A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

authored by: Michal L. Mika, Thomas J.R. Hughes, Dominik Schillinger, Peter Wriggers, René R. Hiemstra
Abstract: The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L²-orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2d dimensional domain, where d, in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C⁰-conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness.
Organisation(s): Institute of Mechanics and Computational Mechanics
Institute of Continuum Mechanics
External Organisation(s): University of Texas at Austin
Type: Article
Journal: Computer Methods in Applied Mechanics and Engineering
Volume: 379
ISSN: 0045-7825
Publication date: 01.06.2021
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Mechanics of Materials, Mechanical Engineering, Physics and Astronomy(all), Computer Science Applications
Electronic version(s): http://arxiv.org/abs/2011.13861 (Access: Open)
https://doi.org/10.1016/j.cma.2021.113730 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{5da16ef6d77646b19a3fce060de82327,
title = "A matrix-free isogeometric Galerkin method for Karhunen–Lo{\`e}ve approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature",
abstract = "The Karhunen–Lo{\`e}ve series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L2-orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2d dimensional domain, where d, in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C0-conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness.",
keywords = "Fredholm integral eigenvalue problem, Isogeometric analysis, Kronecker products, Matrix-free solver, Random fields",
author = "Mika, {Michal L.} and Hughes, {Thomas J.R.} and Dominik Schillinger and Peter Wriggers and Hiemstra, {Ren{\'e} R.}",
note = "Funding Information: M.L. Mika, R.R. Hiemstra and D. Schillinger gratefully acknowledge funding from the German Research Foundation through the DFG Emmy Noether Award SCH 1249/2-1. T.J.R. Hughes and R.R. Hiemstra were partially supported by the National Science Foundation Industry/University Cooperative Research Center (IUCRC) for Efficient Vehicles and Sustainable Transportation Systems (EV-STS), and the United States Army CCDC Ground Vehicle Systems Center (TARDEC/NSF Project # 1650483 AMD 2 ). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors thank Mona Dannert and Udo Nackenhorst for very helpful discussions and comments. ",
year = "2021",
month = jun,
day = "1",
doi = "10.1016/j.cma.2021.113730",
language = "English",
volume = "379",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0045-7825",
publisher = "Elsevier",
}