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arXiv 2023年

作者： Keith, Brendan Surowiec, Thomas M. Division of Applied Mathematics Brown University ProvidenceRI02912 United States Simula Research Laboratory Department of Numerical Analysis and Scientific Computing Kris-tian Augusts gate 23 Oslo0164 Norway

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation;(2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups;and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption. Copyright © 2023, The Authors. All rights reserved.

关键词： Poisson equation

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Achieving High Accuracy with PINNs via Energy Natural Gradient Descent

arXiv

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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

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On The Importance of Fundamental Computational Fluid Dynamics Towards a Robust and Reliable Model of Left Atrial Flows: Is There More Than Meets the Eye?

arXiv

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arXiv 2023年

作者： Khalili, Ehsan Daversin-Catty, Cécile Olivares, Andy L. Mill, Jordi Camara, Oscar Valen-Sendstad, Kristian Department of Computational Physiology Simula Research Laboratory Oslo Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway Department of Information and Communication Technologies Universitat Pompeu Fabra Barcelona Spain

Computational fluid dynamics (CFD) studies of left atrial flows have reached a sophisticated level, e.g., revealing plausible relationships between hemodynamics and stresses with atrial fibrillation. However, little focus has been on fundamental fluid modelling of LA flows. The purpose of this study was to investigate the spatiotemporal convergence, along with the differences between high- (HR) versus normal-resolution/accuracy (NR) solution strategies, respectively. CFD simulations on 12 patient-specific left atrial geometries, obtained from computed tomography scans, were performed by using a second-order accurate and space/time centered solver. The convergence studies showed an average variability of around 30% and 55% for time averaged wall shear stress (WSS), oscillatory shear index (OSI), relative residence time (RRT), and endothelial cell activation potential (ECAP), even between intermediate spatial and temporal resolutions, in the left atrium (LA) and left atrial appendage (LAA), respectively. The comparison between HR and NR simulations showed good correlation in the LA for WSS, RRT, and ECAP (R2 > 0.9), but not for OSI (R2 = 0.63). However, there were poor correlations in the LAA especially for OSI, RRT, and ECAP (R2 = 0.55, 0.63, and 0.61, respectively), except for WSS (R2 = 0.81). The errors are comparable to differences previously reported with disease correlations. To robustly predict atrial hemodynamics and stresses, numerical resolutions of 10M elements and 10k time-steps per cycle seem necessary (i.e., one order of magnitude higher than normally used in both space and time). In conclusion, attention to fundamental numerical aspects is essential towards establishing a plausible, robust, and reliable model of LA flows. Copyright © 2023, The Authors. All rights reserved.

关键词： Sensitivity analysis

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The directional flow generated by peristalsis in perivascular networks – theoretical and numerical reduced-order descriptions

arXiv

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arXiv 2023年

作者： Gjerde, Ingeborg G. Rognes, Marie E. Sánchez, Antonio L. Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Kristian Augusts gate 23 Oslo0164 Norway Department of Mechanical and Aerospace Engineering University of California San Diego La Jolla United States

Directional fluid flow in perivascular spaces surrounding cerebral arteries is hypothesized to play a key role in brain solute transport and clearance. While various drivers for pulsatile flow, such as cardiac or respiratory pulsations, are well quantified, the question remains as to which mechanisms could induce directional flow within physiological regimes. To address this question, we develop theoretical and numerical reduced-order models to quantify the directional (net) flow induceable by peristaltic pumping in periarterial networks. Each periarterial element is modeled as a slender annular space bounded internally by a circular tube supporting a periodic traveling (peristaltic) wave. Under the reasonable assumptions of small Reynolds number flow, small radii, and small-amplitude peristaltic waves, we use lubrication theory and regular perturbation methods to derive theoretical expressions for the directional net flow and pressure distribution in the perivascular network. The reduced model is used to derive closed-form analytical expressions for the net flow for simple network configurations of interest, including single elements, two elements in tandem, and a three element bifurcation, with results compared with numerical predictions. In particular, we provide a computable theoretical estimate of the net flow induced by peristaltic motion in perivascular networks as a function of physiological parameters, notably wave length, frequency, amplitude and perivascular dimensions. Quantifying the maximal net flow for specific physiological regimes, we find that vasomotion may induce net pial periarterial flow velocities on the order of a few to tens of µm/s and that sleep-related changes in vasomotion pulsatility may drive a threefold flow *** Codes 76M45, 76Z05, 76D08, 76M10, © 2023, CC BY-NC-ND.

