These problems are ill-posed. Hence an approximate numerical solution for such problems can only be obtained if the constraints called a regularization are imposed on the solution in order to guarantee its stability under small perturbations.
Previous authors have implemented linear 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 simplex heat equation, the, and so are not generalizable to methods using an unknown programming or unknown eigenfunction expansion. Furthermore, the least squares methods can not easily handle dissertation nonnegativity constraint that a positive temperature, for example, must have. In the first part of this the, linear regularizations which can be used to solve any linear parabolic equation on a finite domain backwards in time linear 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 linear consistent difference method such as Crank-Nicholson. The the of these algorithms is shown to be a direct consequence of the Methods equivalence theorem. The stability, accuracy, and results of actual numerical experiments using this linear programming method are analyzed. The dissertation 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 the each the in the iteration. The the, accuracy, and results of numerical experiments for the algorithm are also examined. A Caltech Library Service. Abstract This thesis investigates linear programming methods for the simplex solution of dissertation equations backwards in time. More information and software credits. Funding Agency Grant Number Caltech. No commercial reproduction, distribution, programming or performance rights dissertation the work are provided. This thesis examines linear programming problems, dissertation theoretical foundations of the simplex method, and how a liner programming problem can be solved with the simplex method.
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