arXiv subjects

Martin Licht

Publications and source records attributed to Martin Licht.

5 recordsLinked to original sources

arXiv2026

Optimal Control in Hilbert Complex Spaces with Finite Element Exterior Calculus

We develop a framework for PDE-constrained optimal control on Hilbert complexes and its structure-preserving discretization by finite element exterior calculus (FEEC), with emphasis on problems where nontrivial topology creates physically meaningful global modes. We separate the gauge-fixed local state from harmonic components representing global circulation or flux, and introduce finite-dimensional topological actuators whose reachable modes are determined jointly by domain topology and actuator rank. We establish well-posedness, weak optimality conditions, reachability results, and FEEC error estimates preserving the harmonic structure. Numerical experiments on contractible and multiply connected domains demonstrate circulation and cavity- flux control, rank-dependent reachability, mesh-independent optimization, and topology-induced singular convergence. We further extend the framework to time-dependent Maxwell control, where harmonic electric circulation and magnetic flux become dynamical states exhibiting explicit conservation and reachability properties.

math.OC↗

arXiv2025

ProtoTopic: Prototypical Network for Few-Shot Medical Topic Modeling

Topic modeling is a useful tool for analyzing large corpora of written documents, particularly academic papers. Despite a wide variety of proposed topic modeling techniques, these techniques do not perform well when applied to medical texts. This can be due to the low number of documents available for some topics in the healthcare domain. In this paper, we propose ProtoTopic, a prototypical network-based topic model used for topic generation for a set of medical paper abstracts. Prototypical networks are efficient, explainable models that make predictions by computing distances between input datapoints and a set of prototype representations, making them particularly effective in low-data or few-shot learning scenarios. With ProtoTopic, we demonstrate improved topic coherence and diversity compared to two topic modeling baselines used in the literature, demonstrating the ability of our model to generate medically relevant topics even with limited data.

cs.LG↗

arXiv2022

Finite Element Methods for Linear Maxwell's Equations in Bianisotropic Media Permitting Polarization Fields and Magnetic Currents

We review Maxwell's equations and constitutive relations for 3D bianisotropic media in a generalized form: we consider all four variables and allow for nonzero polarization or magnetization, and also nonzero nonzero magnetic charge or current. After a discussion of general boundary conditions, we introduce a time-harmonic variational formulation of linear Maxwell's equations within 3D bianisotropic media in terms of the electric and magnetic fields. We showcase a finite element approximation of our variational formulation, using curl-conforming N\'ed\'elec edge elements of the first kind. Numerical examples illustrate the convergence of the method.

physics.comp-ph↗

arXiv2020

A divergence-conforming finite element method for the surface Stokes equation

The Stokes equation posed on surfaces is important in some physical models, but its numerical solution poses several challenges not encountered in the corresponding Euclidean setting. These include the fact that the velocity vector should be tangent to the given surface and the possible presence of degenerate modes (Killing fields) in the solution. We analyze a surface finite element method which provides solutions to these challenges. We consider an interior penalty method based on the well-known Brezzi-Douglas-Marini $H({\rm div})$-conforming finite element space. The resulting spaces are tangential to the surface, but require penalization of jumps across element interfaces in order to weakly maintain $H^1$ conformity of the velocity field. In addition our method exactly satisfies the incompressibility constraint in the surface Stokes problem. Secondly, we give a method which robustly filters Killing fields out of the solution. This problem is complicated by the fact that the dimension of the space of Killing fields may change with small perturbations of the surface. We first approximate the Killing fields via a Stokes eigenvalue problem and then give a method which is asymptotically guaranteed to correctly exclude them from the solution. The properties of our method are rigorously established via an error analysis and illustrated via numerical experiments.

math.NA↗

arXiv2017

Flux Reconstruction for Goal-Oriented A Posteriori Error Estimation

We propose a new heuristic goal-oriented a posteriori error estimator that connects the dual weighted residual method with equilibrated a posteriori error estimation. Our numerical experiments demonstrate the practical reliability of the error estimator, confirming theoretical predictions, as well as optimally convergent adaptivity even over singular domains and coarse meshes. The central algorithm is a localized flux reconstruction, which has been implemented in the finite element library deal.II. For a solid preparation we assess the performance of the equilibrated a posteriori error estimator of the energy norm in numerical experiments. Moreover, we give what seems to be first rigorous discussion in the numerical literature of localized flux reconstruction over quadrilateral meshes with hanging nodes.

math.NA↗