arXiv subjects

Ole Løseth Elvetun

Publications and source records attributed to Ole Løseth Elvetun.

11 recordsLinked to original sources

arXiv2026

Directionally Weighted Total Variation for Inverse Problems

We study weighted total variation (TV) regularization for inverse problems in which the forward operator has a large null space, a setting in which standard (unweighted) TV is known to produce systematic reconstruction artifacts such as spatial bias. To address this issue, we consider spatially varying weights derived from a sensitivity analysis of the forward operator expression via an associated Green's function. These weights rebalance the TV penalty and thereby compensate for the inhomogeneous sensitivity of the forward operator. We introduce and analyze two related models: an isotropic weighted TV functional, where the weight reflects worst-case directional sensitivity, and a directionally weighted refinement in which the weighting depends on the actual jump direction of the solution. In addition to the standard Tikhonov formulation, we also study corresponding basis-pursuit problems. We derive optimality conditions and prove exact recovery results under suitable structural assumptions. Furthermore, we investigate how recoverability depends on the geometry of the solution and its distance to the observation boundary. Numerical experiments for inverse source problems and related applications demonstrate that the proposed weighted formulations significantly reduce artifacts present in unweighted TV reconstructions, yielding improved localization and size recovery. These results highlight the crucial role of spatial weighting in TV regularization for inverse problems with large null spaces.

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arXiv2026

Weighted total variation regularization for inverse problems with significant null spaces

We consider inverse problems with large null spaces, which arise in important applications such as in inverse ECG and EEG procedures. Standard regularization methods typically produce solutions in or near the orthogonal complement of the forward operator's null space. This often leads to inadequate results, where internal sources are mistakenly interpreted as being near the data acquisition sites -- e.g., near or at the body surface in connection with EEG and ECG recordings. To mitigate this, we previously proposed weighting schemes for Tikhonov and sparsity regularization. Here, we extend this approach to total variation (TV) regularization, which is particularly suited for identifying spatially extended regions with approximately constant values. We introduce a weighted TV-regularization method, provide supporting analysis, and demonstrate its performance through numerical experiments. Unlike standard TV regularization, the weighted version successfully recovers the location and size of large, piecewise constant sources away from the boundary, though not their exact shape. Additionally, we explore a hybrid weighted-sparsity and TV regularization approach, which better captures both small and large sources, albeit with somewhat more blurred reconstructions than the weighted TV method alone.

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arXiv2025

Weighted Group Lasso for a static EEG problem

We investigate the weighted Group Lasso formulation for the static inverse electroencephalography (EEG) problem, aiming at reconstructing the unknown underlying neuronal sources from voltage measurements on the scalp. By modelling the three orthogonal dipole components at each location as a single coherent group, we demonstrate that depth bias and orientation bias can be effectively mitigated through the proposed regularization framework. On the theoretical front, we provide concise recovery guarantees for both single and multiple group sources. Our numerical experiments highlight that while theoretical bounds hold for a broad range of weight definitions, the practical reconstruction quality, for cases not covered by the theory, depends significantly on the specific weighting strategy employed. Specifically, employing a truncated Moore-Penrose pseudoinverse for the involved weighting matrix gives a small Dipole Localization Error (DLE). The proposed method offers a robust approach for inverse EEG problems, enabling improved spatial accuracy and a more physiologically realistic reconstruction of neural activity.

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arXiv2025

Weighting operators for sparsity regularization

Standard regularization methods typically favor solutions which are in, or close to, the orthogonal complement of the null space of the forward operator/matrix $\mathsf{A}$. This particular biasedness might not be desirable in applications and can lead to severe challenges when $\mathsf{A}$ is non-injective. We have therefore, in a series of papers, investigated how to "remedy" this fact, relative to a chosen basis and in a certain mathematical sense: Based on a weighting procedure, it turns out that it is possible to modify both Tikhonov and sparsity regularization such that each member of the chosen basis can be almost perfectly recovered from their image under $\mathsf{A}$. In particular, we have studied this problem for the task of using boundary data to identify the source term in an elliptic PDE. However, this weighting procedure involves $\mathsf{A}^\dagger \mathsf{A}$, where $\mathsf{A}^\dagger$ denotes the pseudo inverse of $\mathsf{A}$, and can thus be CPU-demanding and lead to undesirable error amplification. We therefore, in this paper, study alternative weighting approaches and prove that some of the recovery results established for the methodology involving $\mathsf{A}$ hold for a broader class of weighting schemes. In fact, it turns out that "any" linear operator $\mathsf{B}$ has an associated proper weighting defined in terms of images under $\mathsf{B}\mathsf{A}$. We also present a series of numerical experiments, employing different choices of $\mathsf{B}$.

