Fictitious null spaces for improving the solution of injective inverse problems

TL;DR

The paper addresses ill-posed injective inverse problems by exploiting a fictitious null space created by small singular values of the forward operator. It introduces a weighting framework via $W_k$ and a thresholded projection $P_k$ to bias-correct regularization, develops weighted sparsity and weighted Tikhonov formulations, and proves convergence properties under standard source conditions. The approach yields unbiased recovery of basis components and reduces reconstruction bias, with strong empirical improvements in three PDE-driven problems (inverse heat conduction, Cauchy problem for Laplace, and linearized EIT). This method offers a practical path to improved, more faithful reconstructions in injective, highly ill-posed settings, particularly when data are limited or noisy.

Abstract

For linear ill-posed problems with nontrivial null spaces, Tikhonov regularization and truncated singular value decomposition (TSVD) typically yield solutions that are close to the minimum norm solution. Such a bias is not always desirable, and we have therefore in a series of papers developed a weighting procedure which produces solutions with a different and controlled bias. This methodology can also conveniently be invoked when sparsity regularization is employed. The purpose of the present work is to study the potential use of this weighting applied to injective operators. The image under a compact operator of the singular vectors/functions associated with very small singular values will be almost zero. Consequently, one may regard these singular vectors/functions to constitute a basis for a fictitious null space that allows us to mimic the previous weighting procedure. It turns out that this regularization by weighting can improve the solution of injective inverse problems compared with more traditional approaches. We present some analysis of this methodology and exemplify it numerically, using sparsity regularization, for three PDE-driven inverse problems: the inverse heat conduction problem, the Cauchy problem for Laplace's equation, and the (linearized) Electrical Impedance Tomography problem with experimental data.

Figures (10)

• Figure 1: The weights \ref{['A6.01']} associated with the basis functions for two levels of noise. The criterion \ref{['eq:select_k']} was used to determine the size of the truncation parameter $k$.
• Figure 2: Inverse heat conduction example: $1 \permil$ noise and Morozov's discrepancy principle.
• Figure 3: Inverse heat conduction example: $1 \%$ noise and Morozov's discrepancy principle.
• Figure 4: The annulus-shaped domain $\Omega$ for the Cauchy problem. The blue region is the subdomain $\Omega_e$ in which data is recorded.
• Figure 5: The weights \ref{['A6.01']} associated with the basis functions, employed to discretize the forward operator \ref{['eq:forward_Cauchy']}, for two different levels of noise.

Theorems & Definitions (15)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Remark 3.4
• Proposition 3.5
• proof
• Proposition 3.6