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A Modified Steepest Descent Scheme for Solving a Class of Parameter Identification Problems

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Abstract

The parameter identification problems, a class of nonlinear inverse problems, associated with partial differential equations are of utmost importance in real-life applications. Generally, the nonlinear inverse problem can be solved through many standard techniques. But the complexity of the problem increases when the problem is ill-posed and, due to the necessity of computing the Fréchet derivative in deriving the solution. Beyond this, many assumptions are required to establish the convergence and convergence rate of the solution. The steepest descent method employed in literature to solve nonlinear problems is not bolted away from these kinds of limitations. In this paper, we propose a modified steepest descent iterative scheme for parameter identification problems that circumvents these limitations. We follow an operator theory approach and establish the convergence and convergent rate of the scheme with very minimal assumptions. The theoretical results are illustrated through numerical examples.

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Acknowledgements

We profoundly thank the unknown referee(s) for their careful reading of the manuscript and valuable suggestions that significantly improved the presentation of the paper as well.

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This is to declare that this research received no specific grant from any funding agency in the public, commercial or not-for-profit sector.

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  1. School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, Vithura, Thiruvanathapuram, 695551, India

    M. P. Rajan & Niloopher Salam

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  1. M. P. Rajan
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  2. Niloopher Salam
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Correspondence to M. P. Rajan.

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Rajan, M.P., Salam, N. A Modified Steepest Descent Scheme for Solving a Class of Parameter Identification Problems. Results Math 78, 235 (2023). https://doi.org/10.1007/s00025-023-02014-1

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