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Uncertain eigenvalue analysis by the sparse grid stochastic collocation method

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Acta Mechanica Sinica 2015年第4期31卷 545-557页

作者： J.C.Lan X.J.Dong Z.K.Peng W.M.Zhang G.Meng State Key Laboratory of Mechanical Systems and Vibration School of Mechanical Engineering Shanghai Jiao Tong University

In this paper, the eigenvalue problem with multiple uncertain parameters is analyzed by the sparse grid stochastic collocation method. This method provides an interpolation approach to approximate eigenvalues and eigenvectors＇ functional dependencies on uncertain parame- ters. This method repetitively evaluates the deterministic solutions at the pre-selected nodal set to construct a high- dimensional interpolation formula of the result. Taking advantage of the smoothness of the solution in the uncer- tain space, the sparse grid collocation method can achieve a high order accuracy with a small nodal set. Compared with other sampling based methods, this method converges fast with the increase of the number of points. Some numerical examples with different dimensions are presented to demon- strate the accuracy and efficiency of the sparse grid stochastic collocation method.

关键词： Uncertainty quantification Eigenvalue Eigenvector sparse grid Stochastic collocation methodEigenvector pairing

A Numerical Comparison Between Quasi-Monte Carlo and sparse grid Stochastic Collocation Methods

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Communications in Computational Physics 2012年第9期12卷 1051-1069页

作者： Juarez dos Santos Azevedo Saulo Pomponet Oliveira CETEC-UFRB Centro44380-000Cruz das Almas-BABrazil.2 DMAT-UFPRCentro Politecnico81531-980Curitiba-PRBrazil

Quasi-Monte Carlo methods and stochastic collocation methods based on sparse grids have become popular with solving stochastic partial differential equations.These methods use deterministic points for multi-dimensional integration or interpolation without suffering from the curse of dimensionality.It is not evident which method is best,specially on random models of physical phenomena.We numerically study the error of quasi-Monte Carlo and sparse gridmethods in the context of groundwater flow in heterogeneous media.In particular,we consider the dependence of the variance error on the stochastic dimension and the number of samples/collocation points for steady flow problems in which the hydraulic conductivity is a lognormal process.The suitability of each technique is identified in terms of computational cost and error tolerance.

关键词： Karhunen-Loève expansion Monte Carlo quasi-Monte Carlo sparse grid

An ADI sparse grid method for Pricing Efficiently American Options under the Heston Model

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Advances in Applied Mathematics and Mechanics 2021年第6期13卷 1384-1397页

作者： A.Clevenhaus M.Ehrhardt M.Gunther Institute of Mathematical Modelling Analysis and Computational Mathematics(IMACM)Chair of Applied Mathematics and Numerical AnalysisBergische Universitat WuppertalGaußstraße 2042119 WuppertalGermany

One goal of financial research is to determine fair prices on the financial market.As financial models and the data sets on which they are based are becoming ever larger and thus more complex,financial instruments must be further developed to adapt to the new complexity,with short runtimes and efficient use of memory space.Here we show the effects of combining known strategies and incorporating new ideas to further improve numerical techniques in computational finance.In this paper we combine an ADI(alternating direction implicit)scheme for the temporal discretization with a sparse grid approach and the combination technique.The later approach considerably reduces the number of“spatial”grid points.The presented standard financial problem for the valuation of American options using the Heston model is chosen to illustrate the advantages of our approach,since it can easily be adapted to other more complex models.

关键词： sparse grid combination technique American options ADI Heston model

An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

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Communications on Applied Mathematics and Computation 2022年第1期4卷 60-83页

作者： Zhanjing Tao Juntao Huang Yuan Liu Wei Guo Yingda Cheng School of Mathematics Jilin UniversityJilin 130012China Department of Mathematics Michigan State UniversityEast LansingMI 48824USA Department of Mathematics Statistics and PhysicsWichita State UniversityWichitaKS 67260USA Department of Mathematics and Statistics Texas Tech UniversityLubbockTX 70409USA Department of Mathematics Department of Computational MathematicsScience and EngineeringMichigan State UniversityEast LansingMI 48824USA

This paper develops a high-order adaptive scheme for solving nonlinear Schrödinger equa-tions.The solutions to such equations often exhibit solitary wave and local structures,which make adaptivity essential in improving the simulation efficiency.Our scheme uses the ultra-weak discontinuous Galerkin(DG)formulation and belongs to the framework of adaptive multiresolution schemes.Various numerical experiments are presented to demon-strate the excellent capability of capturing the soliton waves and the blow-up phenomenon.

关键词： Multiresolution sparse grid Ultra-weak discontinuous Galerkin method Schrödinger equation Adaptivity

A Stochastic Collocation Method for Delay Differential Equations with Random Input

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Advances in Applied Mathematics and Mechanics 2014年第4期6卷 403-418页

作者： Tao Zhou Institute of Computational Mathematics Academy of Mathematics and Systems SciencesThe Chinese Academy of SciencesBeijing 100190China

In this work,we concern with the numerical approach for delay differential equations with random coefficients.We first show that the exact solution of the problem considered admits good regularity in the random space,provided that the given data satisfy some reasonable assumptions.A stochastic collocation method is proposed to approximate the solution in the random space,and we use the Legendre spectral collocation method to solve the resulting deterministic delay differential equations.Convergence property of the proposed method is analyzed.It is shown that the numerical method yields the familiar exponential order of convergence in both the random space and the time space.Numerical examples are given to illustrate the theoretical results.

关键词： Delay differential equations stochastic collocation sparse grid legendre spectral method