A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems

Pornsawad, Pornsarp; Sapsakul, Nantawan; Böckmann, Christine

doi:10.3390/math7050419

Open AccessArticle

A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems

by

Pornsarp Pornsawad

^1,2,*,

Nantawan Sapsakul

^1,2 and

Christine Böckmann

³

¹

Department of Mathematics, Faculty of Science, Silpakorn University, 6 Rachamakka Nai Rd., Nakhon Pathom 73000, Thailand

²

Centre of Excellence in Mathematics, Mahidol University, Rama 6 Rd., Bangkok 10400, Thailand

³

Institut für Mathematik, Universität Potsdam, Karl-Liebknecht-Str. 24-25, D-14476 Potsdam OT Golm, Germany

^*

Author to whom correspondence should be addressed.

Mathematics 2019, 7(5), 419; https://doi.org/10.3390/math7050419

Submission received: 19 March 2019 / Revised: 25 April 2019 / Accepted: 30 April 2019 / Published: 10 May 2019

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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Abstract

In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of

∥ F (x^{δ} (T)) - y^{δ} ∥ = τ δ^{+}

for some

δ^{+} > δ

, and an appropriate source condition. We yield the optimal rate of convergence.

Keywords: nonlinear operator; regularization; discrepancy principle; asymptotic method; optimal rate

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MDPI and ACS Style

Pornsawad, P.; Sapsakul, N.; Böckmann, C. A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems. Mathematics 2019, 7, 419. https://doi.org/10.3390/math7050419

AMA Style

Pornsawad P, Sapsakul N, Böckmann C. A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems. Mathematics. 2019; 7(5):419. https://doi.org/10.3390/math7050419

Chicago/Turabian Style

Pornsawad, Pornsarp, Nantawan Sapsakul, and Christine Böckmann. 2019. "A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems" Mathematics 7, no. 5: 419. https://doi.org/10.3390/math7050419

APA Style

Pornsawad, P., Sapsakul, N., & Böckmann, C. (2019). A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems. Mathematics, 7(5), 419. https://doi.org/10.3390/math7050419

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A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems

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