In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector f...In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector field, the solution of this stochastic partial differential equation is proved to have moments. The exact p-th moment is represented through integrals with respect to Brownian motions. The basic tool is Girsanov formula.展开更多
In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for ...In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.展开更多
The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for s...The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for solving SPDEs have been used such as, finite difference and finite element schemes [1]-[5]. Also, some practical techniques like the method of lines for boundary value problems have been applied to the linear stochastic partial differential equations, and the outcomes of these approaches have been experimented numerically [7]. In [8]-[10], the author discussed mean square convergent finite difference method for solving some random partial differential equations. Random numerical techniques for both ordinary and partial random differential equations are treated in [4] [10]. As regards applications using explicit analytic solutions or numerical methods, a few results may be found in [5] [6] [11]. This article focuses on solving random heat equation by using Crank-Nicol- son technique under mean square sense and it is organized as follows. In Section 2, the mean square calculus preliminaries that will be required throughout the paper are presented. In Section 3, the Crank-Nicolson scheme for solving the random heat equation is presented. In Section 4, some case studies are showed. Short conclusions are cleared in the end section.展开更多
The generalized master equation for the space-time coupled continuous time random walk is derived analytically, in which the space-time coupling is considered through the correlated function 9(t) ~ t^γ, 0 ≤ γ 〈...The generalized master equation for the space-time coupled continuous time random walk is derived analytically, in which the space-time coupling is considered through the correlated function 9(t) ~ t^γ, 0 ≤ γ 〈 2, and the probability density function ω(t) of a particle's waiting time t follows a power law form for large t: ω(t) ~t^-(1+α), 0 〈 α 〈 1. The results indicate that the expressions of the generalized master equation are determined by the correlation exponent 7 and the long-tailed index α of the waiting time. Moreover, the diffusion results obtained from the generalized master equation are in accordance with the previous known results and the numerical simulation results.展开更多
In this paper we first prove a Darbao type fixed point theorem for a system of continuous random operators with random domains. Thenb, by using the theorem. wegive the existence criteria of solutions for a systems of ...In this paper we first prove a Darbao type fixed point theorem for a system of continuous random operators with random domains. Thenb, by using the theorem. wegive the existence criteria of solutions for a systems of nonlinear random Volterraintegral equations and for the Cauchy problem of a system of nonlinear random differential equations. The existence of extremal random solutions and random comparison results for these systems of random equations are also obtained Our theorems improve and generalize the corresponding results of Vaughn Lakshmikantham Lakshmidantham-Leela De blasi-Myjak and Ding展开更多
The differential transformation method (DTM) is applied to solve the second-order random differential equations. Several examples are represented to demonstrate the effectiveness of the proposed method. The results sh...The differential transformation method (DTM) is applied to solve the second-order random differential equations. Several examples are represented to demonstrate the effectiveness of the proposed method. The results show that DTM is an efficient and accurate technique for finding exact and approximate solutions.展开更多
The concepts of Markov process in random environment and homogeneous random transition functions are introduced. The necessary and sufficient conditions for homogeneous random transition function are given. The main r...The concepts of Markov process in random environment and homogeneous random transition functions are introduced. The necessary and sufficient conditions for homogeneous random transition function are given. The main results in this article are the analytical properties, such as continuity, differentiability, random Kolmogorov backward equation and random Kolmogorov forward equation of homogeneous random transition functions.展开更多
We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,...We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,we construct a small number of reduced basis functions within each coarse grid block,which can then be used to approximate the multiscale finite element basis functions.In the online stage,we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid,without requiring the coarsest grid to resolve the smallestscale of the solution.We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients.Moreover,we provide convergence analysis of the proposed method.Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.展开更多
In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations(RFDEs)generate the random ...In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations(RFDEs)generate the random dynamical system(RDS) if and only if all the solutions are globally defined, and establish the comparison theorem for RFDEs and the random Riesz representation theorem. These three results lead to the Borel measurability of coefficient functions in the Riesz representation of variational equations for quasimonotone RFDEs, which paves the way following the Smith line to establish eventual strong monotonicity for the RDS under cooperative and irreducible conditions. Then strong comparison principles, strong sublinearity theorems and the existence of random attractors for RFDEs are proved. Finally, criteria are presented for the existence of a unique random equilibrium and its global stability in the universe of all the tempered random closed sets of the positive cone. Applications to typical random or stochastic delay models in monotone dynamical systems,such as biochemical control circuits, cyclic gene models and Hopfield-type neural networks, are given.展开更多
