A method for approximating the solution of self-adjoint parabolic equations backwards in time is presented. The technique uses the Green's function of the parabolic equation to formulate the problem as a first-kind Fredholm integral equation, and then does a regularized least-squares solution of the resulting discrete system. Error estimates and examples are presented. Direct knowledge of the kernel of the differential operator is not required.
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Ames et al. (1997) studied this question.
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