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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. [ABSTRACT FROM AUTHOR]

The aim of this work is to reconstruct a spacewise dependent heat source in a degenerate heat equation using the final measurement temperature. Based on the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are established, and the global uniqueness and stability of the minimizer are also obtained from the necessary condition. Finally, a numerical algorithm on the basis of the conjugate gradient method is designed to treat the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm designed in the paper is quite efficient and stable, and the unknown heat source is recovered very well. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the inverse problem of determining a pair in the reaction–diffusion equationwith initial and boundary conditionsfrom the final measurement data , which has important application in a large fields of applied science. Based on the optimal control framework, the existence, uniqueness and stability of the minimizer for the cost functional are established. A necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. The gradient iteration algorithm is applied to the inverse problem and some numerical results are presented for various typical test examples. [Copyright &y& Elsevier]

Abstract: 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 estimate regarding the minimizer is obtained, from which the uniqueness and stability of the minimizer can be deduced immediately. [Copyright &y& Elsevier]

Abstract: This work studies an inverse problem of determining the first-order coefficient of degenerate parabolic equations using the measurement data specified at a fixed internal point. Being different from other ordinary parameter identification problems in parabolic equations, in our mathematical model there exists degeneracy on the lateral boundaries of the domain, which may cause the corresponding boundary conditions to go missing. By the contraction mapping principle, the uniqueness of the solution for the inverse problem is proved. A numerical algorithm on the basis of the predictor–corrector method is designed to obtain the numerical solution and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown function is recovered very well. The results obtained in the paper are interesting and useful, and can be extended to other more general inverse coefficient problems of degenerate PDEs. [Copyright &y& Elsevier]

Abstract: The zero-coupon bond (ZCB) is a special kind of bond without coupons, which is purchased today at a certain price, while at maturity the bond is redeemed for a fixed price. In the ZCB world there is the important quantity which is often called the market price of risk and cannot be observed directly but has a major impact on the ZCB value. In this paper, the inverse problem of determining the market price of risk from the current market prices of ZCB is discussed. Being different from ordinary parameter identification problems in parabolic equations, the mathematical model in the paper belongs to the second order parabolic equations with non-negative characteristic form, i.e., there exists degeneracy on the lateral boundaries in the model. Based on the optimal control framework, the existence, uniqueness and stability of the minimizer for the cost functional are established. A necessary condition which is a coupled system of a parabolic equation and a parabolic variational inequality is deduced. The results obtained in the paper are interesting and useful, and may be applied to a variety of derivatives pricing problems. [Copyright &y& Elsevier]

Abstract: This paper studies an evolutional type inverse problem of identifying the radiative coefficient of heat conduction equation when the over-specified data is given. Problems of this type have important applications in several fields of applied science. Being different from other ordinary inverse coefficient problems, the unknown coefficient in this paper depends on both the space variable x and the time t. Based on the optimal control framework, the inverse problem is transformed into an optimization problem and a new cost functional is constructed in the paper. The existence, uniqueness and stability of the minimizer of the cost functional are proved, and the necessary conditions which must be satisfied by the minimizer are also given. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations. [Copyright &y& Elsevier]

Abstract: This paper deals with the determination of a pair in the nonlinear parabolic equation with initial and boundary conditions from the overspecified data . Based on the optimal control framework, the problem is transformed into a nonlinear optimization problem and the existence of the minimizer for the control functional is established. The necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. Since the optimal control problem is nonconvex, one may not expect a unique solution in general. However, the local uniqueness and stability of the solution are proved, which is also the main contribution of the paper. [Copyright &y& Elsevier]

Abstract: In the Black–Scholes world there is the important quantity of volatility which cannot be observed directly but has a major impact on the option value. In practice, traders usually work with what is known as implied volatility which is implied by option prices observed in the market. In this paper, we use an optimal control framework to discuss an inverse problem of determining the implied volatility when the average option premium, namely the average value of option premium corresponding with a fixed strike price and all possible maturities from the current time to a chosen future time, is known. The issue is converted into a terminal control problem by Green function method. The existence and uniqueness of the minimum of the control functional are addressed by the optimal control method, and the necessary condition which must be satisfied by the minimum is also given. The results obtained in the paper may be useful for those who engage in risk management or volatility trading. [Copyright &y& Elsevier]

An inverse source problem in degenerate parabolic equations is considered, where the diffusion coefficient are zero on the whole boundary. Based on the strong maximum principle for degenerate parabolic equations established in the paper, the uniqueness of the solution is proved. [ABSTRACT FROM AUTHOR]

This paper deals with an inverse problem of identifying the source coefficient in degenerate parabolic equations. On the basis of extremum principle, the uniqueness of the proposed problem is obtained. The difference scheme for the direct problem is got by using the finite volume method and the corresponding stability is also proved. An iteration scheme of the Landweber type is applied to handle the numerical solution. In the end, several numerical experiments are performed to test the validity of the scheme. The numerical results show that the method is stable and the unknown function is recovered very well when iterative parameter α and iterative times are matching appropriately. [ABSTRACT FROM AUTHOR]