Uniqueness and stability of the minimizer for a binary functional arising in an inverse heat conduction problem

The local well-posedness of the minimizer of an optimal control problem is studied in this paper. The optimization problem concerns an inverse problem of simultaneously reconstructing the initial temperature and heat radiative coefficient in a heat conduction equation. Being different from other ordinary optimization problems, the cost functional constructed in the paper is a binary functional which contains two independent variables and two independent regularization parameters. Particularly, since the status of the two unknown coefficients in the cost functional are different, the conjugate theory which is extensively used in single-parameter optimization problems cannot be applied for our problem. The necessary condition which must be satisfied by the minimizer is deduced. By assuming the terminal time T is relatively small, an L² estimate regarding the minimizer is obtained, from which the uniqueness and stability of the minimizer can be deduced immediately.     

An Inverse Problem of Identifying the Radiative Coefficient in a Degenerate Parabolic Equation

The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.     

Identification of the thermal growth characteristics of coagulated tumor tissue in laser‐induced thermotherapy

We consider an inverse problem arising in laser‐induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues are destroyed by coagulation. For the dosage planning, numerical simulations play an important role. To this end, a crucial problem is to identify the thermal growth kinetics of the coagulated zone. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. The solution to this inverse problem is defined as the minimizer of a nonconvex cost functional in this paper. The existence of the minimizer is proven. We derive the Gateaux derivative of the cost functional, which is based on the adjoint system, and use it for a numerical approximation of the optimal coefficient. Numerical implementations are presented to show the validity of the optimization schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

A C finite element method for an inverse problem

A C⁰ finite element method is presented for an inverse problem in which the coefficient in the differential operator is to be determined from the measurement of the solution of a boundary value problem. The unknown in the inverse problem is approximated by a minimizer of a cost function that includes both the output error and equation error. Error estimates in a weighted H⁻¹ norm and L² are given. Numerical examples are presented to show features of the method.     

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

This work studies a nonlinear inverse problem of reconstructing the diffusion coefficient in a parabolic‐elliptic system using the final measurement data, which has important application in a large field of applied science. Being different from other works, which are governed by single partial differential equations, the underlying mathematical model in this paper is a coupled parabolic‐elliptic system, which makes theoretical analysis rather difficult. On the basis of the optimal control framework, the identification problem is transformed into an optimization problem. Then the existence of the minimizer is proved, and the necessary condition that must be satisfied by the minimizer is also given. Since the optimal control problem is nonconvex, one may not expect a unique solution universally. However, the local uniqueness and stability of the minimizer are deduced in this paper.     

Parameter identification by optimization method for a pollution problem in porous media

In the present work, we investigate the inverse problem of reconstructing the parameter of an integro-differential parabolic equation, which comes from pollution problems in porous media, when the final observation is given. We use the optimal control framework to establish both the existence and necessary condition of the minimizer for the cost functional. Furthermore, we prove the stability and the local uniqueness of the minimizer. Some numerical results will be presented and discussed.     

Maximum entropy solution to ill-posed inverse problems with approximately known operator

We consider the linear inverse problem of reconstructing an unknown finite measure μ from a noisy observation of a generalized moment of μ defined as the integral of a continuous and bounded operator Φ with respect to μ. Motivated by various applications, we focus on the case where the operator Φ is unknown; instead, only an approximation Φm to it is available. An approximate maximum entropy solution to the inverse problem is introduced in the form of a minimizer of a convex functional subject to a sequence of convex constraints. Under several assumptions on the convex functional, the convergence of the approximate solution is established.     

