A novel method based on time-dependent stochastic orthogonal bases for stochastic response surface approximation is proposed to overcome the problem of significant errors in the utilization of the generalized polynomi...A novel method based on time-dependent stochastic orthogonal bases for stochastic response surface approximation is proposed to overcome the problem of significant errors in the utilization of the generalized polynomial chaos(GPC) method that approximates the stochastic response by orthogonal polynomials. The accuracy and effectiveness of the method are illustrated by different numerical examples including both linear and nonlinear problems. The results indicate that the proposed method modifies the stochastic bases adaptively, and has a better approximation for the probability density function in contrast to the GPC method.展开更多
Stationarity of a class of stochastically interconnecteil discrete-timesystems is analyzed by utilizins results from ergodic theory of general stateMarkov chains, incorporated with the so called large-scale system app...Stationarity of a class of stochastically interconnecteil discrete-timesystems is analyzed by utilizins results from ergodic theory of general stateMarkov chains, incorporated with the so called large-scale system approach.展开更多
We consider qualitatively robust predictive mappings of stochastic environmental models, where protection against outlier data is incorporated. We utilize digital representations of the models and deploy stochastic bi...We consider qualitatively robust predictive mappings of stochastic environmental models, where protection against outlier data is incorporated. We utilize digital representations of the models and deploy stochastic binary neural networks that are pre-trained to produce such mappings. The pre-training is implemented by a back propagating supervised learning algorithm which converges almost surely to the probabilities induced by the environment, under general ergodicity conditions.展开更多
In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(1/2,1). In order to prove ...In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(1/2,1). In order to prove convergence, we use rough paths techniques. Theoretical bounds are established and numerical simulations are displayed.展开更多
Ocean basin is modeled as a two-dimensional closed,bounded domain in which the fluid flow is governed by the complex partial differential equations in the flow function.Keeping in view that the ocean currents are non-...Ocean basin is modeled as a two-dimensional closed,bounded domain in which the fluid flow is governed by the complex partial differential equations in the flow function.Keeping in view that the ocean currents are non-viscous,no normal flow conditions are used at the basin boundaries.The parameters investigated here are:Coriolis parameter,wind stress coefficient,and latitude.Stochastic differential equations in time scales are solved by deterministic and stochastic methods.Deterministic results concluded that streamlines are symmetric about stagnation point(no flow)for 0<R_(p)<6.57.Stochastic controls are introduced to account for variability in time scales.Euler-Maruyama(direct)and Fokker-Planck equation schemes(indirect)are proposed.It is concluded that stream functions in both direct and indirect methods are of the same qualitatively and quantitatively when 0<R_(p)<79.展开更多
As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans, the time discretization of these equations by an implicit Euler scheme is studied. From the determ...As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans, the time discretization of these equations by an implicit Euler scheme is studied. From the deterministic point of view, the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions. From the probabilistic viewpoint, this paper deals with a wide class of nonlinear, state dependent, white noise forcings which may be interpreted in either the Itor the Stratonovich sense. The proof of convergence of the Euler scheme,which is carried out within an abstract framework, covers the equations for the oceans, the atmosphere, the coupled oceanic-atmospheric system as well as other related geophysical equations. The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense, a result which is new by itself to the best of our knowledge.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.11632011,11572189,and 51421092)the China Postdoctoral Science Foundation(No.2016M601585)
文摘A novel method based on time-dependent stochastic orthogonal bases for stochastic response surface approximation is proposed to overcome the problem of significant errors in the utilization of the generalized polynomial chaos(GPC) method that approximates the stochastic response by orthogonal polynomials. The accuracy and effectiveness of the method are illustrated by different numerical examples including both linear and nonlinear problems. The results indicate that the proposed method modifies the stochastic bases adaptively, and has a better approximation for the probability density function in contrast to the GPC method.
文摘Stationarity of a class of stochastically interconnecteil discrete-timesystems is analyzed by utilizins results from ergodic theory of general stateMarkov chains, incorporated with the so called large-scale system approach.
文摘We consider qualitatively robust predictive mappings of stochastic environmental models, where protection against outlier data is incorporated. We utilize digital representations of the models and deploy stochastic binary neural networks that are pre-trained to produce such mappings. The pre-training is implemented by a back propagating supervised learning algorithm which converges almost surely to the probabilities induced by the environment, under general ergodicity conditions.
基金supported by MATH-AmSud 18-MATH-07 SaS MoTiDep ProjectHERMES project 41305+1 种基金partially supported by the Project ECOS-CONICYT C15E05,REDES 150038,MATH-AmSud 18-MATH-07 SaS MoTiDep Project and Fondecyt(1171335)supported by NSF(Grant DMS-1613163)
文摘In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(1/2,1). In order to prove convergence, we use rough paths techniques. Theoretical bounds are established and numerical simulations are displayed.
基金The author is very thankful to Dutch Research Council for funding the project bearing the number 435063.
文摘Ocean basin is modeled as a two-dimensional closed,bounded domain in which the fluid flow is governed by the complex partial differential equations in the flow function.Keeping in view that the ocean currents are non-viscous,no normal flow conditions are used at the basin boundaries.The parameters investigated here are:Coriolis parameter,wind stress coefficient,and latitude.Stochastic differential equations in time scales are solved by deterministic and stochastic methods.Deterministic results concluded that streamlines are symmetric about stagnation point(no flow)for 0<R_(p)<6.57.Stochastic controls are introduced to account for variability in time scales.Euler-Maruyama(direct)and Fokker-Planck equation schemes(indirect)are proposed.It is concluded that stream functions in both direct and indirect methods are of the same qualitatively and quantitatively when 0<R_(p)<79.
基金supported by the National Science Foundation under the grants NSF-DMS-1206438 and NSF-DHS-1510249,the National Science Foundation under the grants NSF-DMS-1004638 and NSF-DMS-1313272the Research Fund of Indiana University
文摘As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans, the time discretization of these equations by an implicit Euler scheme is studied. From the deterministic point of view, the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions. From the probabilistic viewpoint, this paper deals with a wide class of nonlinear, state dependent, white noise forcings which may be interpreted in either the Itor the Stratonovich sense. The proof of convergence of the Euler scheme,which is carried out within an abstract framework, covers the equations for the oceans, the atmosphere, the coupled oceanic-atmospheric system as well as other related geophysical equations. The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense, a result which is new by itself to the best of our knowledge.
