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.     

Numerical simulations for two‐dimensional stochastic incompressible Navier‐Stokes equations

This article mainly concerns modeling the stochastic input and its propagation in incompressible Navier‐Stokes(N‐S) flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space. A standard Galerkin projection is applied in the random dimension to derive the equations in the weak form. The resulting set of deterministic equations is then solved with standard methods to obtain the mean solution. In this article, the main method employs the Hermite polynomial as the basis in random space. Numerical examples are given and the error analysis is demonstrated for a model problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Numerical simulations for two‐dimensional incompressible Navier‐Stokes equations with stochastic boundary conditions

The article mainly concerns modeling the stochastic input and its propagation in incompressible Navier‐Stokes(N‐S) flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space. A standard Galerkin projection is applied in the random dimension to derive the equations in the weak form. The resulting set of deterministic equations is then solved with standard methods to obtain the mean solution and variance of the stochastic velocity. In this article, the main method employs the Hermite polynomial as the basis in random space. Cavity problems are given to demonstrate the process of numerical simulation. Furthermore, Monte‐Carlo simulation method is applied to illustrate the accurate numerical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Stochastic programming approaches to stochastic scheduling

Practical scheduling problems typically require decisions without full information about the outcomes of those decisions. Yields, resource availability, performance, demand, costs, and revenues may all vary. Incorporating these quantities into stochastic scheduling models often produces diffculties in analysis that may be addressed in a variety of ways. In this paper, we present results based on stochastic programming approaches to the hierarchy of decisions in typical stochastic scheduling situations. Our unifying framework allows us to treat all aspects of a decision in a similar framework. We show how views from different levels enable approximations that can overcome nonconvexities and duality gaps that appear in deterministic formulations. In particular, we show that the stochastic program structure leads to a vanishing Lagrangian duality gap in stochastic integer programs as the number of scenarios increases.This author's work was supported in part by the National Science Foundation under Grants ECS 88-15101, ECS 92-16819, and SES 92-11937.This author's work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A-5489 and by the UK Engineering and Physical Sciences Research Council under Grants J90855 and K17897.     

Stochastic programming approach to optimization under uncertainty

In this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems. We argue that two stage (say linear) stochastic programming problems can be solved with a reasonable accuracy by Monte Carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages. We discuss an extension of coherent risk measures to a multistage setting and, in particular, dynamic programming equations for such problems.      

倒向随机方程期权定价模型的一类随机算法

期权定价问题可以转化为对倒向随机微分方程的求解,进而转化为对相应抛物型偏微分方程的求解.为了求解与倒向随机微分方程相应的二阶拟线性抛物型微分方程初值问题,引入一类新的随机算法-分层方法取代传统的确定性数值算法.这种数值方法理论上是通过弱显式欧拉法,离散其相应随机系统解的概率表示而得到.该随机算法的收敛性在文中得到证明,其稳定性是自然的.并构造了易于数值实现的基于插值的算法,实证研究说明这种算法能很好地提供期权定价模型的数值模拟.     

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。     

Efficient Krylov subspace methods for uncertainty quantification in large Bayesian linear inverse problems

Uncertainty quantification for linear inverse problems remains a challenging task, especially for problems with a very large number of unknown parameters (e.g., dynamic inverse problems) and for problems where computation of the square root and inverse of the prior covariance matrix are not feasible. This work exploits Krylov subspace methods to develop and analyze new techniques for large‐scale uncertainty quantification in inverse problems. In this work, we assume that generalized Golub‐Kahan‐based methods have been used to compute an estimate of the solution, and we describe efficient methods to explore the posterior distribution. In particular, we use the generalized Golub‐Kahan bidiagonalization to derive an approximation of the posterior covariance matrix, and we provide theoretical results that quantify the accuracy of the approximate posterior covariance matrix and of the resulting posterior distribution. Then, we describe efficient methods that use the approximation to compute measures of uncertainty, including the Kullback‐Liebler divergence. We present two methods that use the preconditioned Lanczos algorithm to efficiently generate samples from the posterior distribution. Numerical examples from dynamic photoacoustic tomography demonstrate the effectiveness of the described approaches.     

Modelling of uncertainty in discrete dynamical systems

Dynamic systems that are not Gaussian, stationary and linear are difficult to model by full probabilistic analysis. Sufficient information for practical application can often be obtained by second moment analysis, described in the paper. Alternatively, second moment analysis can be performed using point distributions. Two new methods in this class, one exact for linear systems and one approximate, are described. Examples show the application and illustrate the accuracy.     

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.     

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.     

Stochastic control of SDEs associated with Lévy generators and application to financial optimization

This paper is concerned with the optimal control of jump type stochastic differential equations associated with (general) Lévy generators. The maximum principle is formulated for the solutions of the equations, which is inspired by N. C. Framstad, B. Øsendal and A. Sulem [J. Optim. Theory Appl., 2004, 121: 77―98] (and a continuation, J. Bennett and J. -L. Wu [Front. Math. China, 2007, 2(4): 539―558]). The result is then applied to optimization problems in financial models driven by Lévy-type processes.     

Medium range optimization of copper extraction planning under uncertainty in future copper prices

Deterministic mine planning models along a time horizon have proved to be very effective in supporting decisions on sequencing the extraction of material in copper mines. Some of these models have been developed for, and used successfully by CODELCO, the Chilean state copper company. In this paper, we wish to consider the uncertainty in a very volatile parameter of the problem, namely, the copper price along a given time horizon. We represent the uncertainty by a multistage scenario tree. The resulting stochastic model is then converted into a mixed 0–1 Deterministic Equivalent Model using a compact representation. We first introduce the stochastic model that maximizes the expected profit along the time horizon over all scenarios (i.e., as in a risk neutral environment). We then present several approaches for risk management in a risk averse environment. Specifically, we consider the maximization of the Value-at-Risk and several variants of the Conditional Value-at-Risk (one of them is new), the maximization of the expected profit minus the weighted probability of having an undesirable scenario in the solution provided by the model, and the maximization of the expected profit subject to stochastic dominance constraints recourse-integer for a set of profiles given by the pairs of target profits and bounds on either the probability of failure or the expected profit shortfall. We present an extensive computational experience on the actual problem, by comparing the risk neutral approach, the tested risk averse strategies and the performance of the traditional deterministic approach that uses the expected value of the uncertain parameters. The results clearly show the advantage of using the risk neutral strategy over the traditional deterministic approach, as well as the advantage of using any risk averse strategy over the risk neutral one.     

Stochastic extensions to necessary conditions in the theory of the calculus of variations

A stochastic version of the modified Young's generalized necessary conditions in the calculus of variations is given in this paper. It is based on an extension of Minkowski's theorem on the existence of a flat support for a convex figure, and it generalizes the necessary conditions of Weierstrass and Euler in the classical theory of the calculus of variations to a class of admissible curves which are expressible in terms of a finite number of random parameters. The integrals which we consider here are in the general Denjoy sense, except those with respect to the random parameters, which exist in the Lebesgue sense defined on a probability space. The importance of our stochastic analysis lies in the completion that a minimum not attained in the classical sense may be, and frequently is, attained in the stochastic case.This research was supported in part by the National Science Foundation under Grants Nos. GK-1834X and GK-31229     

不完全偏好下随机选择问题中的极大元定理

提出了不完全偏好的概念.在不完全偏好意义下获得了随机选择问题中的极大元定理和其它一些定理,统一和推广了许多已知结果.