arXiv subjects

Christian Aarset

Publications and source records attributed to Christian Aarset.

9 recordsLinked to original sources

arXiv2026

Optimal experimental design for passive imaging source problems

This work focuses on optimal experimental design (OED) methods for passive imaging. We adopt a Bayesian inverse problem framework for passive imaging source problems, primarily focusing on spatially uncorrelated sources and systems governed by the Helmholtz equation. A major challenge in passive imaging is that the use of correlation data causes the observation dimension to grow quadratically with the number of sensor locations, compounding the computational difficulty of finding optimal designs. To overcome the computational bottleneck of repeated PDE solves in optimal design algorithms, we develop a two-level, low-rank approximation of the A-optimal design objective. This effectively decouples the problem into an offline and an online phase, enabling efficient evaluation of the design objective and its gradient without additional PDE solves. Our numerical results demonstrate that the proposed algorithm efficiently scales to large problems and that the resulting optimal designs significantly outperform random sensor placements in minimizing posterior uncertainty.

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arXiv2026

Data assimilation via model reference adaptation for linear and nonlinear dynamical systems

We address data assimilation for linear and nonlinear dynamical systems via the so-called model reference adaptive system. Continuing our theoretical developments, we deliver the first practical implementation of this approach for online parameter identification with time series data. Our semi-implicit scheme couples a modified state equation with a parameter evolution law that is driven by model-data residuals. We demonstrate four benchmark problems of increasing complexity: the Darcy flow, the Fisher-KPP equation, a nonlinear potential equation and finally, an Allen-Cahn type equation. Across all cases, explicit model reference adaptive system construction, verified assumptions and numerically stable reconstructions underline our proposed method as a reliable, versatile tool for data assimilation and real-time inversion.

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arXiv2025

FEM-based A-optimal sensor placement for heat source inversion from final time measurement

Within the field of optimal experimental design, \emph{sensor placement} refers to the act of finding the optimal locations of data collecting sensors, with the aim to optimise reconstruction of an unknown parameter from finite data. In this work, we investigate sensor placement for the inverse problem of reconstructing a heat source given final time measurements. Employing forward and adjoint analysis of this PDE-driven model, we show how one can leverage the first author's recently invented \emph{redundant-dominant $p$-continuation} algorithm to obtain binary A-optimal sensor placements also for this time-dependent model.

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arXiv2025

Global optimality conditions for sensor placement, with extensions to binary low-rank A-optimal designs

The \emph{sensor placement problem} for stochastic linear inverse problems consists of determining the optimal manner in which sensors can be employed to collect data. Specifically, one wishes to place a limited number of sensors over a large number of candidate locations, quantifying and optimising over the effect this data collection strategy has on the solution of the inverse problem. In this article, we provide a global optimality condition for the sensor placement problem via a subgradient argument, obtaining sufficient and necessary conditions for optimality\revix{, and marking certain sensors as \emph{dominant} or \emph{redundant}, i.e.~always on or always off}. We demonstrate how to take advantage of this optimality criterion to find approximately optimal binary designs, i.e.~designs where no fractions of sensors are placed. Leveraging our optimality criteria, we derive a powerful low-rank formulation of the A-optimal design objective for finite element-discretised function space settings, demonstrating its high computational efficiency, particularly in terms of derivatives, and study globally optimal designs for a Helmholtz-type source problem and extensions towards optimal binary designs.

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arXiv2024

A global optimum-informed greedy algorithm for A-optimal experimental design

Optimal experimental design (OED) concerns itself with identifying ideal methods of data collection, e.g.~via sensor placement. The \emph{greedy algorithm}, that is, placing one sensor at a time, in an iteratively optimal manner, stands as an extremely robust and easily executed algorithm for this purpose. However, it is a priori unclear whether this algorithm leads to sub-optimal regimes. Taking advantage of the author's recent work on non-smooth convex optimality criteria for OED, we here present a framework for rejection of sub-optimal greedy indices, and study the numerical benefits this offers.

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arXiv2024

Bi-level regularization via iterative mesh refinement for aeroacoustics

In this work, we illustrate the connection between adaptive mesh refinement for finite element discretized PDEs and the recently developed \emph{bi-level regularization algorithm}. By adaptive mesh refinement according to data noise, regularization effect and convergence are immediate consequences. We moreover demonstrate its numerical advantages to the classical Landweber algorithm in term of time and reconstruction quality for the example of the Helmholtz equation in an aeroacoustic setting.

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arXiv2023

Unsupervised energy disaggregation via convolutional sparse coding

In this work, a method for unsupervised energy disaggregation in private households equipped with smart meters is proposed. This method aims to classify power consumption as active or passive, granting the ability to report on the residents' activity and presence without direct interaction. This lays the foundation for applications like non-intrusive health monitoring of private homes. The proposed method is based on minimizing a suitable energy functional, for which the iPALM (inertial proximal alternating linearized minimization) algorithm is employed, demonstrating that various conditions guaranteeing convergence are satisfied. In order to confirm feasibility of the proposed method, experiments on semi-synthetic test data sets and a comparison to existing, supervised methods are provided.

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arXiv2023

Learning-informed parameter identification in nonlinear time-dependent PDEs

We introduce and analyze a method of learning-informed parameter identification for partial differential equations (PDEs) in an all-at-once framework. The underlying PDE model is formulated in a rather general setting with three unknowns: physical parameter, state and nonlinearity. Inspired by advances in machine learning, we approximate the nonlinearity via a neural network, whose parameters are learned from measurement data. The later is assumed to be given as noisy observations of the unknown state, and both the state and the physical parameters are identified simultaneously with the parameters of the neural network. Moreover, diverging from the classical approach, the proposed all-at-once setting avoids constructing the parameter-to-state map by explicitly handling the state as additional variable. The practical feasibility of the proposed method is confirmed with experiments using two different algorithmic settings: A function-space algorithm based on analytic adjoints as well as a purely discretized setting using standard machine learning algorithms.

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arXiv2020

Bifurcations in periodic integrodifference equations in $C(Ω)$ I: Analytical results and applications

We study local bifurcations of periodic solutions to time-periodic (systems of) integrodifference equations over compact habitats. Such infinite-dimensional discrete dynamical systems arise in theoretical ecology as models to describe the spatial dispersal of species having nonoverlapping generations. Our explicit criteria allow us to identify branchings of fold- and crossing curve-type, which include the classical transcritical-, pitchfork- and flip-scenario as special cases. Indeed, not only tools to detect qualitative changes in models from e.g. spatial ecology and related simulations are provided, but these critical transitions are also classified. In addition, the bifurcation behavior of various time-periodic integrodifference equations is investigated and illustrated. This requires a combination of analytical methods and numerical tools based on Nyström discretization of the integral operators involved.

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