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sparse-Grid Implementation of Fixed-Point Fast Sweeping WENO Schemes for Eikonal Equations

Communications on Applied Mathematics and Computation 2024年第1期6卷 3-29页

作者： Zachary M.Miksis Yong-Tao Zhang Department of Applied and Computational Mathematics and Statistics University of Notre DameNotre DameIN46556USA

Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.

关键词： Fixed-point fast sweeping methods Weighted essentially non-oscillatory(WENO)schemes sparse grids Static Hamilton-Jacobi(H-J)equations Eikonal equations

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Selected Recent Applications of sparse grids

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Numerical Mathematics(Theory,Methods and Applications) 2015年第1期8卷 47-77页

作者： Benjamin Peherstorfer Christoph Kowitz Dirk Pfluger Hans-Joachim Bungartz Scientific Computing Group Dept.of Computer ScienceBoltzmannstr.385748 GarchingGermany Institutefor Advanced Study Technische UniversitätMünchenLichtenbergstr.2a85748 GarchingGermany Institute for Parallel and Distributed Systems University of StuttgartUni-versitätsstr.3870569 StuttgartGermany

sparse grids have become a versatile tool for a vast range of applications reaching from interpolation and numerical quadrature to data-driven problems and uncertainty quantification.We review four selected real-world applications of sparse grids:financial product pricing with the Black-Scholes model,interactive explo-ration of simulation data with sparse-grid-based surrogate models,analysis of simu-lation data through sparse grid data mining methods,and stability investigations of plasma turbulence simulations.

关键词： sparse grids hierarchical decomposition adaptivity combination technique

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Efficient sparse-Grid Implementation of a Fifth-Order Multi-resolution WENO Scheme for Hyperbolic Equations

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Communications on Applied Mathematics and Computation 2023年第4期5卷 1339-1364页

作者： Ernie Tsybulnik Xiaozhi Zhu Yong-Tao Zhang Department of Applied and Computational Mathematics and Statistics University of Notre DameNotre DameIN46556USA

High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.

关键词： Weighted essentially non-oscillatory(WENO)schemes Multi-resolution WENO schemes sparse grids High spatial dimensions Hyperbolic partial differential equations(PDEs)

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Efficient Solution of Ordinary Differential Equations with High-Dimensional Parametrized Uncertainty

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Communications in Computational Physics 2011年第7期10卷 253-278页

作者： Zhen Gao Jan S.Hesthaven Research Center for Applied Mathematics Ocean University of ChinaQingdao 266071ShandongChina Division of Applied Mathematics Brown University182 George StreetProvidenceRI 02912USA

The important task of evaluating the impact of random parameters on the output of stochastic ordinary differential equations(SODE)can be computationally very demanding,in particular for problems with a high-dimensional parameter space.In this work we consider this problem in some detail and demonstrate that by combining several techniques one can dramatically reduce the overall cost without impacting the predictive accuracy of the output of interests.We discuss how the combination of ANOVA expansions,different sparse grid techniques,and the total sensitivity index(TSI)as a pre-selective mechanism enables the modeling of problems with hundred of parameters.We demonstrate the accuracy and efficiency of this approach on a number of challenging test cases drawn from engineering and science.

关键词： sparse grids stochastic collocation method ANOVA expansion total sensitivity index

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Variance-Based Global Sensitivity Analysis via sparse-Grid Interpolation and Cubature

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Communications in Computational Physics 2011年第3期9卷 542-567页

作者： Gregery T.Buzzard Dongbin Xiu Department of Mathematics Purdue UniversityWest LafayetteIN 47907USA

The stochastic collocation method using sparse grids has become a popular choice for performing stochastic computations in high dimensional(random)parameter space.In addition to providing highly accurate stochastic solutions,the sparse grid collocation results naturally contain sensitivity information with respect to the input random parameters.In this paper,we use the sparse grid interpolation and cubature methods of Smolyak together with combinatorial analysis to give a computationally efficient method for computing the global sensitivity values of Sobol’.This method allows for approximation of all main effect and total effect values from evaluation of f on a single set of sparse grids.We discuss convergence of this method,apply it to several test cases and compare to existing methods.As a result which may be of independent interest,we recover an explicit formula for evaluating a Lagrange basis interpolating polynomial associated with the Chebyshev extrema.This allows one to manipulate the sparse grid collocation results in a highly efficient manner.

关键词： Stochastic collocation sparse grids sensitivity analysis Smolyak Sobol’

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Sequential Multiscale Modeling Using sparse Representation

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Communications in Computational Physics 2008年第10期4卷 1025-1033页

作者： Carlos J.Garcıa-Cervera Weiqing Ren Jianfeng Lu Weinan E Mathematics Department University of CaliforniaSanta BarbaraCA 93106USA Courant Institute of Mathematical Sciences New York UniversityNew YorkNY 10012USA Program in Applied and ComputationalMathematics Princeton UniversityPrincetonNJ 08544USA Department of Mathematics and PACM Princeton UniversityPrincetonNJ 08544USA

The main obstacle in sequential multiscale modeling is the pre-computation of the constitutive relationwhich often involvesmany independent variables.The constitutive relation of a polymeric fluid is a function of six variables,even after making the simplifying assumption that stress depends only on the rate of strain.Precomputing such a function is usually considered too expensive.Consequently the value of sequential multiscale modeling is often limited to“parameter passing”.Here we demonstrate that sparse representations can be used to drastically reduce the computational cost for precomputing functions of many variables.This strategy dramatically increases the efficiency of sequential multiscale modeling,making it very competitive in many situations.

关键词： Multiscale modeling sparse grids

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