Inverse coefficient problems for monotone potential operators, Inverse Problems, 13(1997), pp. 1265-1278. (original) (raw)
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The class of inverse problems for a nonlinear elliptic variational inequality is considered. The nonlinear elliptic operator is assumed to be a monotone potential. The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admissible coefficients which is compact in H 1 (0, ξ * ). It is shown that the nonlinear operator is pseudomonotone for the given class of coefficients. For the corresponding direct problem H 1coefficient convergence is proved. Based on this result the existence of a quasisolution of the inverse problem is obtained. As an important application an inverse diagnostic problem for an axially symmetric elasto-plastic body is considered. For this problem the numerical method and computational results are also presented.
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The coefficient in a linear elliptic partial differential equation can be estimated from interior measurements of the solution. Posing the estimation problem as a constrained optimization problem with the PDE as the constraint allows the use of the augmented Lagrangian method, which is guaranteed to converge. Moreover, the convergence analysis encompasses discretization by finite element methods, so the proposed algorithm can be implemented and will produce a solution to the constrained minimization problem. All of these properties hold in an abstract framework that encompasses several interesting problems: the standard (scalar) elliptic BVP in divergence form, the system of isotropic elasticity, and others. Moreover, the analysis allows for the use of total variation regularization, so rapidly-varying or even discontinuous coefficients can be estimated.
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In this paper, we make a review of some inverse problems in elasticity, in statics and dynamics, in acoustics, thermoelasticity and viscoelasticity. Crack inverse problems have been solved in closed form, by considering a nonlinear variational equation provided by the reciprocity gap functional. This equation involves the unknown geometry of the crack and the boundary data. It results from the symmetry lost between current fields and adjoint fields which is related to their support. The nonlinear equation is solved step by step by considering linear inverse problems. The normal to the crack plane, then the crack plane and finally the geometry of the crack, defined by the support of the crack displacement discontinuity, are determined explicitly. We also consider the problem of a volumetric defect viewed as the perturbation of a material constant in elastic solids which satisfies the nonlinear Calderon?s equation. The nonlinear problem reduces to two successive ones: a source inverse...
Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities
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The paper investigates an inverse problem for a stationary variational-hemivariational inequality. The solution of the variational-hemivariational inequality is approximated by its penalized version. We prove existence of solutions to inverse problems for both the initial inequality problem and the penalized problem. We show that optimal solutions to the inverse problem for the penalized problem converge, up to a subsequence, when the penalty parameter tends to zero, to an optimal solution of the inverse problem for the initial variational-hemivariational inequality. The results are illustrated by a mathematical model of a nonsmooth contact problem from elasticity.
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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
The solution of an elliptic boundary value problem is an infinitely differentiable function of the coefficient in the partial differential equation. When the (coefficient-dependent) energy norm is used, the result is a smooth, convex output least-squares functional. Using total variation regularization, it is possible to estimate discontinuous coefficients from interior measurements. The minimization problem is guaranteed to have a solution, which can be obtained in the limit from finite-dimensional discretizations of the problem. These properties hold in an abstract framework that encompasses several interesting problems: the standard (scalar) elliptic BVP in divergence form, the system of isotropic elasticity, and others. Downloaded from J 1a4 u2 5 R1a2, where J 1a4 u2 4 1122T 1a4 u 1 z4 u 1 z2, and e1a4 u2 4 0 if and only if u 4 F1a2. We could, of course, simply take e1a4 u2 4 F1a2 1 u, but it is advantageous to express the constraint in a less nonlinear form. By the Riesz representation theorem, there is an isomorphism E : V V defined by 1Eu212 4 1u4 2 V for all 8 V . For each 1a4 u2 8 A V , T 1a4 u4 32 1 m defines an element of V . We define e1a4 u2 to be the pre-image under E of this element: 1e1a4 u24 2 V 4 T 1a4 u4 21m12 for all 8 V . Then e1a4 u2 4 0 if and only if T 1a4 u4 2 4 m12 for all 8 V , that is, if and only if u 4 F1a2.
New a Priori Estimations of the Solution of Quasi-Inverse Problem
International Journal of Apllied Mathematics, 2014
In R. Almomani and H. Almefleh [1], the authors formulated the control problem of heat conduction problem with inverse direction of time and integral boundary conditions and they show the non-wellposedness of this problem. In H. Almefleh [2], the author reduced the solution of the control problem of the inhomogeneous heat equation to the homogeneous case. In H. Almefleh, R. Almomani [3] the authors established a priori estimate for the solution of quasi-inverse problem. In this paper we establish a new priori estimate for the same problem and the same order but with another weight function. The solution of our problem plays an important role in optimal control in heat conduction theory and in plasma physics, that is, in those problems where we have an integral restriction on a function.
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Journal of Differential Equations, 2020
In this paper, we consider several geometric inverse problems for linear elliptic systems. We prove uniqueness and stability results. In particular, we show the way that the observation depends on the perturbations of the domain. In some particular situations, this provides a strategy that could be used to compute approximations to the solution of the inverse problem. In the proofs, we use techniques related to (local) Carleman estimates and differentiation with respect to the domain.
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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.