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In this paper, we would like to give an indirect possible way to reconstruct the shape of rotating body by temperature measurements. The mathematical model is governed by a heat conduction equation in which the parameter to determine is denoted by an unknown coefficient of first-order derivative. Firstly, the uniqueness of the solution is proved by using the maximum principle. Then, on the basis of optimal control framework, the inverse problem is transformed into an optimization problem. To ensure the well-posedness of the considered problem, we prove the existence and derive the necessary condition of the minimizer. By carefully analysing the necessary condition, we obtain the local uniqueness and stability on the basis of some a-priori estimates of forward problem. Finally, some numerical implements are performed to show the validity of the proposed method. [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 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 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: This work deals with an inverse problem of identifying the radiative coefficient of heat conduction equations from discrete measurement data. On the basis of the interpolation technique, we find a new way to reconstruct the unknown coefficient by using the optimization method. The results obtained in the work are interesting and useful and may be applied in a variety of fields. [Copyright &y& Elsevier]

Abstract: This paper studies an inverse problem of identifying the coefficient of parabolic equation when the final observation is given, which has important application in a large fields of applied science. Based on the optimal control framework, the existence and necessary condition of the minimum for the control functional are established. Since the optimal control problem is nonconvex, one may not expect a unique solution. However, in this paper the solution is proved to be locally unique. After the necessary condition is transformed into an elliptic bilateral variational inequality, an algorithm and some numerical experiments are proposed in the paper. The numerical results show that the algorithm designed in this paper is stable and that the coefficient is recovered very well. [Copyright &y& Elsevier]

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]

Abstract: This work investigates the inverse problem of reconstructing a spacewise dependent heat source in a two-dimensional heat conduction equation using a final temperature measurement. Problems of this type have important applications in several fields of applied science. Under certain assumptions, this problem can be transformed into a one-dimensional problem where the heat source only depends on the variable r. However, being different from other one-dimensional inverse heat source problems, there exists singularity on the coefficient of our model, which may make the analysis more difficult, regardless of theoretical or numerical. The inverse problem is reduced to an operator equation of the first kind and the corresponding adjoint operator is deuced. For the two dimensional case, i.e., , theoretical analysis can be done by similar derivation. Based on the landweber regularization framework, an iterative algorithm is proposed to obtain the numerical solution. Some typical numerical examples are presented to show the validity of the inversion method. [Copyright &y& Elsevier]

Abstract: This paper deals with an inverse problem of determining a heat source function in heat conduction equations when the solution is known in a discrete point set. Being different from other ordinary inverse source problems which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable x, but also depends on the time t. On the basis of 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. The convergence of the minimizer as the mesh parameters tend to zero is also proved. The conjugate gradient method is applied to the inverse problem and some typical numerical experiments are performed in the paper. The numerical results show that the proposed method is stable and the unknown heat source is recovered very well. [Copyright &y& Elsevier]