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From low-rank retractions to dynamical low-rank approximation and back

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arXiv 2023年

作者： Séguin, Axel Ceruti, Gianluca Kressner, Daniel Institute of Mathematics LausanneCH-1015 Switzerland University of Innsbruck Numerical Analysis and Scientific Computing Innsbruck Austria

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston–Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new *** Codes 65L05, 65L29 Copyright © 2023, The Authors. All rights reserved.

关键词： Differential equations

On the numerical evaluation of wall shear stress using the finite element method

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arXiv 2025年

作者： Brunátová, Jana Dokken, Jørgen Schartum Valen-Sendstad, Kristian Hron, Jaroslav Mathematical Institute Charles University Czech Republic Bernoulli Institute University of Groningen Netherlands Scientific Computing and Numerical Analysis Simula Research Laboratory Norway Computational Physiology Simula Research Laboratory Norway

Wall shear stress (WSS) is a crucial hemodynamic quantity extensively studied in cardiovascular research, yet its numerical computation is not straightforward. This work aims to compare WSS results obtained from two different finite element discretizations, quantify the differences between continuous and discontinuous stresses, and introduce a novel method for WSS evaluation through the formulation of a boundary-flux problem. Two benchmark problems are considered – a 2D Stokes flow on a unit square and a 3D Poiseuille flow through a cylindrical pipe. These are followed by investigations of steady-state Navier–Stokes flow in two patient-specific aneurysms. The study focuses on P1/P1 stabilized and Taylor–Hood P2/P1 mixed finite elements for velocity and pressure. WSS is computed using either the proposed boundary-flux method or as a projection of tangential traction onto First order Lagrange (P1), Discontinuous Galerkin first order (DG-1), or Discontinuous Galerkin zero order (DG-0) space. For the P1/P1 stabilized element, the boundary-flux and P1 projection methods yielded equivalent results. With the P2/P1 element, the boundary-flux evaluation demonstrated faster convergence in the Poiseuille flow example but showed increased sensitivity to pressure field inaccuracies in patient-specific geometries compared to the projection method. In patient-specific cases, the P2/P1 element exhibited superior robustness to mesh size when evaluating average WSS and low shear area (LSA), outperforming the P1/P1 stabilized element. Projecting discontinuous finite element results into continuous spaces can introduce artifacts, such as the Gibbs phenomenon. Consequently, it becomes crucial to carefully select the finite element space for boundary stress calculations – not only in applications involving WSS computations for aneurysms. © 2025, CC BY.

关键词： Shear stress

FULLY DISCRETE analysis OF THE GALERKIN POD NEURAL NETWORK APPROXIMATION WITH APPLICATION TO 3D ACOUSTIC WAVE SCATTERING

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arXiv 2025年

作者： Dölz, Jürgen Henríquez, Fernando Institute for Numerical Simulation University of Bonn Friedrich-Hirzebruch-Allee 7 Bonn53115 Germany Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstrae 8-10 WienA-1040 Austria

In this work, we consider the approximation of parametric maps using the so-called Galerkin POD-NN method. This technique combines the computation of a reduced basis via proper orthogonal decomposition (POD) and artificial neural networks (NNs) for the construction of fast surrogates of said parametric maps. We provide a fully discrete error analysis of this method that accounts for different discretization errors, including the number of reduced basis in the approximation of the solution manifold, truncation in the parameter space, and, most importantly, the number of samples in the computation of the reduced space, together with the effect of the use of NNs in the approximation of the reduced coefficients. Following this error analysis we provide a-priori bounds on the required POD tolerance, the resulting POD ranks, and NN-parameters to maintain the order of convergence of quasi Monte Carlo sampling techniques. The main advantage of Galerkin POD-NN over existing traditional techniques is that the online and offline phases of the projection-based reduced basis method are completely decoupled, making the evaluation of the constructed surrogate in run-time considerably faster without the need for an hyper-reduction technique. We conclude this work by showcasing the applicability of this method through a practical industrial application: the sound-soft acoustic scattering problem by a parametrically defined scatterer in three physical dimensions. Copyright © 2025, The Authors. All rights reserved.

关键词： Monte Carlo methods

Achieving High Accuracy with PINNs via Energy Natural Gradient Descent

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arXiv 2023年

作者： Müller, Johannes Zeinhofer, Marius Max Planck Institute for Mathematics in the Sciences Inselstraße 22 Leipzig04103 Germany Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Kristian Augusts Gate 23 Oslo0164 Norway

We propose energy natural gradient descent, a natural gradient method with respect to a Hessian-induced Riemannian metric as an optimization algorithm for physics-informed neural networks (PINNs) and the deep Ritz method. As a main motivation we show that the update direction in function space resulting from the energy natural gradient corresponds to the Newton direction modulo an orthogonal projection onto the model’s tangent space. We demonstrate experimentally that energy natural gradient descent yields highly accurate solutions with errors several orders of magnitude smaller than what is obtained when training PINNs with standard optimizers like gradient descent, Adam or BFGS, even when those are allowed significantly more computation time. We show that the approach can be combined with deterministic and stochastic discretizations of the integral terms and with deep networks allowing for an application in higher dimensional settings. Copyright © 2023, The Authors. All rights reserved.

