BIRS Workshop Lecture Videos
Regularization of backwards diffusion by fractional time derivatives Kaltenbacher, Barbara
The backwards heat equation is one of the classical inverse problems, related to a wide range of applications and exponentially ill-posed. One of the first and maybe most intuitive approaches to its stable numerical solution was that of quasireversibility, whereby the parabolic operator is replaced by a differential operator for which the backwards problem in time is well posed. After a short overview of approaches in this vein, we will dwell on a new one that relies on replacement of the first time derivative in the PDE by a fractional differential operator, which, due to the asymptotic properties of the Mittag-Leffler function as compared to the exponential function, leads to an only moderately ill-posed problem. Thus the order alpha of (fractional) differentiation acts as a regularization parameter and convergence takes place in the limit as alpha tends to one. We study the regularizing properties of this approach and a regularization parameter choice by the discrepancy principle. Additionally, a substantial numerical improvement can be achieved by exploiting the linearity of the problem by breaking the inversion into distinct frequency bands and using a different fractional order for each. This is joint work with William Rundell.
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