Identifying The Lower Coefficient Of Second Order Parabolic Equation By Using The Optimization Method

The inverse problem for partial differential equation (PDE) is a new research field. Usually, the direct problem for PDE studies how to determine the solution from a given equation and corresponding initial-boundary condition. The inverse problem, which is different from the direct problem, deals with identification of some unknown quantities in the problem by using some information of its solution, i. e., some coefficients in the equation, the region, or some conditions for determining the solution. Most of inverse problems are ill-posed, which is a main difficulty for the research. A problem is called as well-posed when its solution exists, and is unique, and depends on the input data continuously. Otherwise, it is called as ill-posed. Since the data for determining the solution is often obtained by experiments, it is hard to avoid errors. Hence, when a problem is ill-posed, a small perturbation in the input data may result in a big change in the solution. In such case, the obtained solution will be meaningless. The key point for solving an inverse problem is to look for a well-posed condition, i.e., the original problem is well-posed under the new condition.This paper studies an inverse problem of identifying the lower coefficient of a second order parabolic equation when a terminal condition is known. This kind of inverse problem is of great importance in a large fields of applied' science such as finance and physics. According to the difference with the dimension of the unknown coefficient q, there are two kinds of problems denoted as problem P1 and P2 respectively. In problem P1, the unknown coefficient q is independent with the time variable t, but only depends on the space variable x, i.e., q=q(x). In problem P2, the unknown coefficient q not only depends on the space variable x, but also depends on the time variable t, i.e., q=q(x, t). Problem P2 is often called as evolutional type inverse problem. In this paper, the two kinds of problems are analyze thoroughly by using the optimization method.The manuscript is divided into three parts.Chapter 1 concisely introduces problem P1, P2 and correlated research background.Chapter 2 mainly discusses problem P1 from theoretical analysis angle. The strong ill-posedness is analyze in Section 2.1. In Section 2.2, problem P1 is transformed into an optimal control problem, and a control functional is also constructed. In Section 2.3, we discuss the existence of the minimum for the control problem and its approximating problem. The necessary conditions with which the minimums of the control problem and its approximating problem must satisfy are deduced in Section 2. 4. In Section 2. 5, we prove that the solution of the approximating problem must converge to the solution of the original problem. Since the control problem is non-convex, in general case, one may not expect a unique solution. In the last section, we prove the minimum is local unique under the assumption that T is small enough.Chapter 3 mainly deals with problem P2 from theoretical analysis angle. In Section 3. 1, the strong ill-posedness of problem P2 is discussed. Being different from problem P1, it is difficult to conquer the non-stability for numerical computation when one attempt to recover the unknown coefficient q(x, t) by using extra condition directly. In Section 3. 2, the inverse problem of recovering the coefficient q(x, t) is transformed to an issue of recovering functions q_n, n=1, 2,…, N step by step, and the necessary condition with which q_n must satisfy is also established. Function q_n is defined as the optimal control element(OCE). By using the definition of q_n, we also define the discrete OCE q~h(x, t). In Section 3. 3, certain estimates of q~h(x, t) are established which are uniformly bounded, independent of time step length h. In Section 3. 4, we prove that q~h(x, t) must converge to the limiting OCE q(x, t) as h→0. The necessary condition of q(x, t) is also obtained. In the last section of Chapter 3, we prove that q(x, t) is unique in the sense of L~2 space and depends on the condition for determining solution continuously.