Volume 21, Issue 6

Abstract. We consider the inverse scattering problem at fixed and sufficiently large energy for the nonrelativistic and relativistic Newton equation in ℝ n , , with a smooth and short-range electromagnetic field ( V , B ). Proceeding from different known results we obtain, in particular, that the scattering map at a fixed and sufficiently large energy uniquely determines ( V , B ) when B is assumed to be zero in a neighborhood of infinity and V is assumed to be spherically symmetric in a neighborhood of infinity.

Abstract. Because of x-ray dose considerations or practical mechanical restrictions, it may be interesting for some applications to perform limited-angle scanning. So the limited-angle problem is of practical significance in computed tomography (CT). Furthermore, it is an ill-posed problem. Total variation minimization (TVM) and projection on convex sets (POCS) based iterative reconstruction algorithm is comparatively valid for this undersampling CT reconstruction problem and can obtain comparatively good performance. But the reconstruction images have artifacts with gradual change gray nearby edges. In this paper, a simple but efficient method to eliminate such artifacts during the reconstruction is proposed. Based on the same assumption of the TVM algorithm, for the object whose density distribution is approximate piecewise-constant, we develop and investigate an improved iterative reconstruction algorithm for volume image reconstruction from limited-angle cone-beam scan, which is referred to as piecewise-constant modification TVM-POCS (PM-TVM-POCS) algorithm. The grays of voxels with gradually changed gray artifacts are modified in the TVM-POCS iteration process by the piecewise-constant modification algorithm which is regarded as a “collapse” process, and gradually approaches the true piecewise-constant image. The results of simulation experiments show that the presented modification algorithm can improve the quality of the reconstructed image obviously and it is steady to noise. This modification algorithm can also be applied to other reconstruction problems which have artifacts with gradual change gray.

Abstract. We consider the inverse problem of determining the spatial dependence of the source term in a heat equation in assuming known, from a single internal measurements of the solution in . The purpose of this paper is to establish a reconstruction formula for similar to the one obtained by M. Yamamoto in “Stability, Reconstruction formula and regularization for an inverse source hyperbolic problem by a control method”, Inverse Problems 11 (1995), 481–496, for hyperbolic equations using exact controls, and generalized here for parabolic equations. The reconstruction formula is associated to a family of exact controls indexed by . We perform numerical simulations in order to illustrate the feasibility, accuracy and stability of the proposed reconstruction formula.

Abstract. A problem of an approximate solving of Symm's integral equation for an infinitely smooth closed boundary is considered. For a fully discrete projection method, error estimates are found in the metric of Sobolev spaces. A discretization parameter is chosen by balancing principle.

Abstract. In this paper we study an inverse problem for a singular differential equation in a Banach space. We use two projections to reduce the problem to a regular abstract inverse problem. In order to solve the problem, some conditions on the space and the operators are assumed. As a possible application of the abstract theorems, some examples of partial differential equations are given.

Abstract. We prove approximate Lipschitz stability for non-overdetermined inverse scattering at fixed energy with incomplete data in dimension . Our estimates are given in uniform norm for coefficient difference and related stability precision efficiently increases with increasing energy and coefficient difference regularity. In addition, our estimates are rather optimal even in the Born approximation.

Abstract. In this paper, we consider an inverse problem for the linearized Jordan–Moore–Gibson–Thompson equation, which is a third-order (in time) PDE that arises in nonlinear acoustic waves modeling high-intensity ultrasound. Both canonical recovery problems are investigated: (i) uniqueness and (ii) stability, by use of just one boundary measurement. Our approach relies on the dynamical decomposition of the Jordan–Moore–Gibson–Thompson equation given in [Math. Methods Appl. Sci. 35 (2012), 1896–1929].

Abstract. We investigate the continuation problem for the elliptic equation. The continuation problem is formulated in operator form . The singular values of the operator  A are presented and analyzed for the continuation problem for the Helmholtz equation. Results of numerical experiments are presented.

Abstract. We consider the continuation problem from the time-like surface for the 2D Maxwell equation. The problem is formulated in an operator form . We describe and justify gradient methods for minimizing the cost functional for the continuation and coefficient inverse problems. The results of a computational experiment are presented.