Stochastic approaches to uncertainty quantification in CFD simulations

This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.     

基于嵌入式多项式混沌展开法的随机边界下流动与传热问题不确定性量化

提出了一种嵌入式多项式混沌展开(polynomial chaos expansion, PCE)的随机边界条件下流动与传热问题不确定性量化方法及有限元程序框架.该方法利用Karhunen-Loeve展开表达随机输入边界条件,以及嵌入式多项式混沌展开法表达输出随机场;同时利用谱分解技术将控制方程转化为一组确定性控制方程,并对每个多项式混沌进行求解得到其统计特征.与Monte-Carlo法相比,该方法能够准确高效地预测随机边界条件下流动与传热问题的不确定性特征,同时可以节省大量计算资源.     

Volatility Uncertainty Quantification in a Stochastic Control Problem Applied to Energy

Methodology and Computing in Applied Probability - This work designs a methodology to quantify the uncertainty of a volatility parameter in a stochastic control problem arising in energy...     

Stochastic Lagrangian Relaxation Applied to Power Scheduling in a Hydro-Thermal System under Uncertainty

A dynamic (multi-stage) stochastic programming model for the weekly cost-optimal generation of electric power in a hydro-thermal generation system under uncertain demand (or load) is developed. The model involves a large number of mixed-integer (stochastic) decision variables and constraints linking time periods and operating power units. A stochastic Lagrangian relaxation scheme is designed by assigning (stochastic) multipliers to all constraints coupling power units. It is assumed that the stochastic load process is given (or approximated) by a finite number of realizations (scenarios) in scenario tree form. Solving the dual by a bundle subgradient method leads to a successive decomposition into stochastic single (thermal or hydro) unit subproblems. The stochastic thermal and hydro subproblems are solved by a stochastic dynamic programming technique and by a specific descent algorithm, respectively. A Lagrangian heuristics that provides approximate solutions for the first stage (primal) decisions starting from the optimal (stochastic) multipliers is developed. Numerical results are presented for realistic data from a German power utility and for numbers of scenarios ranging from 5 to 100 and a time horizon of 168 hours. The sizes of the corresponding optimization problems go up to 200000 binary and 350000 continuous variables, and more than 500000 constraints.     

An Adaptive Hybrid Spectral Method for Stochastic Helmholtz Problems

The implementation of an adaptive hybrid spectral method for Helmholtz equations with random parameters is addressed in this work. New error indicators for generalized polynomial chaos for stochastic approximations and spectral element methods for physical approximations are developed, and systematic adaptive strategies are proposed associated with these error indicators. Numerical results show that these error indicators provide effective estimates for the approximation errors, and the overall adaptive procedure results in efficient approximation method for the stochastic Helmholtz equations.     

随机广义KdV方程的随机精确解

通过一个辅助性方程和埃米尔特变换研究广义随机KdV方程的随机雅克比椭圆函数类波解.此外,还通过椭圆函数在模数取极限m→0和m→1的情况,给出方程的随机类孤子解和随机三角函数波解,所得结果丰富了广义随机KdV方程的精确解.     

Competitive Advertising Under Uncertainty: A Stochastic Differential Game Approach

We analyze optimal advertising spending in a duopolistic market where each firm's market share depends on its own and its competitor's advertising decisions, and is also subject to stochastic disturbances. We develop a differential game model of advertising in which the dynamic behavior is based on the Sethi stochastic advertising model and the Lanchester model of combat. Particularly important to note is the morphing of the sales decay term in the Sethi model into decay caused by competitive advertising and noncompetitive churn that acts to equalize market shares in the absence of advertising. We derive closed-loop Nash equilibria for symmetric as well as asymmetric competitors. For all cases, explicit solutions and comparative statics are presented.     

Stochastic modelling and optimization for environmental management

In modelling and managing complex environmental systems, inherent uncertainties of all relevant natural processes are to be taken into consideration. In the present paper diverse stochastic modelling and optimization approaches for handling such problems (primarily in the field of water quality analysis and control) are highlighted, drawing on the findings of case studies and real-world applications.     

A Multi-Stage Stochastic Integer Programming Approach for Capacity Expansion under Uncertainty

This paper addresses a multi-period investment model for capacity expansion in an uncertain environment. Using a scenario tree approach to model the evolution of uncertain demand and cost parameters, and fixed-charge cost functions to model the economies of scale in expansion costs, we develop a multi-stage stochastic integer programming formulation for the problem. A reformulation of the problem is proposed using variable disaggregation to exploit the lot-sizing substructure of the problem. The reformulation significantly reduces the LP relaxation gap of this large scale integer program. A heuristic scheme is presented to perturb the LP relaxation solutions to produce good quality integer solutions. Finally, we outline a branch and bound algorithm that makes use of the reformulation strategy as a lower bounding scheme, and the heuristic as an upper bounding scheme, to solve the problem to global optimality. Our preliminary computational results indicate that the proposed strategy has significant advantages over straightforward use of commercial solvers.     