关键词： Stochastic systems

THE LATENT VARIABLE PROXIMAL POINT ALGORITHM FOR VARIATIONAL PROBLEMS WITH INEQUALITY CONSTRAINTS

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arXiv 2025年

作者： Dokken, Jørgen S. Farrell, Patrick E. Keith, Brendan Papadopoulos, Ioannis P.A. Surowiec, Thomas M. Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Oslo Norway Mathematical Institute University of Oxford Oxford United Kingdom Mathematical Institute Faculty of Mathematics and Physics Charles University Prague Czech Republic Division of Applied Mathematics Brown University ProvidenceRI United States Weierstrass Institute Berlin Germany

The latent variable proximal point (LVPP) algorithm is a framework for solving infinite-dimensional variational problems with pointwise inequality constraints. The algorithm is a saddle point reformulation of the Bregman proximal point algorithm. At the continuous level, the two formulations are equivalent, but the saddle point formulation is more amenable to discretization because it introduces a structure-preserving transformation between a latent function space and the feasible set. Working in this latent space is much more convenient for enforcing inequality constraints than the feasible set, as discretizations can employ general linear combinations of suitable basis functions, and nonlinear solvers can involve general additive updates. LVPP yields numerical methods with observed mesh-independence for obstacle problems, contact, fracture, plasticity, and others besides;in many cases, for the first time. The framework also extends to more complex constraints, providing means to enforce convexity in the Monge–Ampère equation and handling quasi-variational inequalities, where the underlying constraint depends implicitly on the unknown solution. In this paper, we describe the LVPP algorithm in a general form and apply it to twelve problems from across mathematics. © 2025, CC BY.

关键词： Consensus algorithm

ON THE IMPROVED CONVERGENCE OF LIFTED DISTRIBUTIONAL GAUSS CURVATURE FROM REGGE ELEMENTS

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arXiv 2024年

作者： Gopalakrishnan, Jay Neunteufel, Michael Schöberl, Joachim Wardetzky, Max Portland State University PO Box 751 PortlandOR97201 United States Institute of Analysis and Scientific Computing TU Wien Wiedner Hauptstr. 8-10 Wien1040 Austria Institute of Numerical and Applied Mathematics University of Göttingen Lotzestr. 16-18 Göttingen37083 Germany

Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor approximation in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the approximate metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degree and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical *** Codes 65N30 (Primary) 53A70, 83C27 (Secondary) Copyright © 2024, The Authors. All rights reserved.

关键词： Tensors

Risk-averse optimal control of random elliptic variational inequalities

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arXiv 2022年

作者： Alphonse, Amal Geiersbach, Caroline Hintermüller, Michael Surowiec, Thomas M. Weierstrass Institute Mohrenstrasse 39 Berlin10117 Germany Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Kristian Augusts gate 23 Oslo0164 Norway

We consider a risk-averse optimal control problem governed by an elliptic variational inequality (VI) subject to random inputs. By deriving KKT-type optimality conditions for a penalised and smoothed problem and studying convergence of the stationary points with respect to the penalisation parameter, we obtain two forms of stationarity conditions. The lack of regularity with respect to the uncertain parameters and complexities induced by the presence of the risk measure give rise to new challenges unique to the stochastic setting. We also propose a path-following stochastic approximation algorithm using variance reduction techniques and demonstrate the algorithm on a modified benchmark problem. Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems

MEDICAL IMAGE REGISTRATION USING OPTIMAL CONTROL OF A LINEAR HYPERBOLIC TRANSPORT EQUATION WITH A DG DISCRETIZATION

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arXiv 2023年

作者： Zapf, Bastian Haubner, Johannes Baumgärtner, Lukas Schmidt, Stephan Expert Analytics AS Oslo Norway Department of Mathematics University of Oslo Oslo Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway Institute of Mathematics and Scientific Computing University of Graz Austria Institut of Mathematics Humboldt University of Berlin Berlin10099 Germany Department of Mathematics University of Trier Trier54296 Germany

Patient specific brain mesh generation from MRI can be a time consuming task and require manual corrections, e.g., for meshing the ventricular system or defining subdomains. To address this issue, we consider an image registration approach. The idea is to use the registration of an input magnetic resonance image (MRI) to a respective target in order to obtain a new mesh from a template mesh. To obtain the transformation, we solve an optimization problem that is constrained by a linear hyperbolic transport equation. We use a higher-order discontinuous Galerkin finite element method for discretization and motivate the numerical upwind scheme and its limitations from the continuous weak space–time formulation of the transport equation. We present a numerical implementation that builds on the finite element packages FEniCS and dolfin-adjoint. To demonstrate the efficacy of the proposed approach, numerical results for the registration of an input to a target MRI of two distinct individuals are presented. Moreover, it is shown that the registration transforms a manually crafted input mesh into a new mesh for the target subject whilst preserving mesh quality. Challenges of the algorithm are *** Codes 35D30, 35L02, 35Q49, 35Q93, 35R30, 49M05, 49M41, 49N45, 65D18, 65K10, 65M32, 65M50, 68T99 © 2023, CC BY.

关键词： numerical methods

Natural convection in the horizontal annulus: critical Rayleigh number for the steady problem

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arXiv 2025年

作者： Passerini, Arianna Rummler, Bernd Růžička, Michael Thäter, Gudrun Department of Mathematics and Scientific Computing Università di Ferrara Via Saragat 1 Ferrara44100 Italy Institute for Analysis and Numerics Otto-von-Guericke-Universität Magdeburg PF 4120 Magdeburg39016 Germany Institute for Applied Mathematics Universität Freiburg Eckerstr. 1 Freiburg79104 Germany Institute for Applied & Numerical Mathematics KIT Karlsruhe76128 Germany

For the 2D Oberbeck-Boussinesq system in an annulus we are looking for the critical Rayleigh number for which the (nonzero) basic flow loses stability. For this we consider the corresponding Euler-Lagrange equations and construct a precise functional analytical frame for the Laplace- and the Stokes problem as well as the Bilaplacian operator in this domain. With this frame and the right set of basis functions it is then possible to construct and apply a numerical scheme providing the critical Rayleigh number. These results will be published in ZAMM in *** Codes 76E09, 76E06, 76D03, 35Q35 © 2025, CC BY.

关键词： Rayleigh number