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Data-driven geometric parameter optimization for PD-GMRES

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arXiv 2025年

作者： Duvenbeck, Lennart Riethmüller, Cedric Rohde, Christian Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart70569 Germany

Restarted GMRES is a robust and widely used iterative solver for linear systems. The control of the restart parameter is a key task to accelerate convergence and to prevent the well-known stagnation phenomenon. We focus on the Proportional-Derivative GMRES (PD-GMRES), which has been derived using control-theoretic ideas in [Cuevas Núñez, Schaerer, and Bhaya (2018)] as a versatile method for modifying the restart parameter. Several variants of a quadtree-based geometric optimization approach are proposed to find a best choice of PD-GMRES parameters. We show that the optimized PD-GMRES performs well across a large number of matrix types and we observe superior performance as compared to major other GMRES-based iterative solvers. Moreover, we propose an extension of the PD-GMRES algorithm to further improve performance by controlling the range of values for the restart parameter. Copyright © 2025, The Authors. All rights reserved.

关键词： Linear systems

A NON-DEGENERACY THEOREM FOR INTERACTING ELECTRONS IN ONE DIMENSION

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arXiv 2025年

作者： Corso, Thiago Carvalho Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

In this paper, we show that the ground-state of many-body Schrödinger operators for electrons in one dimension is non-degenerate. More precisely, we consider Schrödinger operators of the form (Formula Presented)) acting on ∧N L2 ([0,1]),where the external and interaction potentials υ and w belong to a large class of distributions. In this setting, we show that the ground-state of the system with Fermi statistics and local boundary conditions is non-degenerate and does not vanish on a set of positive measure. In the case of periodic and anti-periodic (or more general non-local) boundary conditions, we show that the same result holds whenever the number of particles is odd and even, respectively. This non-degeneracy result seems to be new even for regular potentials υ and w. As an immediate application of this result, we prove eigenvalue inequalities and the strong unique continuation property for eigenfunctions of the single-particle one-dimensional operators ℎ(υ) = −Δ + υ. In addition, we prove strict inequalities between the lowest eigenvalues of different self-adjoint realizations of HN (υ, w).MSC Codes Primary: 81Q10 Secondary: 35J10, 81V74 © 2025, CC BY-NC-SA.

关键词： Density functional theory

A hybrid-dimensional Stokes–Brinkman–Darcy model for arbitrary flows to the fluid–porous interface

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arXiv 2025年

作者： Ruan, Linheng Rybak, Iryna Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

Mathematical modelling of coupled flow systems containing a free-flow region in contact with a porous medium is challenging, especially for arbitrary flow directions to the fluid–porous interface. Transport processes in the free flow and porous medium are typically described by distinct equations: the Stokes equations and Darcy’s law, respectively, with an appropriate set of coupling conditions at the common interface. Classical interface conditions based on the Beavers–Joseph condition are not accurate for general flows. Several generalisations are recently developed for arbitrary flows at the interface, some of them are however only theoretically formulated and still need to be validated. In this manuscript, we propose an alternative to couple free flow and porous-medium flow, namely, the hybrid-dimensional Stokes–Brinkman–Darcy model. Such formulation incorporates the averaged Brinkman equations within a complex interface between the free-flow and porous-medium regions. The complex interface acts as a buffer zone facilitating storage and transport of mass and momentum and the model is applicable for arbitrary flow directions. We validate the proposed hybrid-dimensional model against the pore-scale resolved model in multiple examples and compare numerical simulation results also with the classical and generalised coupling conditions from the literature. The proposed hybrid-dimensional model demonstrates its applicability to describe arbitrary coupled flows and shows its advantages in comparison to other generalised coupling *** Codes 35Q35, 65N08, 76D07, 76S05 Copyright © 2025, The Authors. All rights reserved.

关键词： Buffer storage

V-REPRESENTABILITY AND HOHENBERG-KOHN THEOREM FOR NON-INTERACTING SCHRÖDINGER OPERATORS WITH DISTRIBUTIONAL POTENTIALS IN THE ONE-DIMENSIONAL TORUS

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arXiv 2025年

作者： Corso, Thiago Carvalho Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

In this paper, we show that the ground-state density of any non-interacting Schrödinger operator on the one-dimensional torus with potentials in a certain class of distributions is strictly positive. This result together with recent results from [SPR+24] provides a complete characterization of the set of non-interacting v-representable densities on the torus. Moreover, we prove that, for said class of non-interacting Schrödinger operators with distributional potentials, the Hohenberg-Kohn theorem holds, i.e., the external potential is uniquely determined by the ground-state density. In particular, the density-to-potential Kohn-Sham map is single-valued, and the non-interacting Lieb functional is differentiable at every point in this space of υ-representable densities. These results contribute to establishing a solid mathematical foundation for the Kohn-Sham scheme in this simplified *** Codes Primary: 81Q10, secondary: 81V74, 34L40 © 2025, CC BY-NC-SA.

关键词： Schrodinger equation

A hyperbolic model for two-layer thin film flow with a perfectly soluble anti-surfactant

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arXiv 2025年

作者： Barthwal, Rahul Rohde, Christian Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pffafenwaldring 57 Stuttgart Germany

We consider the motion of a two-phase thin film that consists of two immiscible viscous fluids and is endowed with an anti-surfactant solute. The presence of such solute particles induces variations of the surface tension and interfacial stress and can drive a Marangoni-type flow. We first analyze a lubrication limit and derive one-dimensional evolution equations of film heights and solute concentrations. Then, under the assumption that the capillarity and diffusion effects are negligible and the solute is perfectly soluble, we obtain a conservative first-order system in terms of film heights and concentration gradients. This reduced system is found to be strictly hyperbolic for a certain set of states and to admit an entire class of entropy/entropy-flux pairs. We also provide a strictly convex entropy for the hyperbolic system. Thus, the well-posedness for the Cauchy problem is given. Moreover, the system is almost a Temple-class system which allows to provide explicit solutions of the Riemann problem. The paper concludes with some numerical experiments using a Godunov-type finite volume method which relies on the exact Riemann *** Codes 35L40, 35L45, 35L65, 76A20, 78M35 © 2025, CC BY.

