Inverse Hyperbolic Problem Research Articles

Inverse Hyperbolic Conduction Problem in Estimating Two Unknown Surface Heat Fluxes Simultaneously

An inverse hyperbolic heat conduction problem is solved in the present study by an iterative regularization method, that is, the conjugate gradient method, to estimate simultaneously two unknown boundary heat fluxes based on the interior temperature measurements. The inverse solutions will be justified based on the numerical experiments in which two specific cases in determining the unknown boundary heat flux distributions are examined. Results show that the inverse solutions can always be obtained with any arbitrary initial guesses of the boundary heat fluxes and that the position of each sensor should be as close to each boundary as possible to obtain accurate estimations. Finally, it is concluded that accurate boundary heat fluxes can be estimated in the present study when large measurement errors are considered.

On inverse problems in secondary oil recovery

We review simple models of oil reservoirs and suggest some ideas for theoretical and numerical study of this important inverse problem. These models are formed by a system of an elliptic and a parabolic (or first-order hyperbolic) quasilinear partial differential equations. There are and probably there will be serious theoretical and computational difficulties mainly due to the degeneracy of the system. The practical value of the problem justifies efforts to improve the methods for its solution. We formulate ‘history matching’ as a problem of identification of two coefficients of this system. We consider global and local versions of this inverse problem and propose some approaches, including the use of the inverse conductivity problem and the structure of fundamental solutions. The global approach looks for properties of the ground in the whole domain, while the local one is aimed at recovery of these properties near wells. We discuss the use of the model proposed by Muskat which is a difficult free boundary problem. The inverse Muskat problem combines features of inverse elliptic and hyperbolic problems. We analyse its linearisation about a simple solution and show uniqueness and exponential instability for the linearisation.

Determination of a coefficient in the wave equation with a single measurement

We consider an inverse problem of finding the coefficient of the second-order derivatives in a second-order hyperbolic equation with variable coefficients. Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement of Neumann data on a suitable sub-boundary. Moreover we show that our uniqueness yields the Lipschitz stability estimate in L 2 space for solution to the inverse problem. The key is a Carleman estimate for a hyperbolic operator with variable coefficients.

A new approach to hyperbolic inverse problems II: global step

We study the inverse problem for the second-order self-adjoint hyperbolic equation with the boundary data given on a part of the boundary. This paper is the continuation of the author's paper (Eskin 2006 A new approach to hyperbolic inverse problems Inverse Problems 22 815–33), in which we presented the crucial local step of the proof. In this paper, we prove the global step. Our method is a modification of the BC method with some new ideas. In particular, the method of determination of the metric is new.

Lipschitz stability for a hyperbolic inverse problem by finite local boundary data

In this article we consider the inverse problem of determining the potential q in a wave equation in a bounded smooth domain Ω in from a finite number of data of the hyperbolic Dirichlet to Neumann map and we prove the Lipschitz stability in determining q. Our main result is stated as follows. Let T> diam Ω and Γ0⊂∂Ω. For any k-dimensional space X in L ∞(Ω), there exist 2k-functions f 1, …, f 2k on (0,T)×∂Ω such that , provided that q1, q2∈X are uniformly bounded in a suitable Sobolev space. Here Λq is the Dirichlet to Neumann map for the coefficient q(x).

A new approach to hyperbolic inverse problems

We present a modification of the BC method in inverse hyperbolic problems. The main novelty is the study of the restrictions of the solutions to the characteristic surfaces instead of the fixed time hyperplanes. The main result is that the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the self-adjoint hyperbolic operator up to a diffeomorphism and a gauge transformation. In this paper, we prove the crucial local step. The global step of the proof will be presented in a forthcoming paper.

Direct and Inverse Solutions of the Hyperbolic Heat Conduction Problems

A sequential method for estimating boundary conditions in the field of hyperbolic heat-conduction problems is proposed. An inverse solution is deduced from a finite-difference method, the concept of future time, and a modified Newton‐Raphson method. The undetermined boundary condition at each time step is denoted as an unknown variable in a set of nonlinear equations, which are formulated from the measured temperature and the calculated temperature. Then an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. Two examples are used to demonstrate the characteristics of the proposed method. In the first example, a well-known problem is used to demonstrate the validity of the proposed direct method and then an inverse solution is evaluated. In the second example, a larger value of the relaxation time is implemented in the direct solutions and the inverse solutions. The close agreement between the exact values and the estimated results is used to confirm the validity and accuracy of the proposed method. The results show that the proposed method is an accurate and stable method for determining the boundary conditions in inverse hyperbolic heat-conduction problems.

