Liu Yang

This work deals with an inverse problem of identifying the refractive index in terrain parabolic equations, which has important applications in radio systems and other applied scientific fields. Unlike other inverse coefficient problems arising in the real field, the underlying mathematical model is discussed in the complex field which may lead to many mathematical tools not available, and the corresponding theoretical analysis rather difficult. Based on the optimal control framework, the recognition problem is transformed into an optimization problem, and the existence and necessary condition of the minimizer for the cost functional are deduced. The local uniqueness and stability of the minimizer are derived from the necessary condition. Finally, a gradient descent iterative scheme is designed to solve the numerical solution of the inverse problem. The validity of the proposed algorithm is illustrated by some typical numerical examples.

In this paper, we study the well-posedness of the solution of an optimal control problem related to a multi-parameters identification problem in degenerate parabolic equations. Problems of this type have important applications in several fields of applied science. Unlike other inverse coefficient problems for classical parabolic equations, the mathematical model discussed in the paper is degenerate on both lateral boundaries of the domain. Moreover, the status of the two unknown coefficients are different, namely that the reconstruction of the source term is mildly ill-posed, while the inverse initial value problem is severely ill-posed. On the basis of optimal control framework, the problem is transformed into an optimization problem. The existence of the minimizer is proved and the necessary conditions which must be satisfied by the minimizer are also established. Due to the difference between ill-posedness degrees of the two unknown coefficients, the extensively used conjugate theory for parabolic equations cannot be directly applied for our problem. By carefully analyzing the necessary conditions and the direct problem, the uniqueness, stability and convergence of the minimizer are obtained. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations with degenerate coefficients.

A semi-infinite inverse source problem in heat conduction equations is considered, where the source term is assumed to be compactly supported in the region. After introducing a suitable artificial boundary, the semi-infinite problem is transformed into a bounded one and the corresponding exact expression of the boundary condition is derived. Then we rigorously prove the uniqueness of the solution of original problem, together with the stability of the corresponding optimal control solution.

Drift rate is a very important parameter in the evolution of stock price, which has significant impact on the corresponding option pricing. This paper deals with an inverse problem of recovering the drift function by current market prices of options. Different from the usual inverse volatility problem, our mathematical model does not tend to zero at infinity, which may bring great trouble to theoretical analysis and numerical calculation. To overcome this difficulty, we use an artificial boundary and homogenization technique to transform the original problem into a homogeneous initial boundary value problem on a bounded domain. Then, based on the optimal control framework, we construct the corresponding optimization problem and strictly prove the well-posedness of the minimizer. Finally, we design an iterative algorithm to obtain the numerical solution. We give some typical examples to verify the validity of our method.

In this paper, we study the inverse problem of identifying the initial value of a two-dimensional degenerate parabolic equation, which often appears in the fields of engineering, physics, and computer image processing. Firstly, the difference scheme of forward problem is established by using the finite volume method. Then stability and convergence of the difference equations are proved rigorously. Finally, the Landweber iteration and conjugate gradient method are used to solve the inverse problem, and some typical numerical examples are shown to verify the validity of our iterative algorithm. Numerical results show that the algorithm is stable and efficient.