Data assimilation in heat conduction

In this thesis we study data assimilation in heat conduction of reconstructing the initial condition in a heat transfer process from either 1) the observation of the temperature at the final time moment, or 2) interior integral observations which are regarded as interior measurements, or 3) boundary observations. The first problem is new in the sense that the coefficients of the equation describing the heat transfer process are depending on time, and up to now there are very few studies devoted to it. The second problem is a new setting for such kind of problems in data assimilation: interior observations are important, but related studies are devoted to the case of point wise observations which are not realistic in practice. The use of integral observations is more practical. The third problem is very hard as the observation is only on the boundary and up to now there have been very few studies for this case. We reformulate these problems as a variational problem aiming at minimizing a misfit functional in the least squares sense. We prove that the functional is Frechet differentiable and derive a formula for its gradient via an adjoint problem and as a by-product of the method, we propose a very natural and easy method for estimating the degree of ill-posedness of the reconst met ion problem. For numerically solving the problems, we discretize the direct and adjoint problems by the splitting finite difference method for getting the gradient of the discretized variational problems and then apply the conjugate gradient method for solving them. We note that since the solutions in the thesis are understood in the weak sense, the finite difference method for them is not trivial. With respect to the discretization in space variables we prove convergence results for the discretization methods. We test our method on computer for various numerical examples to show the efficiency of our approach.