First and second order numerical differentiation with Tikhonov regularization

This work deals with the numerical differentiation for an unknown smooth function whose data on a given set are available. The numerical differentiation is an ill-posed problem. In this work, the first and second derivatives of the smooth function are approximated by using the Tikhonov regularization method. It is proved that the approximate function can be chosen as a minimizer to a cost functional. The existence and uniqueness theory of the minimizer is established. Errors in the derivatives between the smooth unknown function and the approximate function are obtained, which depend on the mesh size of the grid and the noise level in the data. The numerical results are provided to support the theoretical analysis of this work. Selected from Numerical Mathematics (A Journal of Chinese Universities), 2004, 26(1):62–74     

A further study on inverse linear programming problems

In this paper we continue our previous study (Zhang and Liu, J. Comput. Appl. Math. 72 (1996) 261–273) on inverse linear programming problems which requires us to adjust the cost coefficients of a given LP problem as less as possible so that a known feasible solution becomes the optimal one. In particular, we consider the cases in which the given feasible solution and one optimal solution of the LP problem are 0–1 vectors which often occur in network programming and combinatorial optimization, and give very simple methods for solving this type of inverse LP problems. Besides, instead of the commonly used l₁ measure, we also consider the inverse LP problems under l_∞ measure and propose solution methods.     

Identification of the zeroth-order coefficient and fractional order in a time-fractional reaction-diffusion-wave equation

In this paper, we investigate an inverse problem of recovering the zeroth-order coefficient and fractional order simultaneously in a time-fractional reaction-diffusion-wave equation by using boundary measurement data from both of uniqueness and numerical method. We prove the uniqueness of the considered inverse problem and the Lipschitz continuity of the forward operator. Then the inverse problem is formulated into a variational problem by the Tikhonov-type regularization. Based on the continuity of the forward operator, we prove that the minimizer of the Tikhonov-type functional exists and converges to the exact solution under an a priori choice of regularization parameter. The steepest descent method combined with Nesterov acceleration is adopted to solve the variational problem. Three numerical examples are presented to support the efficiency and rationality of our proposed method.     

Dirichlet problem on locally finite graphs

In this paper, we study the existence and uniqueness of solutions to the vertex-weighted Dirichlet problem on locally finite graphs. Let B be a subset of the vertices of a graph G. The Dirichlet problem is to find a function whose discrete Laplacian on G?B and its values on B are given. Each infinite connected component of G?B is called an end of G relative to B. If there are no ends, then there is a unique solution to the Dirichlet problem. Such a solution can be obtained as a limit of an averaging process or as a minimizer of a certain functional or as a limit-solution of the heat equation on the graph. On the other hand, we show that if G is a locally finite graph with l ends, then the set of solutions of any Dirichlet problem, if non-empty, is at least l-dimensional.     

Inverse problem for the reaction diffusion system by optimization method

In this paper, we study an inverse problem of reconstructing two time independent coefficients in the reaction diffusion system from the final measurement. First the given problem is transformed into an optimization problem by using optimal control framework and the existence of the minimizer for the control functional is established. Then we prove the stability estimate for two coefficients with the upper bound given by some Sobolev norms of the final measurement.     

An Optimal Control Method for Nonlinear Inverse Diffusion Coefficient Problem

This paper investigates the solution of a parameter identification problem associated with the two-dimensional heat equation with variable diffusion coefficient. The singularity of the diffusion coefficient results in a nonlinear inverse problem which makes theoretical analysis rather difficult. Using an optimal control method, we formulate the problem as a minimization problem and prove the existence and uniqueness of the solution in weighted Sobolev spaces. The necessary conditions for the existence of the minimizer are also given. The results can be extended to more general parabolic equations with singular coefficients.     

The inverse Fermat–Weber problem

Given n points in the plane with nonnegative weights, the inverse Fermat–Weber problem consists in changing the weights at minimum cost such that a prespecified point in the plane becomes the Euclidean 1-median. The cost is proportional to the increase or decrease of the corresponding weight. In case that the prespecified point does not coincide with one of the given n points, the inverse Fermat–Weber problem can be formulated as linear program. We derive a purely combinatorial algorithm which solves the inverse Fermat–Weber problem with unit cost using O(n) greedy-like iterations where each of them can be done in constant time if the points are sorted according to their slopes. If the prespecified point coincides with one of the given n points, it is shown that the corresponding inverse problem can be written as convex problem and hence is solvable in polynomial time to any fixed precision.     

