Inverse boundary value problems with unknown boundaries: Optimal stability (original) (raw)
On an inverse boundary value problem* Mathematical subject classification
This paper is a reprint of the original work by A. P. Calderón published by the Brazilian Mathematical Society (SBM) in ATAS of SBM (Rio de Janeiro), pp. 65-73, 1980. The original paper had no abstract, so this reprint to be truthful to the original work is published with no abstract. Mathematical subject classification: 35J25, 35Q60, 47F05, 86A20, 86A22.
Two regularization methods for a class of inverse boundary value problems of elliptic type
Boundary Value Problems, 2013
This paper deals with the problem of determining an unknown boundary condition u(0) in the boundary value problem u yy (y) -Au(y) = 0, u(0) = f , u(+∞) = 0, with the aid of an extra measurement at an internal point. It is well known that such a problem is severely ill-posed, i.e., the solution does not depend continuously on the data. In order to overcome the instability of the ill-posed problem, we propose two regularization procedures: the first method is based on the spectral truncation, and the second is a version of the Kozlov-Maz'ya iteration method. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution. MSC: 35R25; 65J20; 35J25
Stability for Some Inverse Problems for Transport Equations
SIAM Journal on Mathematical Analysis, 2016
In this article, we consider inverse problems of determining a source term and a coefficient of a first-order partial differential equation and prove conditional stability estimates with minimum boundary observation data and relaxed condition on the principal part.
Transactions of the American Mathematical Society, 2001
In this paper, we investigate an inverse problem of determining a shape of a part of the boundary of a bounded domain in R 3 by a solution to a Cauchy problem of the Laplace equation. Assuming that the unknown part is a Lipschitz continuous surface, we give a logarithmic conditional stability estimate in determining the part of boundary under reasonably a priori information of an unknown part. The keys are the complex extension and estimates for a harmonic measure.
Journal of Computational and Applied Mathematics, 2007
Stationary thermography can be used for investigating the functional form of a nonlinear cooling law that describes heat exchanges through an inaccessible part of the boundary of a conductor. In this paper, we obtain a logarithmic stability estimate for the associated nonlinear inverse problem. This stability estimate is obtained from the convergence and sensitivity analysis of a finite difference method for the numerical solution of the Cauchy problem for Laplace's equation, based on the Störmer-Verlet scheme.
Boundary Value Problems, 2015
In the present study, the inverse problem for a multidimensional elliptic equation with mixed boundary conditions and overdetermination is considered. The first and second orders of accuracy in t and the second order of accuracy in space variables for the approximate solution of this inverse problem are constructed. Stability, almost coercive stability, and coercive stability estimates for the solution of these difference schemes are established. For the two-dimensional inverse problems with mixed boundary value conditions, numerical results are presented in test examples.
Lipschitz Stability in an Inverse Hyperbolic Problem by Boundary Observations
More Progresses in Analysis - Proceedings of the 5th International ISAAC Congress, 2009
Let u = u(q) satisfy a hyperbolic equation with impulsive input: ∂ 2 t u(x, t) − u(x, t) + q(x)u(x, t) = δ(x 1)δ (t) and let u| t<0 = 0. Then we consider an inverse problem of determining q(x), x ∈ Ω from data u(q)| S T and (∂u(q)/∂ν
Solution of inverse problems by boundary integral equations without residual minimization
International Journal of Solids and Structures, 2000
In this paper the solution of some inverse problems for potential ®elds is tackled. The aim is to compute the position and shape of an unknown¯aw within a body, using some experimental measures as additional data. By the linearization of the dierence between the Boundary Integral Equation for the actual con®guration and the same equation for an assumed con®guration, an integral equation for the variations is deduced. This integral equation is carried to the boundary by a limiting process and a solution procedure is devised to compute an approximation to the actual¯aw. The solution method proceeds iteratively, solving a direct and an inverse problem in every step, but no minimization algorithm is involved. The performance of the method is shown in several numerical examples.
On the existence of solution for an inverse problem
We consider a boundary detection problem. We present physical motivations. We formulate the problem as a shape optimization problem by introducing the Neumann condition of the accessible part in a cost functional to be minimized, which complicates the study of continuity state that requires more regularity of the free boundary. We show the existence of the optimal solution of the problem by the J. Haslinger and P. Neittaanmäki principle.
Lipschitz stability in inverse problems for a Kirchhoff plate equation
Asymptotic Analysis
In this paper, we prove a Carleman estimate for a Kirchhoff plate equation and apply the Carleman estimate to inverse problems of determining spatially varying two Lamé coefficients and the mass density by a finite number of boundary observations. Our main results are Lipschitz stability estimates for the inverse problems under suitable conditions of initial values and boundary val-1 ues, which are satisfied, in particular, by paraboloid initial displacements.