arXiv subjects

Jyrki Jauhiainen

Publications and source records attributed to Jyrki Jauhiainen.

9 recordsLinked to original sources

arXiv2026

Observability as a constraint qualification for a nonsmooth bilevel PDE-constrained problem

Motivated by experiment design for parameter identification in image reconstruction, we analyze a bilevel control problem involving sparse inversion at the lower level. Qualified optimality conditions in explicit multiplier form are shown to hold under either an observability condition for a coupled system or the Lebesgue nullity of a contact set induced by the lower-level adjoint variable. We prove that these conditions are independent of one another and that they cannot, in general, both be omitted. The approach is based on the characterization of local first-order geometric objects (tangent and normal cones) associated with a set of lifted constraints. Based on the resulting optimality conditions, we formulate a reduced gradient method and illustrate it through numerical experiments.

math.OC↗

arXiv2026

Dynamic inverse problems: Single-loop online algorithms

We study efficient online methods for dynamic inverse problems with infinite time horizon. We concentrate, in particular, on problems whose forward model arises from a PDE. Our motivating application is flow monitoring with Electrical Impedance Tomography (EIT). The idea of such online methods is to take single steps of of standard optimisation algorithms, on each time index; each data frame. A predictor, based on problem dynamics, is used to transfer iterates one from time index to the next one. If we monitor a fast flow with a correspondingly fast measurement modality, such as EIT, basic methods are unable to solve the PDE before new data arrives. Our idea, then, is to not solve it, and instead, on each iteration, each time index, take single or few steps of standard iterative solvers towards the solution of both the PDE and an adjoint PDE. This is what ``single loop'' refers to. To the overall problem, we apply standard online optimisation methods, at the outside developed for exact gradients $\nabla E_k(x^k)$ of the iteration-dependent data fidelity $E_k$ that incorporates the PDE. We replace the gradient by a single-loop estimate $\tilde\grad E_k(x^k)$ that satisfies standard smoothness properties with summable errors. This allows standard regret proofs to go through. Our numerical experiments on dynamic EIT validate the theoretical predictions and highlight the potential of the proposed approach for the real-time solution of PDE-constrained dynamic inverse problems.

math.OC↗

arXiv2026

Dynamic inverse problems: Online regularisation theory

We develop regularisation theory for dynamic inverse problems, solved using online methods with an infinite time horizon. Using concepts of subregularity to treat nonsmooth regularisers, we prove that time-averaged reconstruction errors converge to zero as noise, algorithmic errors, and regularisation vanish as the horizon grows. We illustrate the theory numerically with a dynamic electrical impedance tomography example.

math.NA↗

arXiv2025

Online optimisation for dynamic electrical impedance tomography

Online optimisation studies the convergence of optimisation methods as the data embedded in the problem changes. Based on this idea, we propose a primal dual online method for nonlinear time-discrete inverse problems. We analyse the method through regret theory and demonstrate its performance in real-time monitoring of moving bodies in a fluid with Electrical Impedance Tomography (EIT). To do so, we also prove the second-order differentiability of the Complete Electrode Model (CEM) solution operator on $L^\infty$.

math.OC↗

arXiv2024

Prediction techniques for dynamic imaging with online primal-dual methods

Online optimisation facilitates the solution of dynamic inverse problems, such as image stabilisation, fluid flow monitoring, and dynamic medical imaging. In this paper, we improve upon previous work on predictive online primal-dual methods on two fronts. Firstly, we provide a more concise analysis that symmetrises previously unsymmetric regret bounds, and relaxes previous restrictive conditions on the dual predictor. Secondly, based on the latter, we develop several improved dual predictors. We numerically demonstrate their efficacy in image stabilisation and dynamic positron emission tomography.

math.OC↗

arXiv2022

Mumford-Shah regularization in electrical impedance tomography with complete electrode model

In electrical impedance tomography, we aim to solve the conductivity within a target body through electrical measurements made on the surface of the target. This inverse conductivity problem is severely ill-posed, especially in real applications with only partial boundary data available. Thus regularization has to be introduced. Conventionally regularization promoting smooth features is used, however, the Mumford--Shah regularizer familiar for image segmentation is more appropriate for targets consisting of several distinct objects or materials. It is, however, numerically challenging. We show theoretically through $Γ$-convergence that a modification of the Ambrosio--Tortorelli approximation of the Mumford--Shah regularizer is applicable to electrical impedance tomography, in particular the complete electrode model of boundary measurements. With numerical and experimental studies, we confirm that this functional works in practice and produces higher quality results than typical regularizations employed in electrical impedance tomography when the conductivity of the target consists of distinct smoothly-varying regions.

math.OC↗

arXiv2020

Non-planar sensing skins for structural health monitoring based on electrical resistance tomography

Electrical resistance tomography (ERT) -based distributed surface sensing systems, or sensing skins, offer alternative sensing techniques for structural health monitoring, providing capabilities for distributed sensing of, for example, damage, strain and temperature. Currently, however, the computational techniques utilized for sensing skins are limited to planar surfaces. In this paper, to overcome this limitation, we generalize the ERT-based surface sensing to non-planar surfaces covering arbitrarily shaped three-dimensional structures; We construct a framework in which we reformulate the image reconstruction problem of ERT using techniques of Riemannian geometry, and solve the resulting problem numerically. We test this framework in series of numerical and experimental studies. The results demonstrate that the feasibility of the proposed formulation and the applicability of ERT-based sensing skins for non-planar geometries.

physics.comp-ph↗

arXiv2020

Relaxed Gauss-Newton methods with applications to electrical impedance tomography

As second-order methods, Gauss--Newton-type methods can be more effective than first-order methods for the solution of nonsmooth optimization problems with expensive-to-evaluate smooth components. Such methods, however, often do not converge. Motivated by nonlinear inverse problems with nonsmooth regularization, we propose a new Gauss--Newton-type method with inexact relaxed steps. We prove that the method converges to a set of disjoint critical points given that the linearisation of the forward operator for the inverse problem is sufficiently precise. We extensively evaluate the performance of the method on electrical impedance tomography (EIT).

math.OC↗

arXiv2020

Primal-dual block-proximal splitting for a class of non-convex problems

We develop block structure adapted primal-dual algorithms for non-convex non-smooth optimisation problems whose objectives can be written as compositions $G(x)+F(K(x))$ of non-smooth block-separable convex functions $G$ and $F$ with a non-linear Lipschitz-differentiable operator $K$. Our methods are refinements of the non-linear primal-dual proximal splitting method for such problems without the block structure, which itself is based on the primal-dual proximal splitting method of Chambolle and Pock for convex problems. We propose individual step length parameters and acceleration rules for each of the primal and dual blocks of the problem. This allows them to convergence faster by adapting to the structure of the problem. For the squared distance of the iterates to a critical point, we show local $O(1/N)$, $O(1/N^2)$ and linear rates under varying conditions and choices of the step lengths parameters. Finally, we demonstrate the performance of the methods on practical inverse problems: diffusion tensor imaging and electrical impedance tomography.

math.OC↗