Uniform-cost inverse absolute and vertex center location problems with edge length variations on trees

This article considers the inverse absolute and the inverse vertex 1-center location problems with uniform cost coefficients on a tree network T with n+1 vertices. The aim is to change (increase or reduce) the edge lengths at minimum total cost with respect to given modification bounds such that a prespecified vertex s becomes an absolute (or a vertex) 1-center under the new edge lengths. First an O(nlogn) time method for solving the height balancing problem with uniform costs is described. In this problem the height of two given rooted trees is equalized by decreasing the height of one tree and increasing the height of the second rooted tree at minimum cost. Using this result a combinatorial O(nlogn) time algorithm is designed for the uniform-cost inverse absolute 1-center location problem on tree T. Finally, the uniform-cost inverse vertex 1-center location problem on T is investigated. It is shown that the problem can be solved in O(nlogn) time if all modified edge lengths remain positive. Dropping this condition, the general model can be solved in O(r_vnlogn) time where the parameter r_v is bounded by ⌈n/2⌉. This corrects an earlier result of Yang and Zhang.     

Simultaneous identification of parameters and initial datum of reaction diffusion system by optimization method

In this paper, we study an inverse problem of reconstructing two time independent coefficients and the initial data in the linear reaction diffusion system from the arbitrary subdomain measurement and final measurement. First the given problem is transformed into an optimization problem by using optimal control framework and we establish the existence of the minimizer for the control functional. Further the necessary optimality condition is established which is the key ingredient to establish the stability estimate.     

Joint additive Kullback–Leibler residual minimization and regularization for linear inverse problems

For the approximate solution of ill‐posed inverse problems, the formulation of a regularization functional involves two separate decisions: the choice of the residual minimizer and the choice of the regularizor. In this paper, the Kullback–Leibler functional is used for both. The resulting regularization method can solve problems for which the operator and the observational data are positive along with the solution, as occur in many inverse problem applications. Here, existence, uniqueness, convergence and stability for the regularization approximations are established under quite natural regularity conditions. Convergence rates are obtained by using an a priori strategy. Copyright © 2007 John Wiley & Sons, Ltd.     

Identification of a dissipation coefficient by a variational method

A generalized inverse problem for the identification of the absorption coefficient for a hyperbolic system is considered. The well-posedness of the problem is examined. It is proved that the regular part of the solution is an L ₂ function, which reduces the inverse problem to minimizing the error functional. The gradient of the functional is determined in explicit form from the adjoint problem, and approximate formulas for its calculation are derived. A regularization algorithm for the solution of the inverse problem is considered. Numerical results obtained for various excitation sources are displayed.     

A numerical approach to variational problems subject to convexity constraint

Summary. We describe an algorithm to approximate the minimizer of an elliptic functional in the form on the set of convex functions u in an appropriate functional space X. Such problems arise for instance in mathematical economics [4]. A special case gives the convex envelope of a given function . Let be any quasiuniform sequence of meshes whose diameter goes to zero, and the corresponding affine interpolation operators. We prove that the minimizer over is the limit of the sequence , where minimizes the functional over . We give an implementable characterization of . Then the finite dimensional problem turns out to be a minimization problem with linear constraints. Received November 24, 1999 / Published online October 16, 2000     

A constrained minimization problem and its application to guided cylindrical TM-modes in an anisotropic self-focusing dielectric

The first part of this paper establishes the existence of a minimizer of problem: where The essential features of the integrand are that where We show that the minimizer satisfies an Euler- Lagrange equation and estimates are given for the Lagrange multiplier as a function of d. In the second part of the paper, we use this result to establish the existence of guided TM-modes propagating through a self-focusing anisotropic dielectric. These are special solutions of Maxwell's equations with a nonlinear constitutive relation of a type commonly used in nonlinear optics when treating the propagation of waves in a cylindrical wave-guide. In TM-modes, the magnetic field has the form \[ {\bf B}=w(r)\cos (kz-\omega t)i_{\theta } \] when expressed in cylindrical polar co-ordinates The amplitude w is given by where is a minimizer of the problem (0.1) for a function which is determined by the constitutive relation through a Legendre transformation. Received: 4 April 2001 / Accepted: 29 November 2001 / Published online: 28 February 2002