂ Ω, x 1 < t < T + x 1 } and T > 0. For suitable T > 0, we prove an L 2 (Ω)-size estimation of q : || q || L 2 (Ω) ≤ C {||ƒ|| H 1 ( S T ) + || g || L 2 ( S T )}, provided that q satisfies a priori uniform boundedness conditions. We use an inequality of Carleman type in our proof.

We consider the problem of analytic continuation of the solution of the multidimensional Lame system in infinite domains through known values of the solution and the corresponding strain tensor on a part of the boundary, i.e., the Cauchy problem.

Necessary and sufficient conditions of convergence of the L -regularization methods for solution of nonlinear operator equations in Hilbert spaces are found.