Error Estimates for Arnoldo-Tikhonov Regularization for Ill-Posed Operator Equations

2018/12/04 by Ronny Ramlau, Ramlau, Ronny, Lothar Reichel +1 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1812.01560

openalex publication_date 2018/12/04 · openalex created_date 2018/12/11 · openalex updated_date 2026/07/28

Abstract

Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by discretization. This paper discusses the influence of the discretization error on the computed solution. We consider the situation when the discretization used yields an algebraic linear system of equations with a large matrix. An approximate solution of this system is computed by first determining a reduced system of fairly small size by carrying out a few steps of the Arnoldi process. Tikhonov regularization is applied to the reduced problem and the regularization parameter is determined by the discrepancy principle. Errors incurred in each step of the solution process are discussed. Computed examples illustrate the error bounds derived.

Cited by

• Randomized Krylov-Projected Iterated Tikhonov Regularization for Large-Scale Ill-posed Problems Under A Posteriori Stopping Rule
• The iterated Golub-Kahan-Tikhonov method

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• Embedded techniques for choosing the parameter in Tikhonov regularization
• Analysis of the discrepancy principle for Tikhonov regularization under low order source conditions
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