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A variational regularization method for solving the non-characteristic Cauchy problem in multiple dimensions

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Abstract

In this manuscript, we consider the non-characteristic Cauchy problem, which is naturally a general form including inverse heat conduction problem and sideways parabolic equation. This problem is a severely ill-posed problem, that is, the solution (if it exists) does not depend continuously on the data. We introduce a simple and powerful variational regularization method which links to the regularization of Truncated Fourier operators pseudo-inverses. Under smoothness condition with p-norm, we deduce the error estimation, which can be expressed by Hölder type, between the exact solution and regularized approximation in the practical case where the data is noisy. Meanwhile, we propose an a posteriori parameter choice rule based on the Morozov principle and obtain the error estimation. At last, two numerical examples are given to illustrate the efficiency of the proposed method.

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Acknowledgements

The authors thank the editors and reviewers for their constructive comments on an earlier version of this manuscript. The research was supported by funds for Innovative Fundamental Research Group Project of Gansu Province.

Funding

Funds for Innovative Fundamental Research Group Project of Gansu Province (Grant No. 23JRRA684).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, Gansu, China

    Xiangtuan Xiong & Jingjing Han

  2. Basic Disciplines of Mathematics and Statistics, Gansu Provincial Research Center, Lanzhou, 730070, Gansu, China

    Xiangtuan Xiong

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  1. Xiangtuan Xiong
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X.T. Xiong and J.J. Han wrote jointly the manuscript and prepared all figures (tables). All authors reviewed the manuscript.

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Correspondence to Xiangtuan Xiong.

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Xiong, X., Han, J. A variational regularization method for solving the non-characteristic Cauchy problem in multiple dimensions. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01805-z

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