Volume 57, pp. 67-79, 2022.

The Levenberg–Marquardt regularization for the backward heat equation with fractional derivative

Pornsarp Pornsawad, Christine Böckmann, and Wannapa Panitsupakamon


The backward heat problem with time-fractional derivative in Caputo's sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg–Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a Hölder-type source condition. Numerical examples for one and two dimensions are provided.

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Key words

ill-posed problems, time-fractional derivative, backward heat problem, Levenberg–Marquardt method, a posteriori stopping rule, optimal order

AMS subject classifications

26A33, 47A52, 65R30, 65M30

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