Regularization of a backwards parabolic equation by fractional operators

Barbara Kaltenbacher; William Rundell

doi:10.3934/ipi.2019020

Article Contents

2019, Volume 13, Issue 2: 401-430. Doi: 10.3934/ipi.2019020

This issue Previous Article An inverse obstacle problem for the wave equation in a finite time domain Next Article

Regularization of a backwards parabolic equation by fractional operators

Barbara Kaltenbacher^1, , and
William Rundell^2,

1.
Department of Mathematics, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria
2.
Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA

* corresponding author
* corresponding author

Received: June 2018

Revised: September 2018

Published: April 2019

The work of the first author was supported by the Austrian Science Fund FWF under the grants I2271 and P30054 as well as partially by the Karl Popper Kolleg "Modeling-Simulation-Optimization", funded by the Alpen-Adria-Universität Klagenfurt and by the Carinthian Economic Promotion Fund (KWF)
The work of the second author was supported in part by the National Science Foundation through award DMS-1620138.

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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 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.

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Mathematics Subject Classification: Primary: 35R30, 65M32; Secondary: 35R11.

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Figure 1. Amplification factor $ A(\lambda_k, \alpha) $

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Figure 2. Reconstructions from single and double split frequency method

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Figure 3. Reconstructions from SVD and double split frequency method

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Figure 4. Reconstructions from SVD as well as double and triple split frequency method

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