Inverse Boundary Value Problems For Parabolic Equations With Numerical Solutions

The diffusion processes are quite popular in many engineering areas and also social sciences. Mathematically, the diffusion processes can be modeled by the initial boundary value problems for parabolic equations. In practical situations, not all the boundary values can be specified by direct measurements,and therefore some extra measurable data related to the diffusion system are required for reconstructing the unknown boundary ingredients for the completely determination of the diffusion process. Such problems are called inverse boundary value problems for diffusion equations.The boundary ingredients for the diffusion system mainly include the boundary shape of the interested domain in the higher dimensional cases and different kinds of boundary conditions such as Dirichlet boundary condition (DBC), Neumann boundary condition (N-BC) and Robin boundary condition (R.BC). Although the inverse problems for the diffusion system with DBC and NBC have been studied thoroughly, both the theoretical studies and the numerical implementations for the inverse problems with RBC are still in the initial stages of the researches. Physically, the Robin coefficients represent the exchange ability of the concentrations of related physical quantity such as contaminations on the medium boundary and therefore are of great importance in engineering for the descriptions of the diffusion process. Since the reconstruction of Robin coefficient is nonlinear, the correspond-ing inverse problems are much more difficult theoretically and numerically. Due to these reasons, the studies on the reconstruction of Robin coefficients are not too much, especially for the reconstructions of discontinuous Robin coefficients. We mainly consider the recon-structions of Robin impedance coefficients and the related inverse problems for diffusion equations in our dissertation. More precisely, our researches in this thesis are stated as follows.Firstly, we consider the reconstruction of the Robin impedance coefficient of a heat conduction system in a two-dimensional spatial domain from the time-average measurement specified on the boundary. In many physical situations, due to the severe restrictions on measuring equipments and technologies, it is very difficult to obtain the related physical fields in the point-wise sense. Alternatively,we overcome these experimental difficulties by specifying some integral measurements which is nonlocal. Such kinds of inversion inputs are physically related to some average data by repeating measurements. Due to the mathemat-ical importance and the physical flexibility of such kinds of models, many researchers begin to consider these inverse problems, which are also a part of our researches. By using the potential representation of the solution for direct problems, this nonlinear inverse problem is transformed into an ill-posed integral system coupling the unknown density function for potential and the unknown boundary impedance. The uniqueness as well as the conditional stability of this inverse problem is established from this coupled system. For regularization technique, we propose to find the boundary impedance by solving a non-convex regulariz-ing optimization problem. The well-posedness of this optimization problem together with the convergence property of the minimizer is analyzed. Compared with the existing works,we firstly established the conditional stability of this inverse problem and constructed the two-parameter cost functional based on the density function and Robin coefficient which is different from the classical cost functional. Moreover, we give the numerical realizations.Our alternative iterations are more efficient for numerical implementations, compared with the nonlinear conjugate method which have been used to this inverse problem widely.Secondly, we consider the heat conduction process with heat flux exchanges on the boundary in a bounded 2-dimensional spatial domain. The aim is to identify both the Robin coefficient and the initial distribution simultaneously from the final measurement data of the heat field. There is still not so much research works on the simultaneous reconstructions of the Robin coefficient and the initial value. We firstly establish the uniqueness for this nonlinear inverse problem for the positive exchange coefficient and initial heat distribution,by applying the maximal principle and the eigenfunction expansions for the solution of the direct problem. Then a regularizing scheme combining the data mollification technique and quasi-reversibility method along time direction together is proposed to recover two unknowns, with the choice strategies for the regularizing parameters and the error estimates on the regularizing solutions. The basic idea of dealing with this inverse problem is to recover the Robin coefficient firstly which is a nonlinear problem and then to identify the initial value with known Robin coefficient. However, since the Robin coefficient is from reconstruction process, its unavoidable error will contaminate the recovery of initial value nonlinearly. We must optimize the error for Robin coefficient reconstruction and then analyze the error propagation for recovering initial value, especially when we apply the noisy input data. The reconstruction implementations are carried out in terms of the potential representation of the heat field, with numerical examples showing the validity of the proposed scheme. The quantitative analysis on the error propagation and the efficient reconstruction scheme for initial value axe the main contributions of in this part.Finally, we consider the reconstruction of non-smooth Robin coefficient for 1-dimension-al diffusion model. By now the studies on t he reconstructions of nonsmooth Robin coefficient are still rare. The engineering backgrounds on this inverse problem are as follows. When electronic devices are in operation, the sharp change of the temperature on devices sur-face can be considered as an indicator of devices faults. This heat transfer process can be modeled by a 1-dimensional parabolic system with the boundary heat dissipation (Robin coefficient) varying with time. Since the Dirichlet data can be measured at one end of the 1-dimensional medimn,while the boundary heat dissilpation (Robin coefficiert) to be recon-structed is defined the other end of the medium,we consider to detect, the nonsmooth heat dissipat ion coefficient from the Dirichlet data specified at one end. We establish the uniqueness in the admissible set of L2 for the nonsmooth dissipation coefficient and then prove the convergence property of the minimizer of the regularizing cost functional for the inverse problem theoretically. Then a double-iteration schheme minimizing the data-match term and the regularization term alternatively is proposed to implement the reconstruction process.The uniqueness in L2 sense is the main contributions of our work in theoretical as-pect. As for the numerical implementations, the basic idea of the double-iteration scheme proposed is to optimize the data matching term and to optimize the regularizing term alter-natively. Such a double recursion strategy avoids the choice of the regularizing parameter,and generates the iterative solution efficiently compared with optimizing the data match-ing term and regularizing term together. Moreover, this double-iteration scheme can be generalized to deal with inverse problems with multiple nonlinear terms.