A regularization operator for source identification for elliptic PDEs

Ole Løseth Elvetun; Bjørn Fredrik Nielsen

doi:10.3934/ipi.2021006

Article Contents

2021, Volume 15, Issue 4: 599-618. Doi: 10.3934/ipi.2021006

This issue Previous Article Next Article Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem

A regularization operator for source identification for elliptic PDEs

Ole Løseth Elvetun and
Bjørn Fredrik Nielsen^,

Faculty of Science and Technology, Norwegian University of Life Sciences, P.O. Box 5003, NO-1432 Ås, Norway

^* Corresponding author: B. F. Nielsen
^* Corresponding author: B. F. Nielsen

Received: July 31, 2020

Revised: September 30, 2020

Early access: December 2020

Published: August 2021

This work was supported by The Research Council of Norway, project number 239070

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Abstract

We study a source identification problem for a prototypical elliptic PDE from Dirichlet boundary data. This problem is ill-posed, and the involved forward operator has a significant nullspace. Standard Tikhonov regularization yields solutions which approach the minimum $ L^2 $-norm least-squares solution as the regularization parameter tends to zero. We show that this approach 'always' suggests that the unknown local source is very close to the boundary of the domain of the PDE, regardless of the position of the true local source.

We propose an alternative regularization procedure, realized in terms of a novel regularization operator, which is better suited for identifying local sources positioned anywhere in the domain of the PDE. Our approach is motivated by the classical theory for Tikhonov regularization and yields a standard quadratic optimization problem. Since the new methodology is derived for an abstract operator equation, it can be applied to many other source identification problems. This paper contains several numerical experiments and an analysis of the new methodology.

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Mathematics Subject Classification: Primary: 35R30, 47A52; Secondary: 65F22.

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Figure 1. Comparison of the true source and the inverse solution using standard Tikhonov regularization with $ \alpha = 10^{-3} $

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Figure 2. Recovered source, Example 1, with the regularization parameter $ \alpha = 10^{-3} $. The true source is depicted in panel (a) in Figure 1

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Figure 3. L-shaped domain, Example 2. Comparison of the true source and the inverse solutions, using the regularization parameter $ \alpha = 10^{-3} $

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Figure 4. Source at the boundary, Example 3. Comparison of the true source and the inverse solutions, using the regularization parameter $ \alpha = 10^{-4} $

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Figure 5. Source at the boundary, Example 3. Inverse solution computed with standard Tikhonov regularization, $ \alpha = 10^{-4} $

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Figure 6. Vector field of $ \sigma $

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Figure 7. State equation with a tensor, Example 4. Comparison of the true source and the inverse solutions, using the regularization parameter $ \alpha = 10^{-4} $

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Figure 8. Two disjoint sources, Example 5. Comparison of the true sources and the inverse solutions, using the regularization parameter $ \alpha = 10^{-3} $

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Figure 9. Three disjoint sources, Example 5. Comparison of the true sources and the inverse solutions, using the regularization parameter $ \alpha = 10^{-3} $

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Figure 10. Example 6, $ 5 \% $ and $ 20\% $ noise. The true source is shown in panel (a) in Figure 8

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Figure 11. Inhomogeneous Helmholtz equation with $ \epsilon = -1 $. Comparison of the inverse solutions, using the regularization parameter $ \alpha = 10^{-3} $. The true source is displayed in Figure 1a

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Figure 12. Inhomogeneous Helmholtz equation with $ \epsilon = -100 $. Comparison of the inverse solutions, using the regularization parameter $ \alpha = 10^{-3} $. The true source is displayed in Figure 1a

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