A Stable Marching Scheme for an Ill-Posed Initial Value Problem
Alfred S. Carasso · Birkhäuser Basel eBooks · 1983
We develop and analyze a marching procedure for the numerical computation of backwards parabolic equations with variable coefficients and noisy initial data. The scheme is stable (but inconsistent) and leads to error bounds of logarithmic convexity type for t bounded away from the line t = T, where the solution is only of class L2. The scheme is a two step procedure where the solution is appropriately filtered in the frequency domain, at every alternate step. The procedure assumes a constraint on the class of solutions which is stronger than the usual L2 bound at t = T. This stronger constraint is equivalent to the usual constraint in the constant coefficient case.