In this paper we consider a random evolution equation with small perturbations, and show how to construct coupled solutions to the equation. As applications, we prove the Feller continuity of the solutions and the exi...In this paper we consider a random evolution equation with small perturbations, and show how to construct coupled solutions to the equation. As applications, we prove the Feller continuity of the solutions and the existence and uniqueness of invariant measures. Furthermore, we establish a large deviations principle for the family of invariant measures as the perturbations tend to zero.展开更多
In this paper, we apply stochastic Galerkin finite element methods to the optimal control problem governed by an elliptic integral-differential PDEs with random field. The control problem has the control constraints o...In this paper, we apply stochastic Galerkin finite element methods to the optimal control problem governed by an elliptic integral-differential PDEs with random field. The control problem has the control constraints of obstacle type. A new gradient algorithm based on the pre-conditioner conjugate gradient algorithm (PCG) is developed for this optimal control problem. This algorithm can transform a part of the state equation matrix and co-state equation matrix into block diagonal matrix and then solve the optimal control systems iteratively. The proof of convergence for this algorithm is also discussed. Finally numerical examples of a medial size are presented to illustrate our theoretical results.展开更多
In this paper, a hybird approximation scheme for an optimal control problem governed by an elliptic equation with random field in its coefficients is considered. The random coefficients are smooth in the physical spac...In this paper, a hybird approximation scheme for an optimal control problem governed by an elliptic equation with random field in its coefficients is considered. The random coefficients are smooth in the physical space and depend on a large number of random variables in the probability space. The necessary and sufficient optimality conditions for the optimal control problem are obtained. The scheme is established to approximate the optimality system through the discretization by using finite volume element method for the spatial space and a sparse grid stochastic collocation method based on the Smolyak approximation for the probability space, respectively. This scheme naturally leads to the discrete solutions of an uncoupled deterministic problem. The existence and uniqueness of the discrete solutions are proved. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.展开更多
This paper is concerned with the application of generalized polynomial chaos (gPC) method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given fo...This paper is concerned with the application of generalized polynomial chaos (gPC) method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given for the error arising from finite-dimensional noise (FDN) assumption, projection error, aliasing error and discretization error. In the end, with several numerical experiments, the theoretical results are further illustrated.展开更多
The perturbations of nonholonomic mechanical systems under the Gauss white noises are studied in this paper. It is proved that the differential equations of the first-order moments of the solution process coincide wit...The perturbations of nonholonomic mechanical systems under the Gauss white noises are studied in this paper. It is proved that the differential equations of the first-order moments of the solution process coincide with the corresponding equations in the non-perturbational case, and that there are e2 -terms but no e-terms in the differential equations of the second-order moments. Two propositions are obtained. Finally, an example is given to illustrate the application of the results.展开更多
This study aimed to explore traffic safety climate by quantifying driving conditions and driving behaviour.To achieve the objective,the random parameter structural equation model was proposed so that driver action and...This study aimed to explore traffic safety climate by quantifying driving conditions and driving behaviour.To achieve the objective,the random parameter structural equation model was proposed so that driver action and driving condition can address the safety climate by integrating crash features,vehicle profiles,roadway conditions and environment conditions.The geo-localized crash open data of Las Vegas metropolitan area were collected from 2014 to 2016,including 27 arterials with 16827 injury samples.By quantifying the driving conditions and driving actions,the random parameter structural equation model was built up with measurement variables and latent variables.Results revealed that the random parameter structural equation model can address traffic safety climate quantitatively,while driving conditions and driving actions were quantified and reflected by vehicles,road environment and crash features correspondingly.The findings provide potential insights for practitioners and policy makers to improve the driving environment and traffic safety culture.展开更多
In this paper,we first investigate some basic properties of asymptotically mean almost periodic random sequences on Z + and then show some properties of asymptotically mean almost periodic solutions to random differen...In this paper,we first investigate some basic properties of asymptotically mean almost periodic random sequences on Z + and then show some properties of asymptotically mean almost periodic solutions to random difference equations.展开更多
