American Institute of Mathematical Sciences

August  2021, 15(4): 599-618. doi: 10.3934/ipi.2021006

A regularization operator for source identification for elliptic PDEs

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

* Corresponding author: B. F. Nielsen

Received  August 2020 Revised  October 2020 Published  August 2021 Early access  December 2020

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

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.

Citation: Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, 2021, 15 (4) : 599-618. doi: 10.3934/ipi.2021006
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References:
Comparison of the true source and the inverse solution using standard Tikhonov regularization with $\alpha = 10^{-3}$
">Figure 2.  Recovered source, Example 1, with the regularization parameter $\alpha = 10^{-3}$. The true source is depicted in panel (a) in Figure 1
L-shaped domain, Example 2. Comparison of the true source and the inverse solutions, using the regularization parameter $\alpha = 10^{-3}$
Source at the boundary, Example 3. Comparison of the true source and the inverse solutions, using the regularization parameter $\alpha = 10^{-4}$
Source at the boundary, Example 3. Inverse solution computed with standard Tikhonov regularization, $\alpha = 10^{-4}$
Vector field of $\sigma$
State equation with a tensor, Example 4. Comparison of the true source and the inverse solutions, using the regularization parameter $\alpha = 10^{-4}$
Two disjoint sources, Example 5. Comparison of the true sources and the inverse solutions, using the regularization parameter $\alpha = 10^{-3}$
Three disjoint sources, Example 5. Comparison of the true sources and the inverse solutions, using the regularization parameter $\alpha = 10^{-3}$
">Figure 10.  Example 6, $5 \%$ and $20\%$ noise. The true source is shown in panel (a) in Figure 8
">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
">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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