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,...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.展开更多
Stochastic collocation methods as a promising approach for solving stochastic partial differential equations have been developed rapidly in recent years.Similar to Monte Carlo methods,the stochastic collocation method...Stochastic collocation methods as a promising approach for solving stochastic partial differential equations have been developed rapidly in recent years.Similar to Monte Carlo methods,the stochastic collocation methods are non-intrusive in that they can be implemented via repetitive execution of an existing deterministic solver without modifying it.The choice of collocation points leads to a variety of stochastic collocation methods including tensor product method,Smolyak method,Stroud 2 or 3 cubature method,and adaptive Stroud method.Another type of collocation method,the probabilistic collocation method(PCM),has also been proposed and applied to flow in porous media.In this paper,we discuss these methods in terms of their accuracy,efficiency,and applicable range for flow in spatially correlated random fields.These methods are compared in details under different conditions of spatial variability and correlation length.This study reveals that the Smolyak method and the PCM outperform other stochastic collocation methods in terms of accuracy and efficiency.The random dimensionality in approximating input random fields plays a crucial role in the performance of the stochastic collocation methods.Our numerical experiments indicate that the required random dimensionality increases slightly with the decrease of correlation scale and moderately from one to multiple physical dimensions.展开更多
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-dimensiona...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.展开更多
基金the National Natural Science Foundation of China(No.91130003 and No.11201461).
文摘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.
基金The authors are grateful to the supports by Natural Science Foundation of China through grant 50688901the Chinese National Basic Research Program through grant 2006CB705800+1 种基金the U.S.National Science Foundation through grant 0801425The first author acknowledges the support by China Scholarship Council through grant 2007100458.
文摘Stochastic collocation methods as a promising approach for solving stochastic partial differential equations have been developed rapidly in recent years.Similar to Monte Carlo methods,the stochastic collocation methods are non-intrusive in that they can be implemented via repetitive execution of an existing deterministic solver without modifying it.The choice of collocation points leads to a variety of stochastic collocation methods including tensor product method,Smolyak method,Stroud 2 or 3 cubature method,and adaptive Stroud method.Another type of collocation method,the probabilistic collocation method(PCM),has also been proposed and applied to flow in porous media.In this paper,we discuss these methods in terms of their accuracy,efficiency,and applicable range for flow in spatially correlated random fields.These methods are compared in details under different conditions of spatial variability and correlation length.This study reveals that the Smolyak method and the PCM outperform other stochastic collocation methods in terms of accuracy and efficiency.The random dimensionality in approximating input random fields plays a crucial role in the performance of the stochastic collocation methods.Our numerical experiments indicate that the required random dimensionality increases slightly with the decrease of correlation scale and moderately from one to multiple physical dimensions.
基金国家自然科学基金重点项目(90307017)国家自然科学基金(60676018,60806031)+3 种基金国家“九七三”重点基础研究发展规划项目(2005CB321701)教育部跨世纪优秀人才培养计划基金教育部高等学校博士学科点专项科研基金(20050246082)US National Science Foundation grants(CCF-0727791)
基金support of the China Scholarship Committee(No.2008633049)for this researchsupport by OSD/AFOSR FA9550-09-1-0613,NSF,and DoE.
文摘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.