Cross-diffusion is a ubiquitous phenomenon in complex networks, but it is often neglected in the study of reaction–diffusion networks. In fact, network connections are often random. In this paper, we investigate patt...Cross-diffusion is a ubiquitous phenomenon in complex networks, but it is often neglected in the study of reaction–diffusion networks. In fact, network connections are often random. In this paper, we investigate pattern dynamics of random networks with cross-diffusion by using the method of network analysis and obtain a condition under which the network loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation for the network and prove the stability of the amplitude equation which is also an effective tool to investigate pattern dynamics of the random network with cross diffusion. In the meantime, the pattern formation consistently matches the stability of the system and the amplitude equation is verified by simulations. A novel approach to the investigation of specific real systems was presented in this paper. Finally, the example and simulation used in this paper validate our theoretical results.展开更多
A computer software named VDAS (Vehicle Dynamic Analysis Simulation) is presented.Based on the mathematical modelling for a military vehicle as a 3-D dynamic system, the softwarecreates random excitation resulting fro...A computer software named VDAS (Vehicle Dynamic Analysis Simulation) is presented.Based on the mathematical modelling for a military vehicle as a 3-D dynamic system, the softwarecreates random excitation resulting from double side road-surface roughness by statisticalsimulation, and solves for the time series of the system response from the system state equations,and gives out the results in probability statistics as well as performance evaluations.展开更多
A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented.The uncertain parameters are modeled as random variables,and the governing equations are treated as sto...A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented.The uncertain parameters are modeled as random variables,and the governing equations are treated as stochastic.The solutions,or quantities of interests,are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters.A high-order stochastic collocation method is employed to solve the solution statistics,and more importantly,to reconstruct the polynomial expansion.While retaining the high accuracy by polynomial expansion,the resulting“pseudo-spectral”type algorithm is straightforward to implement as it requires only repetitive deterministic simulations.An estimate on error bounded is presented,along with numerical examples for problems with relatively complicated forms of governing equations.展开更多
Missing data and time-dependent covariates often arise simultaneously in longitudinal studies,and directly applying classical approaches may result in a loss of efficiency and biased estimates.To deal with this proble...Missing data and time-dependent covariates often arise simultaneously in longitudinal studies,and directly applying classical approaches may result in a loss of efficiency and biased estimates.To deal with this problem,we propose weighted corrected estimating equations under the missing at random mechanism,followed by developing a shrinkage empirical likelihood estimation approach for the parameters of interest when time-dependent covariates are present.Such procedure improves efficiency over generalized estimation equations approach with working independent assumption,via combining the independent estimating equations and the extracted additional information from the estimating equations that are excluded by the independence assumption.The contribution from the remaining estimating equations is weighted according to the likelihood of each equation being a consistent estimating equation and the information it carries.We show that the estimators are asymptotically normally distributed and the empirical likelihood ratio statistic and its profile counterpart follow central chi-square distributions asymptotically when evaluated at the true parameter.The practical performance of our approach is demonstrated through numerical simulations and data analysis.展开更多
文摘In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector field, the solution of this stochastic partial differential equation is proved to have moments. The exact p-th moment is represented through integrals with respect to Brownian motions. The basic tool is Girsanov formula.
文摘In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.
文摘The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for solving SPDEs have been used such as, finite difference and finite element schemes [1]-[5]. Also, some practical techniques like the method of lines for boundary value problems have been applied to the linear stochastic partial differential equations, and the outcomes of these approaches have been experimented numerically [7]. In [8]-[10], the author discussed mean square convergent finite difference method for solving some random partial differential equations. Random numerical techniques for both ordinary and partial random differential equations are treated in [4] [10]. As regards applications using explicit analytic solutions or numerical methods, a few results may be found in [5] [6] [11]. This article focuses on solving random heat equation by using Crank-Nicol- son technique under mean square sense and it is organized as follows. In Section 2, the mean square calculus preliminaries that will be required throughout the paper are presented. In Section 3, the Crank-Nicolson scheme for solving the random heat equation is presented. In Section 4, some case studies are showed. Short conclusions are cleared in the end section.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11605003 and 11547231
文摘The generalized master equation for the space-time coupled continuous time random walk is derived analytically, in which the space-time coupling is considered through the correlated function 9(t) ~ t^γ, 0 ≤ γ 〈 2, and the probability density function ω(t) of a particle's waiting time t follows a power law form for large t: ω(t) ~t^-(1+α), 0 〈 α 〈 1. The results indicate that the expressions of the generalized master equation are determined by the correlation exponent 7 and the long-tailed index α of the waiting time. Moreover, the diffusion results obtained from the generalized master equation are in accordance with the previous known results and the numerical simulation results.
