[Paper Review] Modified Tikhonov regularization for identifying several sources

Ole Løseth Elvetun, Bjørn Fredrik Nielsen|arXiv (Cornell University)|Nov 9, 2020

Numerical methods in inverse problems15 references4 citations

TL;DR

This paper proposes a modified Tikhonov regularization framework to identify multiple local sources from Dirichlet boundary data in elliptic PDEs, using a problem-specific regularization operator W that enhances source distinguishability. The method enables precise recovery of single sources and improves localization of multiple sources through a three-stage algorithm, with theoretical and numerical evidence showing that identifiability depends critically on domain geometry and source position relative to the boundary.

ABSTRACT

We study whether a modified version of Tikhonov regularization can be used to identify several local sources from Dirichlet boundary data for a prototypical elliptic PDE. This paper extends the results presented in [5]. It turns out that the possibility of distinguishing between two, or more, sources depends on the smoothing properties of a second or fourth order PDE. Consequently, the geometry of the involved domain, as well as the position of the sources relative to the boundary of this domain, determines the identifiability. We also present a uniqueness result for the identification of a single local source. This result is derived in terms of an abstract operator framework and is therefore not only applicable to the model problem studied in this paper. Our schemes yield quadratic optimization problems and can thus be solved with standard software tools. In addition to a theoretical investigation, this paper also contains several numerical experiments.

Motivation & Objective

• To extend previous results on single-source identification to the case of multiple local sources in elliptic PDE inverse problems.
• To establish theoretical conditions under which multiple sources can be distinguished from boundary measurements, particularly based on the PDE's order and domain geometry.
• To develop a numerical algorithm that improves source localization by combining Tikhonov regularization with post-processing via radius optimization.
• To prove a uniqueness result for single-source identification in an abstract operator framework applicable beyond the specific PDE model.
• To demonstrate through numerical experiments that source recovery performance depends significantly on domain shape and source location relative to the boundary.

Proposed method

• A modified Tikhonov regularization approach is formulated as a quadratic optimization problem: minimize the data misfit in L²(∂Ω) plus a regularization term involving a linear operator W on the source f.
• The regularization operator W is defined such that Wϕi = ||Pϕi||X ϕi for each basis function ϕi, where P is the orthogonal projection onto the range of the forward operator's adjoint.
• The method uses a finite-dimensional subspace Fh of L²(Ω) to represent the source, with basis functions having local support to enhance spatial resolution.
• A three-stage algorithm is proposed: first, solve the modified Tikhonov problem to obtain an initial estimate; second, locate local maxima of the solution; third, refine the source geometry by optimizing over radii of circular sources centered at these maxima.
• For the case ε=0 (Poisson’s equation), the source is modified by subtracting its mean to satisfy the compatibility condition ∫Ω f dx = 0, ensuring well-posedness of the forward problem.
• The forward operator K_h maps a source f to the Dirichlet trace of the solution u on ∂Ω, with the solution normalized to have zero mean on the boundary to handle the non-uniqueness due to constant solutions.

Experimental results

Research questions

• RQ1 Can a modified Tikhonov regularization scheme distinguish between multiple local sources based on Dirichlet boundary data?
• RQ2 How does the order of the PDE (second vs. fourth order) affect the identifiability of multiple sources?
• RQ3 To what extent does the geometry of the domain and the relative position of sources to the boundary influence source recovery performance?
• RQ4 Can the proposed three-stage algorithm improve source localization compared to standard Tikhonov regularization?
• RQ5 Under what conditions is the minimum norm least squares solution equivalent to the true source in the inverse source problem?

Key findings

• The modified Tikhonov regularization with the proposed W operator enables precise recovery of a single local source, as proven via an abstract operator framework.
• The ability to distinguish multiple sources depends critically on the smoothing properties of the PDE, with fourth-order problems offering better source resolution than second-order ones.
• Numerical experiments show that non-convex domains can yield better source recovery than convex domains of similar size, particularly when sources are located near corners or edges.
• The three-stage algorithm—initial Tikhonov solve, localization of maxima, and radii optimization—significantly improves source reconstruction, especially for small or weak sources.
• For Poisson’s equation (ε=0), the method remains valid after subtracting the mean of the source, and the minimum norm solution automatically satisfies ∫Ω f dx = 0.
• Consistent localization across Method I and the radii optimization method is a key indicator of reliability, suggesting that Method I should be used first for initial source location.