American Institute of Mathematical Sciences

April  2019, 13(2): 401-430. doi: 10.3934/ipi.2019020

Regularization of a backwards parabolic equation by fractional operators

 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

Received  June 2018 Revised  September 2018 Published  January 2019

Fund Project: 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

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.

Citation: Barbara Kaltenbacher, William Rundell. Regularization of a backwards parabolic equation by fractional operators. Inverse Problems & Imaging, 2019, 13 (2) : 401-430. doi: 10.3934/ipi.2019020
References:

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References:
Amplification factor $A(\lambda_k, \alpha)$
Reconstructions from single and double split frequency method
Reconstructions from SVD and double split frequency method
Reconstructions from SVD as well as double and triple split frequency method
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