arXiv subjects

Jaakko Kultima

Publications and source records attributed to Jaakko Kultima.

3 recordsLinked to original sources

arXiv2026

Efficient TV regularization of large-scale linear inverse problems via the SCD semismooth* Newton method with applications in tomography

In this paper, we consider the efficient numerical minimization of Tikhonov functionals resulting from total-variation (TV) regularization of linear inverse problems. Since the TV penalty is non-smooth, this is typically done either via smooth approximations, which are inexact, or using non-smooth optimization techniques, which can often be numerically expensive, in particular for large-scale problems. Here, we present a numerically efficient minimization approach based on the recently proposed semismooth* Newton method, which employs a novel concept of graphical derivatives and exhibits locally superlinear convergence. The proposed approach is specifically tailored to TV regularization, suitable for large-scale inverse problems, and supported by strong mathematical convergence guarantees. Furthermore, we demonstrate its performance on two (large-scale) tomographic imaging problems and compare our results to those obtained via other state-of-the-art TV regularization approaches. Finally, an open-source Matlab implementation of the proposed method is made available online at https://github.com/HGfrerer/TVReg-2D-Semismoothstar-Newton.

math.NA↗

arXiv2025

Fast reconstruction approaches for photoacoustic tomography with smoothing Sobolev/Matérn priors

In photoacoustic tomography (PAT), the computation of the initial pressure distribution within an object from its time-dependent boundary measurements over time is considered. This problem can be approached from two well-established points of view: deterministically using regularisation methods, or stochastically using the Bayesian framework. Both approaches frequently require the solution of a variational problem. In the paper we elaborate the connection between these approaches by establishing the equivalence between a smoothing Mat{é}rn class of covariance operators and Sobolev embedding operator $E_s: H^s \hookrightarrow L^2$. We further discuss the use of a Wavelet-based implementation of the adjoint operator $E_s^*$ which also allows for efficient evaluations for certain Mat{é}rn covariance operators, leading to efficient implementations both in terms of computational effort as well as memory requirements. The proposed methods are validated with reconstructions for the photoacoustic problem.

math.NA↗

arXiv2022

Recovery of singularities from fixed angle scattering data for biharmonic operator in dimensions two and three

The inverse fixed angle problem for operator $Δ^2 u + V(x,|u|) u$ is considered in dimensions $n=2,3$. We prove that the difference between an inverse fixed angle Born approximation and the function $V(\cdot,1)$ is smoother than the function $V$ itself in some Sobolev scale. This allows us to conclude that the main singularities of the perturbation $V$ can be reconstructed from the knowledge of the scattering amplitude with some fixed incident angle.

math.AP↗