关键词： Solute transport

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

arXiv

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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

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Risk-averse optimal control of random elliptic variational inequalities

arXiv

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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

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A DISCONTINUOUS GALERKIN METHOD FOR THE EXTRACELLULAR MEMBRANE INTRACELLULAR MODEL

arXiv

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arXiv 2024年

作者： Masri, Rami Kirk, Keegan L.A. Hauge, Eirill Kuchta, Miroslav Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway Division of Applied Mathematics Brown University ProvidenceRI02906 United States Department of Mathematical Sciences Center for Mathematics and Artificial Intelligence George Mason University FairfaxVA22030 United States

We formulate and analyze interior penalty discontinuous Galerkin methods for coupled elliptic PDEs modeling excitable tissue, represented by intracellular and extracellular domains sharing a common interface. The PDEs are coupled through a dynamic boundary condition, posed on the interface, that relates the normal gradients of the solutions to the time derivative of their jump. This system is referred to as the Extracellular Membrane Intracellular model or the cell-by-cell model. Due to the dynamic nature of the interface condition and to the presence of corner singularities, the analysis of discontinuous Galerkin methods is non-standard. We prove the existence and uniqueness of solutions by a reformulation of the problem to one posed on the membrane. Convergence is shown by utilizing face-to-element lifting operators and notions of weak consistency suitable for solutions with low spatial regularity. Further, we present parameter-robust preconditioned iterative solvers. numerical examples in idealized geometries demonstrate our theoretical findings, and simulations in multiple cells portray the robustness of the *** Codes 65N30, 65M60 Copyright © 2024, The Authors. All rights reserved.

关键词： Galerkin methods

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SMART: Spatial Modeling Algorithms for Reactions and Transport

arXiv

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arXiv 2023年

作者： Laughlin, Justin G. Dokken, Jørgen S. Finsberg, Henrik N.T. Francis, Emmet A. Lee, Christopher T. Rognes, Marie E. Rangamani, Padmini Department of Mechanical and Aerospace Engineering University of California La Jolla San DiegoCA United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway Department of Computational Physiology Simula Research Laboratory Oslo Norway

Recent advances in microscopy and 3D reconstruction methods have allowed for characterization of cellular morphology in unprecedented detail, including the irregular geometries of intracellular subcompartments such as membrane-bound organelles. These geometries are now compatible with predictive modeling of cellular function. Biological cells respond to stimuli through sequences of chemical reactions generally referred to as cell signaling pathways. The propagation and reaction of chemical substances in cell signaling pathways can be represented by coupled nonlinear systems of reaction-transport equations. These reaction pathways include numerous chemical species that react across boundaries or interfaces (e.g., the cell membrane and membranes of organelles within the cell) and domains (e.g., the bulk cell volume and the interior of organelles). Such systems of multi-dimensional partial differential equations (PDEs) are notoriously difficult to solve because of their high dimensionality, non-linearities, strong coupling, stiffness, and potential instabilities. In this work, we describe Spatial Modeling Algorithms for Reactions and Transport (SMART), a high-performance finite-element-based simulation package for model specification and numerical simulation of spatially-varying reaction-transport processes. SMART is based on the FEniCS finite element library, provides a symbolic representation framework for specifying reaction pathways, and supports geometries in 2D and 3D including large and irregular cell geometries obtained from modern ultra-structural characterization methods. © 2023, CC BY.

关键词： Cell signaling

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

arXiv

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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

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PARAMETER-ROBUST PRECONDITIONER FOR STOKES-DARCY COUPLED PROBLEM WITH LAGRANGE MULTIPLIER

arXiv

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arXiv 2025年

作者： Hu, Xiaozhe Kuchta, Miroslav Mardal, Kent-Andre Wang, Xue Department of Mathematics Tufts University MedfordMA02155 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway Simula Research Laboratory Oslo0164 Norway Department of Mathematics University of Oslo Blindern Oslo0316 Norway School of Mathematics Shandong University Jinan250100 China

In this paper, we propose a parameter-robust preconditioner for the coupled Stokes-Darcy problem equipped with various boundary conditions, enforcing the mass conservation at the interface via a Lagrange multiplier. We rigorously establish that the coupled system is well-posed with respect to physical parameters and mesh size and provide a framework for constructing parameter-robust preconditioners. Furthermore, we analyze the convergence behavior of the Minimal Residual method in the presence of small outlier eigenvalues linked to specific boundary conditions, which can lead to slow convergence or stagnation. To address this issue, we employ deflation techniques to accelerate the convergence. Finally, numerical experiments confirm the effectiveness and robustness of the proposed approach. Copyright © 2025, The Authors. All rights reserved.

关键词： Lagrange multipliers

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