Regularization of a backwards parabolic equation by fractional operators

TL;DR

This paper explores regularizing the ill-posed backwards diffusion problem using fractional operators, breaking the inversion into frequency bands with different fractional orders, and analyzing the method with numerical examples.

Contribution

It introduces a novel fractional regularization approach for backwards parabolic equations, utilizing frequency band decomposition and the discrepancy principle for parameter selection.

Findings

The fractional regularization improves stability of the backwards diffusion problem.

Frequency band decomposition enhances the accuracy of the inversion.

Numerical examples demonstrate the effectiveness of the proposed method.

Abstract

The backwards diffusion equation is one of the classical ill-posed inverse problems, related to a wide range of applications, and has been extensively studied over the last 50 years. One of the first methods was that of {\it quasireversibility\/} whereby the parabolic operator is replaced by a differential operator for which the backwards problem in time is well posed. This is in fact the direction we will take but will do so with a nonlocal operator; an equation of fractional order in time for which the backwards problem is known to be ``almost well posed.'' We shall look at various possible options and strategies but our conclusion for the best of these will exploit the linearity of the problem to break the inversion into distinct frequency bands and to use a different fractional order for each. The fractional exponents will be chosen using the discrepancy principle under the assumption we have an estimate of the noise level in the data. An analysis of the method is provided as are some illustrative numerical examples.