多项式混沌方法在偶然不确定度量化中的应用

复杂工程建模与模拟中必然存在误差与不确定度,分析与辨识其不确定度的来源,对不确定度进行量化,对建模与模拟可信度评估具有重要意义。本文给出建模与模拟中误差与不确定度的概念及不确定度的量化过程,并以质量弹簧阻尼系统为例说明量化偶然不确定度的过程,验证了非嵌入多项式混沌方法在非光滑系统不确定度量化中的有效性,对建模与模拟中不确定度量化具有重要的参考价值。     

Modelling Fruit Characteristics During Apple Maturation: A Stochastic Approach

At present, mathematical models to predict the change of fruit quality attributes during apple maturation are deterministic and do not take into account the large natural variability of fruit quality attributes during the growing season. In this work a stochastic system approach was developed to describe the quality evolution of fruit. The basic dynamics of fruit quality evolution was represented by means of a stochastic system, in which the initial conditions and the model parameters were specified as random variables together with their probability density functions. A fundamental approach from stochastic systems theory was used to compute the propagation of the probability density functions of fruit quality attributes, which requires the numerical solution of the Fokker–Planck equation.     

从超布朗运动到随机偏微分方程

随机偏微分方程(SPDE)是目前国内外广泛关注研究进展迅速的一个活跃的学术研究领域.该主题的研究涉及概率论(随机分析、随机场)、偏微分方程、调和分析等诸多分支学科方向.特别是随机偏微分方程其背景更多地源于现代物理学、化学、生物学、经济学等应用性学科,这使得该领域的研究显示出较强的意义和活力.本文从超布朗运动研究出发,发展性地提出有较强背景意义的典型类随机偏微分方程,并进而过渡到一般及更广泛类的随机偏微分方程的研究.同时我们系统地总结了关于高阶随机偏微分方程和随机波动方程的研究成果.     

基于多项式混沌展开的结构动力特性高阶统计矩计算

结构参数的不确定性会导致其动力特性的不确定性, 量化动力特性的不确定性能为结构动力设计分析提供准确的动力信息.统计矩是表征结构动力特性不确定性非常重要的统计量, 比如均值和方差.传统的Monte Carlo(蒙特 卡洛)模拟方法需要大量次数的模型运算来保证结果的收敛, 其用于复杂结构的动力特性统计矩计算因耗时太高而使用受限.该文采用多项式混沌展开替代模型来取代计算花费高的有限元模型, 然后在替代模型框架下快速有效地计算结构动力特性的统计矩.该方法在建立替代模型之初只需要少量次数有限元分析, 后续的统计矩计算无需有限元模型, 因此从根本上解决了动力特性统计矩计算花费高的难题.该文的多项式混沌展开方法适用于参数服从任意概率分布, 能够有效地计算高阶统计矩, 为量化结构动力特性不确定性提供更多统计矩信息.最后通过平铝板算例验证了此方法的有效性.     

Stochastic flows of mappings

In this paper,the stochastic flow of mappings generated by a Feller convolution semigroup on a compact metric space is studied.This kind of flow is the generalization of superprocesses of stochastic flows and stochastic diffeomorphism induced by the strong solutions of stochastic differential equations.     

不确定线性微分对策的鲁棒性及设计

由于存在各种干扰、噪声和恶劣环境,以及模型也不应过分精确的实际,所以考虑微分对策的鲁棒性是必然和重要的.主要就微分对策中存在随机干扰和模型本身参数不确定这两类不确定性问题研究其鲁棒对策问题,并给出其相应的解.     

Stochastic flows of diffeomorphisms on an open set in rn

In this paper, we will give sufficient conditions for the solution to a stochastic differential equation (SDE) on an open set D in R" to define a stochastic flow of diffeomorphisms of D onto itself. Since a necessary and sufficient condition for the solution to determine a stochastic flow of diffeomorphisms is that the original SDE and its adjoint SDE are both strictly conservative, we will concentrate our attention on finding sufficient conditions for the SDE to be strictly conservative. It will be etablished that the strict conservativeness follows if the vector fields governing the SDE decay suitably near the boundary dD in the direction transversal to 3D and some additional assumptions are satisfied.