Inverse Problem For Hyperbolic Equation Research Articles

A time-nonlocal inverse problem for a hyperbolic equation with an integral overdetermination condition

This article is concerned with the study of the unique solvability of a timenonlocal inverse boundary value problem for second-order hyperbolic equation with an integral overdetermination condition. To study the solvability of the inverse problem, we first reduce the considered problem to an auxiliary system with trivial data and prove its equivalence (in a certain sense) to the original problem. Then using the Banach fixed point principle, the existence and uniqueness of a solution to this system is shown. Further, on the basis of the equivalency of these problems the existence and uniqueness theorem for the classical solution of the inverse coefficient problem is proved for the smaller value of time.

Preconditioning inverse problems for hyperbolic equations with applications to photoacoustic tomography

This paper is concerned with the preconditioning of wave equation-constrained linear inverse problems from boundary observation data, such as, for instance, photoacoustic tomography, which is considered as an application. The main result of this paper is a concept for regularization parameter robust preconditioning. That is, we propose a preconditioner for the PDE-constrained optimization problem such that the condition number of the preconditioned optimality system is uniformly bounded with respect to the regularization parameter. Using an augmented Lagrangian formulation for the discretized optimality system, we employ a discretization strategy which renders the discretized preconditioned system robust with respect to both the regularization parameter and the mesh size. Numerical experiments complement our analysis.

Solvability of Nonlinear Inverse Problem for Hyperbolic Equation

We consider the nonlinear inverse problem for a second order hyperbolic equation with unknown coefficient depending on the time. We establish the existence and uniqueness of regular solutions which are used for constructing a soltuion to the inverse problem under consideration.

ON CERTAIN CONTROL PROBLEM OF DISPLACEMENT AT ONE ENDPOINT OF A THIN BAR

In this paper, we study an inverse problem for hyperbolic equation. This problem arises when we consider vibration of a thin bar if one endpoint is fixed but behavior of the other is unknown and is the subject to find. Overdetermination is given in the form of integral with respect to spacial variable. The problem is reduced to the second kind Volterra integral equation. Special case is considered.

A PROBLEM ON VIBRATION OF A BAR WITH UNKNOWN BOUNDARY CONDITION ON A PART OF THE BOUNDARY

In this paper, we study an inverse problem for hyperbolic equation. This problem arises when we consider vibration of a nonhomogeneous bar if one endpoint is fixed by spring but behavior of the other is unknown and is the subject to find. Overdetermination is given in the form of integral with respect to spacial variable. Unique solvability of this problem is proved under some conditions on data. The proof is based on a priori estimates in Sobolev space.

The Inverse Problem for a General Class of Multidimensional Hyperbolic Equations

Inverse problems for hyperbolic equations are found in geophysical prospecting and seismology, and their multidimensional analogues are especially important for applied work. However, whereas results have been established for the some narrow classes of hyperbolic equations, no results exist for more general classes. This paper proves the solvability of the inverse problem for a general class of multidimensional hyperbolic equations. Our approach consists of properly choosing the shape of the overidentifying condition that is needed to determine the right‐hand side of the hyperbolic PDE and then applying the Fourier series method. We are then able to establish the results of the existence of solution for the cases when the unknown right‐hand side is time‐ independent or space independent.

Stability Estimates in Inverse Problems for Hyperbolic Equations

We discuss here a method for investigating inverse problems for hyperbolic equations based on combining an asymptotic expansion of solutions to the equations in a neighborhood of a characteristic surface, the connections between coefficients of the expansion and the unknown coefficients of the equation and the estimates of a solution to the Cauchy problem in terms of the data on a time-like surface. We give some applications of this method to a number of inverse problems. Stability results to the inverse problems are given.

Methods for solving dynamic inverse problems for hyperbolic equations

Methods for solving dynamic inverse problems for hyperbolic equations

Methods for solving dynamic inverse problems for hyperbolic equations

The paper is a brief exposition of the plenary report presented by the author at the conference devoted to seventy-fifth anniversary of Academician A. S. Alekseev.

Solution of inverse problems for hyperbolic equations by the Monte Carlo method

Solution of inverse problems for hyperbolic equations by the Monte Carlo method

The optimizational method for solving the discrete inverse problem for hyperbolic equation

Article The optimizational method for solving the discrete inverse problem for hyperbolic equation was published on January 1, 1996 in the journal Journal of Inverse and Ill-posed Problems (volume 4, issue 6).

Stability estimates in inverse problems for hyperbolic equations

Stability estimates in inverse problems for hyperbolic equations

Uniqueness of the Continuation Across a Time-Like Hyperplane and Related Inverse Problems for Hyperbolic Equations

On considere le cas ou les donnees de Cauchy sont prescrites sur une partie d'un hyperplan du genre temps. On obtient des theoremes de stabilite (et d'unicite) pour l'equation d'onde