This thesis investigates linear programming methods for the numerical solution of parabolic equations backwards in time. These problems are ill-posed. Hence an approximate numerical solution for such problems can only be obtained if additional constraints (called a regularization) are imposed on the solution in order to guarantee its stability under small perturbations. Previous authors have implemented regularizations on the backward heat equation which used (linear or nonlinear) least squares, or linear programming. These regularizations use the exact form of the kernel for the heat equation, however, and so are not generalizable to problems with an unknown kernel or unknown eigenfunction expansion. Furthermore, the least squares methods can not easily handle the nonnegativity constraint that a positive temperature, for example, must have.
In the first part of this thesis, linear regularizations which can be used to solve any linear parabolic equation on a finite domain backwards in time are introduced. It is then shown how a numerical approximation to the solution of the regularized problem can be obtained by using linear programming and any stable and consistent difference method (such as Crank-Nicholson). The convergence of these algorithms is shown to be a direct consequence of the Lax equivalence theorem. The stability, accuracy, and results of actual numerical experiments using this linear programming method are analyzed.
The second part of this thesis shows how these regularizations can be used on weakly nonlinear equations. This is done by introducing a successive approximation method, and solving a linear program at each step in the iteration. The stability, accuracy, and results of numerical experiments for this algorithm are also examined.