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arXiv2024

Fictitious null spaces for improving the solution of injective inverse problems

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.

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arXiv2023

Identifying the source term in the potential equation with weighted sparsity regularization

We explore the possibility for using boundary measurements to recover a sparse source term f(x) in the potential equation. Employing weighted sparsity regularization and standard results for subgradients, we derive simple-to-check criteria which assure that a number of sinks (f(x) < 0) and sources (f(x) > 0) can be identified. Furthermore, we present two cases for which these criteria always are fulfilled: a) well-separated sources and sinks, and b) many sources or sinks located at the boundary plus one interior source/sink. Our approach is such that the linearity of the associated forward operator is preserved in the discrete formulation. The theory is therefore conveniently developed in terms of Euclidean spaces, and it can be applied to a wide range of problems. In particular, it can be applied to both isotropic and anisotropic cases. We present a series of numerical experiments. This work is motivated by the observation that standard methods typically suggest that internal sinks and sources are located close to the boundary.

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arXiv2023

Weighted sparsity regularization for source identification for elliptic PDEs

This investigation is motivated by PDE-constrained optimization problems arising in connection with electrocardiograms (ECGs) and electroencephalography (EEG). Standard sparsity regularization does not necessarily produce adequate results for these applications because only boundary data/observations are available for the identification of the unknown source, which may be interior. We therefore study a weighted $\ell^1$-regularization technique for solving inverse problems when the forward operator has a significant null space. In particular, we prove that a sparse source, regardless of whether it is interior or located at the boundary, can be exactly recovered with this weighting procedure as the regularization parameter $α$ tends to zero. Our analysis is supported by numerical experiments for cases with one and several local sources. The theory is developed in terms of Euclidean spaces, and our results can therefore be applied to many problems.

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arXiv2023

Box constraints and weighted sparsity regularization for identifying sources in elliptic PDEs

We explore the possibility for using boundary data to identify sources in elliptic PDEs. Even though the associated forward operator has a large null space, it turns out that box constraints, combined with weighted sparsity regularization, can enable rather accurate recovery of sources with constant magnitude/strength. In addition, for sources with varying strength, the support of the inverse solution will be a subset of the support of the true source. We present both an analysis of the problem and a series of numerical experiments. Our work only addresses discretized problems. The reason for introducing the weighting procedure is that standard (unweighted) sparsity regularization fails to provide adequate results for the source identification task considered in this paper. This investigation is also motivated by applications, e.g., recovering mass distributions from measurements of gravitational fields and inverse scattering. We develop the methodology and the analysis in terms of Euclidean spaces, and our results can therefore be applied to many problems. For example, the results are equally applicable to models involving the screened Poisson equation as to models using the Helmholtz equation, with both large and small wave numbers.

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arXiv2020

Modified Tikhonov regularization for identifying several sources

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.

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arXiv2020

A regularization operator for source identification for elliptic PDEs

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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arXiv2014

Analysis of the Diffuse Domain Method for second order elliptic boundary value problems

The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper we study the diffuse domain method for approximating second order elliptic boundary value problems posed on bounded domains, and show convergence and rates of the approximations generated by the diffuse domain method to the solution of the original second order problem when complemented by Robin, Dirichlet or Neumann conditions. The main idea of the diffuse domain method is to relax these boundary conditions by introducing a family of phase-field functions such that the variational integrals of the original problem are replaced by a weighted average of integrals of perturbed domains. From an functional analytic point of view, the phase-field functions naturally lead to weighted Sobolev spaces for which we present trace and embedding results as well as various type of Poincaré inequalities with constants independent of the domain perturbations. Our convergence analysis is carried out in such spaces as well, but allows to draw conclusions also about unweighted norms applied to restrictions on the original domain. Our convergence results are supported by numerical examples.

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