文摘In this paper we first prove a Darbao type fixed point theorem for a system of continuous random operators with random domains. Thenb, by using the theorem. wegive the existence criteria of solutions for a systems of nonlinear random Volterraintegral equations and for the Cauchy problem of a system of nonlinear random differential equations. The existence of extremal random solutions and random comparison results for these systems of random equations are also obtained Our theorems improve and generalize the corresponding results of Vaughn Lakshmikantham Lakshmidantham-Leela De blasi-Myjak and Ding
文摘The differential transformation method (DTM) is applied to solve the second-order random differential equations. Several examples are represented to demonstrate the effectiveness of the proposed method. The results show that DTM is an efficient and accurate technique for finding exact and approximate solutions.
基金Supported by the NNSF of China (10371092)the Foundation of Wuhan University.
文摘The concepts of Markov process in random environment and homogeneous random transition functions are introduced. The necessary and sufficient conditions for homogeneous random transition function are given. The main results in this article are the analytical properties, such as continuity, differentiability, random Kolmogorov backward equation and random Kolmogorov forward equation of homogeneous random transition functions.
基金partially supported by the Hong Kong Ph D Fellowship Schemesupported by the Hong Kong RGC General Research Funds(Projects 27300616,17300817,and 17300318)+2 种基金National Natural Science Foundation of China(Project 11601457)Seed Funding Programme for Basic Research(HKU)Basic Research Programme(JCYJ20180307151603959)of the Science,Technology and Innovation Commission of Shenzhen Municipality。
文摘We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,we construct a small number of reduced basis functions within each coarse grid block,which can then be used to approximate the multiscale finite element basis functions.In the online stage,we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid,without requiring the coarsest grid to resolve the smallestscale of the solution.We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients.Moreover,we provide convergence analysis of the proposed method.Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.
基金supported by National Natural Science Foundation of China (Grant Nos.12171321, 11771295, 11371252 and 31770470)。
文摘In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations(RFDEs)generate the random dynamical system(RDS) if and only if all the solutions are globally defined, and establish the comparison theorem for RFDEs and the random Riesz representation theorem. These three results lead to the Borel measurability of coefficient functions in the Riesz representation of variational equations for quasimonotone RFDEs, which paves the way following the Smith line to establish eventual strong monotonicity for the RDS under cooperative and irreducible conditions. Then strong comparison principles, strong sublinearity theorems and the existence of random attractors for RFDEs are proved. Finally, criteria are presented for the existence of a unique random equilibrium and its global stability in the universe of all the tempered random closed sets of the positive cone. Applications to typical random or stochastic delay models in monotone dynamical systems,such as biochemical control circuits, cyclic gene models and Hopfield-type neural networks, are given.
基金Partially supported by the National Natural Science Foundation of China,Grant No.69874003
文摘In this paper we consider a random evolution equation with small perturbations, and show how to construct coupled solutions to the equation. As applications, we prove the Feller continuity of the solutions and the existence and uniqueness of invariant measures. Furthermore, we establish a large deviations principle for the family of invariant measures as the perturbations tend to zero.
基金This work was supported by National Natural Science Foundation of China (No. 11501326).
文摘In this paper, we apply stochastic Galerkin finite element methods to the optimal control problem governed by an elliptic integral-differential PDEs with random field. The control problem has the control constraints of obstacle type. A new gradient algorithm based on the pre-conditioner conjugate gradient algorithm (PCG) is developed for this optimal control problem. This algorithm can transform a part of the state equation matrix and co-state equation matrix into block diagonal matrix and then solve the optimal control systems iteratively. The proof of convergence for this algorithm is also discussed. Finally numerical examples of a medial size are presented to illustrate our theoretical results.
基金The work is partly supported by the Natural Science Foundation of China (Grant No. 11271231, 11301300, 61572297), by the Shandong Province Outstanding Young Scientists Research Award Fhnd Project (Grant No. BS2013DX010), by the Natural Science Foundation of Shandong Province, China (Grant No. ZR2014FM003), and by the Shandong Academy of Sciences Youth Fund Project (Grant No. 2013QN007).
文摘In this paper, a hybird approximation scheme for an optimal control problem governed by an elliptic equation with random field in its coefficients is considered. The random coefficients are smooth in the physical space and depend on a large number of random variables in the probability space. The necessary and sufficient optimality conditions for the optimal control problem are obtained. The scheme is established to approximate the optimality system through the discretization by using finite volume element method for the spatial space and a sparse grid stochastic collocation method based on the Smolyak approximation for the probability space, respectively. This scheme naturally leads to the discrete solutions of an uncoupled deterministic problem. The existence and uniqueness of the discrete solutions are proved. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.
