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Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes

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arXiv 2018年

作者： Croci, Matteo Giles, Michael B. Rognes, Marie E. Farrell, Patrick E. Mathematical Institute University of Oxford Oxford United Kingdom Department for Numerical Analysis and Scientific Computing Simula Research Laboratory P.O. Box 134 Lysaker1325 Norway

When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in a MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In a MLMC setting, a good coupling is enforced and the telescoping sum is *** Codes 65C05, 60G60, 65N30, 60H35, 35R60 Copyright © 2018, The Authors. All rights reserved.

关键词： Partial differential equations

Robust preconditioning for coupled Stokes-Darcy problems with the Darcy problem in primal form

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arXiv 2020年

作者： Holter, Karl Erik Kuchta, Miroslav Mardal, Kent-Andre Department of Mathematics Division of Mechanics University of Oslo Oslo Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Simula-UCSD-University of Oslo Research Norwegian Ministry of Education and Research

The coupled Darcy-Stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part. In this work we consider preconditioners for monolitic solution algorithms of the coupled Darcy-Stokes problem, where the Darcy problem is in primal form. We employ the operator preconditioning framework and utilize a fractional solver at the interface between the problems to obtain order optimal schemes that are robust with respect to the material parameters, i.e. the permeability, viscosity and Beavers-Joseph-Saffman condition. Our approach is similar to that of [1], but since the Darcy problem is in primal form, the mass conservation at the interface introduces some challenges. These challenges will be specifically addressed in this paper. numerical experiments illustrating the performance are provided. The preconditioner is posed in non-standard Sobolev spaces which may be perceived as an obstacle for its use in applications. However, we detail the implementational aspects and show that the preconditioner is quite feasible to realize in practice. Copyright © 2020, The Authors. All rights reserved.

关键词： Sobolev spaces

Deep neural networks on diffeomorphism groups for optimal shape reparameterization

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arXiv 2022年

作者： Celledoni, Elena Glöckner, Helge Riseth, Jørgen Schmeding, Alexander Department of Mathematics NTNU Trondheim Norway Institut für Mathematik Universität Paderborn Germany Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Norway Department of Mathematics University of Oslo Norway

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting *** Codes 65K10, 58D05, 46T10 © 2022, CC BY.

关键词： Deep neural networks

OPTIMAL CONTROL OF THE LANDAU-DE GENNES MODEL OF NEMATIC LIQUID CRYSTALS

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arXiv 2023年

作者： Surowiec, Thomas M. Walker, Shawn W. Simula Research Laboratory Department of Numerical Analysis and Scientific Computing Kristian Augusts gate 23 Oslo0164 Norway Louisiana State University Baton RougeLA70803 United States

We present an analysis and numerical study of an optimal control problem for the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs), which is a crucial component in modern technology. They exhibit long range orientational order in their nematic phase, which is represented by a tensor-valued (spatial) order parameter Q = Q(x). Equilibrium LC states correspond to Q functions that (locally) minimize an LdG energy functional. Thus, we consider an L2-gradient flow of the LdG energy that allows for finding local minimizers and leads to a semi-linear parabolic PDE, for which we develop an optimal control framework. We then derive several a priori estimates for the forward problem, including continuity in space-time, that allow us to prove existence of optimal boundary and external "force" controls and to derive optimality conditions through the use of an adjoint equation. Next, we present a simple finite element scheme for the LdG model and a straightforward optimization algorithm. We illustrate optimization of LC states through numerical experiments in two and three dimensions that seek to place LC defects (where Q(x) = 0) in desired locations, which is desirable in *** Codes 49M25, 35K91, 65N30 © 2023, CC BY.

关键词： Finite element method

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?

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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

Convergence of finite difference methods for the wave equation in two space dimensions

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arXiv 2017年

作者： Wang, Siyang Nissen, Anna Kreiss, Gunilla Division of Scientific Computing Department of Information Technology Uppsala University UppsalaSE-751 05 Sweden Division of Numerical Analysis Department of Mathematics Royal Institute of Technology StockholmSE-100 44 Sweden Division of Scientific Computing Department of Information Technology Uppsala University UppsalaSE-751 05 Sweden

When using a finite difference method to solve an initial–boundary–value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many previous literatures for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. numerical experiments corroborate our accuracy analysis.65M12, 65M06 Copyright © 2017, The Authors. All rights reserved.

关键词： Finite difference method

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

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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

Feasibility of Acousto-Electric Tomography

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arXiv 2019年

作者： Jensen, Bjørn Kirkeby, Adrian Knudsen, Kim Faculty of Information Technology University of Jyväskylä Finland Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway Department of Applied Mathematics and Computer Science Technical University of Denmark Kgs Lyngby Denmark

In acousto-electric tomography the goal is to reconstruct the electric conductivity in a domain from electrostatic boundary measurements of corresponding currents and voltages, while the domain is penetrated by a time-dependent acoustic wave. In this paper we provide an in-silico model of the entire coupled physics scenario and perform computational experiments. We propose a complete inversion framework for acousto-electric tomography in two steps: First the interior power density is obtained from boundary measurements by solving a linear inverse problem;second the interior conductivity is reconstructed from the power density by solving a non-linear, fairly well-posed problem. We develop a numerical model with realistically chosen parameters inspired by medical imaging. The critical signal strength is analyzed and the omnipresent Johnson-Nyquist noise is estimated. With this setup we perform numerical experiments on synthetic data with noise and demonstrate the feasibility of the method. However, our findings are a mix of positive and negative. We reconstruct features even under severe noise conditions, but we also find that the required Signal-to-Noise ratio yet remains infeasible for practical *** Codes 35R30, 65N21 Copyright © 2019, The Authors. All rights reserved.

关键词： Inverse problems

PROXIMAL GALERKIN: A STRUCTURE-PRESERVING FINITE ELEMENT METHOD FOR POINTWISE BOUND CONSTRAINTS

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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