Global uniqueness for a 3D/2D inverse conductivity problem via the modified method of Carleman estimates

Let Ω ⊂ be a bounded domain and {(x1, x2, x3) | (x1, x3) ∈ Ω, x2 ∈ (−∞, ∞)} be a cylinder. Assume that the stationary electric field u induced by a pointwise current modelled by the function f (x2)S(x1, x3) is governed by an elliptic differential equation ∇ · (σ∇u) = −f(x2)S(x1, x3). Also, assume that a pair of the Dirichlet and Neumann conditions is given on an arbitrary part of the lateral surface of the cylinder. The problem of determining the 2D electric conductivity σ(x1, x3) in this cylinder from the boundary incomplete data is considered. Using the direct Fourier and inverse Laplace transforms, the original inverse problem is reduced to an auxiliary inverse problem for a hyperbolic equation . The main difficulty is due to the presence of derivatives of the function σ (x1, x3). To overcome this difficulty, the method of Carleman estimates is significantly modified. This allows for establishing the Lipschitz stability for the auxiliary hyperbolic inverse problem. In turn, the stability result implies the global uniqueness theorem for the original inverse conductivity problem. The proposed formulation and uniqueness theorem extend both the formulation of the 1D inverse problem of electrical prospecting and uniqueness result established by Tikhonov in 1949.

Global uniqueness for a 3D/2D inverse conductivity problem via the modified method of Carleman estimates

Let Ω ⊂ be a bounded domain and {( x 1 , x 2 , x 3 ) | ( x 1 , x 3 ) ∈ Ω, x 2 ∈ (−∞, ∞)} be a cylinder. Assume that the stationary electric field u induced by a pointwise current modelled by the function f ( x 2 ) S ( x 1 , x 3 ) is governed by an elliptic differential equation ∇ · (σ∇ u ) = − f ( x 2 ) S ( x 1 , x 3 ). Also, assume that a pair of the Dirichlet and Neumann conditions is given on an arbitrary part of the lateral surface of the cylinder. The problem of determining the 2D electric conductivity σ( x 1 , x 3 ) in this cylinder from the boundary incomplete data is considered. Using the direct Fourier and inverse Laplace transforms, the original inverse problem is reduced to an auxiliary inverse problem for a hyperbolic equation . The main difficulty is due to the presence of derivatives of the function σ ( x 1 , x 3 ). To overcome this difficulty, the method of Carleman estimates is significantly modified. This allows for establishing the Lipschitz stability for the auxiliary hyperbolic inverse problem. In turn, the stability result implies the global uniqueness theorem for the original inverse conductivity problem. The proposed formulation and uniqueness theorem extend both the formulation of the 1D inverse problem of electrical prospecting and uniqueness result established by Tikhonov in 1949.

Justification of optimization methods for inverse integro-differential hyperbolic problems

Article Justification of optimization methods for inverse integro-differential hyperbolic problems was published on October 1, 2004 in the journal Journal of Inverse and Ill-posed Problems (volume 12, issue 5).

A uniqueness result in an inverse hyperbolic problem with analyticity

We prove the uniqueness for the inverse problem of determining a coefficient and outside a ball. The proof depends on Fritz John's global Holmgren theorem and the uniqueness by a Carleman estimate.

Justification of optimization methods for inverse integro-differential hyperbolic problems

As is well-known, optimization is the most widely used method for solving numerically inverse problems in applications. The main theoretical difficulty consists in justifying the existence of the cost (misfit) function in an appropriate space. Indeed, in order to prove the existence of the trace of the solution to the direct problem one should suppose that the desired coefficients are sufficiently (and, sometimes, even too much!) smooth. In the present paper we overcome (in some sense) this gap treating our inverse problem as a nonlinear system of integral equations in L2. More exactly, first we deal with the problem of determining a pair of functions u : ∆(l) → R and h : [0, l]→ R, where

Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients

One of the basic inverse problems in an anisotropic media is the determination of coefficients in a bounded domain with a single measurement. We consider the problem of finding the coefficient of the second derivatives in a second-order hyperbolic equation with variable coefficients. Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement. Moreover we show that our uniqueness results yield the Lipschitz stability estimate in L 2 space for solution to the inverse problem under consideration.

Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation

We study the global stability in determination of a coefficient of the zeroth-order term in a second-order hyperbolic equation from data of the solution in a subboundary over a time interval. Providing regular initial data, without any assumption on the dynamics (i.e. without the geometric optics condition for the observability), we prove the uniqueness in multidimensional hyperbolic inverse problems with a single measurement. Moreover, we show that our uniqueness results yield the logarithm stability estimate in L2 space for solution of the inverse problem under consideration.

Global Uniqueness and Stability for a Class of Multidimensional Inverse Hyperbolic Problems with Two Unknowns

In this paper we obtain the global uniqueness and stability estimate for a class of multidimensional inverse hyperbolic problems of determining a source term and an initial value from a single measurement of boundary values or interior values. By means of a suitable transformation, we reduce the problem to the observability inequalities for nonconservative hyperbolic equations with memory. Then, using a compactness/uniqueness argument, we can prove the uniqueness and the stability by a new kind of unique continuation property of a nonlocal hyperbolic equation.

LOCAL EXISTENCE AND GLOBAL UNIQUENESS IN ONE DIMENSIONAL NONLINEAR HYPERBOLIC INVERSE PROBLEMS

We prove local existence and global uniqueness in one dimensional nonlinear hyperbolic inverse problems. The basic key for showing the local existence of inverse solution is the principle of contracted mapping. As an application, we consider a hyperbolic inverse problem with damping term.

Hyperbolic inverse boundary-value problem and time-continuation of the non-stationary Dirichlet-to-Neumann map

Hyperbolic inverse boundary-value problem and time-continuation of the non-stationary Dirichlet-to-Neumann map

Hyperbolic inverse boundary-value problem and time-continuation of the non-stationary Dirichlet-to-Neumann map

Let M be a compact Riemannian manifold with non-empty boundary M. In this paper we consider an inverse problem for the second-order hyperbolic initial-boundary-value problem utt + but + a(x, D)u = 0 in M R+, u|MR+ = f, u|t=0 = ut|t=0 = 0. Our goal is to determine (M, g), b and a(x, D) from the knowledge of the non-stationary Dirichlet-to-Neumann map (the hyperbolic response operator) RT, with sufficiently large T 0. The response operator RT is the map , where is the normal derivative of the solution of the initial-boundary-value problem.More specifically, we show the following. (i)It is possible to determine Rt for any t 0 if we know RT for sufficiently large T and some geometric condition upon the geodesic behaviour on (M, g) is satisfied.(ii)It is then possible to determine (M, g) and b uniquely and the elliptic operator a(x, D) modulo generalized gauge transformations.

GlobalLipschitz stability in an inverse hyperbolic problem by interiorobservations

For the solution u(p) = u(p)(x,t) to ∂t2u(x,t)-Δu(x,t)-p(x)u(x,t) = 0 in Ω×(0,T) and(∂u/∂ν)|∂Ω×(0,T) = 0 with given u(·,0) and ∂tu(·,0), we consider an inverse problem of determiningp(x), x∊Ω, from data u|ω×(0,T).Here Ω⊂ℝn, n = 1,2,3, is a bounded domain,ω is a sub-domain of Ω and T>0. For suitableω⊂Ω and T>0, we prove an upper andlower estimate of Lipschitz type between ||p-q||L2(Ω) and ||∂t(u(p)-u(q))||L2(ω×(0,T)) + ||∂t2(u(p)-u(q))||L2(ω×(0,T)).

A computational method for hyperbolic inverse problem

An approach for solving hyperbolic inverse problems numerically is proposed. The interest on this problem is strongly motivated by applications to seismic prospecting. In many applications it is important to recover the (discontinuous) coefficient a from boundary measurements of solutions of a hyperbolic equation. The inverse problem is severely ill-posed and nonlinear, therefore, numerical solution of this problem is quite difficult. We, first, linearize it around a constant a and then solve numerically the linear ill-posed problem using regularization.