基金Supported by the National Natural Science Foundation of China(No.11501427,11571128)
文摘This paper is concerned with the application of generalized polynomial chaos (gPC) method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given for the error arising from finite-dimensional noise (FDN) assumption, projection error, aliasing error and discretization error. In the end, with several numerical experiments, the theoretical results are further illustrated.
文摘The perturbations of nonholonomic mechanical systems under the Gauss white noises are studied in this paper. It is proved that the differential equations of the first-order moments of the solution process coincide with the corresponding equations in the non-perturbational case, and that there are e2 -terms but no e-terms in the differential equations of the second-order moments. Two propositions are obtained. Finally, an example is given to illustrate the application of the results.
基金supported by National Natural Science Foundation of China(No.52072214).
文摘This study aimed to explore traffic safety climate by quantifying driving conditions and driving behaviour.To achieve the objective,the random parameter structural equation model was proposed so that driver action and driving condition can address the safety climate by integrating crash features,vehicle profiles,roadway conditions and environment conditions.The geo-localized crash open data of Las Vegas metropolitan area were collected from 2014 to 2016,including 27 arterials with 16827 injury samples.By quantifying the driving conditions and driving actions,the random parameter structural equation model was built up with measurement variables and latent variables.Results revealed that the random parameter structural equation model can address traffic safety climate quantitatively,while driving conditions and driving actions were quantified and reflected by vehicles,road environment and crash features correspondingly.The findings provide potential insights for practitioners and policy makers to improve the driving environment and traffic safety culture.
基金Supported by the NNSF of China(No.11171191)the NSF of Shandong Province(No.ZR2010AL011)
文摘In this paper,we first investigate some basic properties of asymptotically mean almost periodic random sequences on Z + and then show some properties of asymptotically mean almost periodic solutions to random difference equations.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11272277,11572278,and 11572084)the Innovation Scientists and Technicians Troop Construction Projects of Henan Province,China(Grant No.2017JR0013)
文摘Cross-diffusion is a ubiquitous phenomenon in complex networks, but it is often neglected in the study of reaction–diffusion networks. In fact, network connections are often random. In this paper, we investigate pattern dynamics of random networks with cross-diffusion by using the method of network analysis and obtain a condition under which the network loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation for the network and prove the stability of the amplitude equation which is also an effective tool to investigate pattern dynamics of the random network with cross diffusion. In the meantime, the pattern formation consistently matches the stability of the system and the amplitude equation is verified by simulations. A novel approach to the investigation of specific real systems was presented in this paper. Finally, the example and simulation used in this paper validate our theoretical results.
文摘A computer software named VDAS (Vehicle Dynamic Analysis Simulation) is presented.Based on the mathematical modelling for a military vehicle as a 3-D dynamic system, the softwarecreates random excitation resulting from double side road-surface roughness by statisticalsimulation, and solves for the time series of the system response from the system state equations,and gives out the results in probability statistics as well as performance evaluations.
文摘A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented.The uncertain parameters are modeled as random variables,and the governing equations are treated as stochastic.The solutions,or quantities of interests,are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters.A high-order stochastic collocation method is employed to solve the solution statistics,and more importantly,to reconstruct the polynomial expansion.While retaining the high accuracy by polynomial expansion,the resulting“pseudo-spectral”type algorithm is straightforward to implement as it requires only repetitive deterministic simulations.An estimate on error bounded is presented,along with numerical examples for problems with relatively complicated forms of governing equations.
基金supported by the NNSF of China(No.11271347)the Fundamental Research Funds for the Central Universities
文摘Missing data and time-dependent covariates often arise simultaneously in longitudinal studies,and directly applying classical approaches may result in a loss of efficiency and biased estimates.To deal with this problem,we propose weighted corrected estimating equations under the missing at random mechanism,followed by developing a shrinkage empirical likelihood estimation approach for the parameters of interest when time-dependent covariates are present.Such procedure improves efficiency over generalized estimation equations approach with working independent assumption,via combining the independent estimating equations and the extracted additional information from the estimating equations that are excluded by the independence assumption.The contribution from the remaining estimating equations is weighted according to the likelihood of each equation being a consistent estimating equation and the information it carries.We show that the estimators are asymptotically normally distributed and the empirical likelihood ratio statistic and its profile counterpart follow central chi-square distributions asymptotically when evaluated at the true parameter.The practical performance of our approach is demonstrated through numerical simulations and data analysis.