关键词： Finite volume method

Optimized Schwarz method for the Stokes–Darcy problem with generalized interface conditions

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arXiv 2025年

作者： Strohbeck, Paula Discacciati, Marco Rybak, Iryna Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany Department of Mathematical Sciences Loughborough University Epinal Way LoughboroughLE11 3TU United Kingdom

Due to their wide appearance in environmental settings as well as industrial and medical applications, the Stokes–Darcy problems with different sets of interface conditions establish an active research area in the community of mathematical modelers and computational scientists. For numerical simulation of such coupled problems in applications, robust and efficient computational algorithms are needed. In this work, we consider a generalization of the Beavers–Joseph interface condition recently developed using homogenization and boundary layer theory. This extension is applicable not only for the parallel flows to the fluid–porous interface as its predecessor, but also for arbitrary flow directions. To solve the Stokes–Darcy problem with these generalized interface conditions efficiently, we develop and analyze a Robin–Robin domain decomposition method using Fourier analysis to identify optimal weights in the Robin interface conditions. We study efficiency and robustness of the proposed method and provide numerical simulations which confirm the obtained theoretical results. Copyright © 2025, The Authors. All rights reserved.

关键词： Fourier analysis

A hyperbolic relaxation approximation of the incompressible Navier-Stokes equations with artificial compressibility

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Journal of Differential Equations 2025年 438卷

作者： Huang, Qian Rohde, Christian Yong, Wen-An Zhang, Ruixi Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart 70569 Germany Department of Mathematical Sciences Tsinghua University Beijing 100084 China Beijing Institute of Mathematical Sciences and Applications Beijing 101408 China

We introduce a new hyperbolic approximation to the incompressible Navier-Stokes equations by incorporating a first-order relaxation and using the artificial compressibility method. With two relaxation parameters in the model, we rigorously prove the asymptotic limit of the system towards the incompressible Navier-Stokes equations as both parameters tend to zero. Notably, the convergence of the approximate pressure variable is achieved by the help of a linear ‘auxiliary’ system and energy-type error estimates of the differences between the two-parameter model and the Navier-Stokes equations. © 2025 The Author(s)

关键词： Artificial compressibility Energy estimates Hyperbolic singular perturbations Incompressible Navier-Stokes equations Relaxation approximations

MATHEMATICAL JUSTIFICATION OF A BAER–NUNZIATO MODEL FOR A COMPRESSIBLE VISCOUS FLUID WITH PHASE TRANSITION

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arXiv 2025年

作者： Rohde, Christian Wendt, Florian Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart70569 Germany

In this work, we justify a Baer–Nunziato system including appropriate closure terms as the macroscopic description of a compressible viscous fluid that can occur in a liquid or a vapor phase in the isothermal framework. As a mathematical model for the two-phase fluid on the detailed scale we chose a non-local version of the Navier–Stokes–Korteweg equations in the one-dimensional and periodic setting. Our justification relies on anticipating the macroscopic description of the two-phase fluid as the limit system for a sequence of solutions with highly oscillating initial densities. Interpreting the density as a parametrized measure, we extract a limit system consisting of a kinetic equation for the parametrized measure and a momentum equation for the velocity. Under the assumption that the initial density distributions converge in the limit to a convex combination of Dirac-measures, we show by a uniqueness result that the parametrized measure also has to be a convex combination of Dirac-measures and, that the limit system reduces to the Baer–Nunziato system. This work extends existing results concerning the justification of Baer–Nunziato models as the macroscopic description of multi-fluid models in the sense, that we allow for phase transition effects on the detailed scale. This work also includes a new global-in-time well-posedness result for the Cauchy problem of the non-local Navier–Stokes–Korteweg *** Codes 35Q35, 76M50, 76N06, 76T10 © 2025, CC BY.

关键词： Two phase flow

A symmetry-preserving and transferable representation for learning the Kohn-Sham density matrix

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arXiv 2025年

作者： Zhang, Liwei Mazzeo, Patrizia Nottoli, Michele Cignoni, Edoardo Cupellini, Lorenzo Stamm, Benjamin Institut für Geometrie und Praktische Mathematik RWTH Aachen University Templergraben 55 Aachen52062 Germany Dipartimento di Chimica e Chimica Industriale Università di Pisa Pisa56124 Italy Universität Stuttgart Institute of Applied Analysis and Numerical Simulation Pfaffenwaldring 57 Stuttgart70569 Germany

The Kohn-Sham (KS) density matrix is one of the most essential properties in KS density functional theory (DFT), from which many other physical properties of interest can be derived. In this work, we present a parameterized representation for learning the mapping from a molecular configuration to its corresponding density matrix using the Atomic Cluster Expansion (ACE) framework, which preserves the physical symmetries of the mapping, including isometric equivariance and Grassmannianity. Trained on several typical molecules, the proposed representation is shown to be systematically improvable with the increase of the model parameters and is transferable to molecules that are not part of and even more complex than those in the training set. The models generated by the proposed approach are illustrated as being able to generate reasonable predictions of the density matrix to either accelerate the DFT calculations or to provide approximations to some properties of the molecules. © 2025, CC BY-NC-SA.

关键词： Density functional theory