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.     

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.     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

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     

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.     

A stochastic collocation method for the second order wave equation with a discontinuous random speed

In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.     

A Sparse Grid Stochastic Collocation Discontinuous Galerkin Method for Constrained Optimal Control Problem Governed by Random Convection Dominated Diffusion Equations

A sparse grid stochastic collocation method combined with discontinuous Galerkin method is developed for solving convection dominated diffusion optimal control problem with random coefficients. By the optimal control theory, an optimality system is obtained for the problem, which consists of a state equation, a co-state equation and an inequality. Based on finite dimensional noise assumption of random field, the random coefficients are assumed to have finite term expansions depending on a finite number of mutually independent random variables in the probability space. An approximation scheme is established by using a discontinuous Galerkin method for the physical space and a sparse grid stochastic collocation method based on the Smolyak construction for the probability space, which leads to the solution of uncoupled deterministic problems. A priori error estimates are derived for the state, co-state and control variables. Numerical experiments are presented to illustrate the theoretical results.     

An efficient cubic B-spline and bicubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

This paper aims to develop a novel numerical approach on the basis of B-spline collocation method to approximate the solution of one-dimensional and two-dimensional nonlinear stochastic quadratic integral equations. The proposed approach is based on the hybrid of collocation method, cubic B-spline, and bi-cubic B-spline interpolation and Itô approximation. Using this method, the problem solving turns into a nonlinear system solution of equations that is solved by a suitable numerical method. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.     

Particle methods for PDEs arising in financial modeling

We numerically study convection–diffusion equations arising in financial modeling. We focus on the convection-dominated cases, in which the diffusion coefficients are relatively small. Both finite-difference and Monte-Carlo methods which are widely used in the problems of this kind might be inefficient due to severe restrictions on the meshsize and the number of realizations needed to achieve high resolution.We propose an alternative approach based on particle methods which have extremely low numerical diffusion and thus do not have the aforementioned restrictions. Our approach is based on the operator splitting: The hyperbolic steps are made using the method of characteristics, while the parabolic steps are performed using either a special discretization of the integral representation of the solution (which leads to a deterministic particle method) or a stochastic random walk approach.We apply the designed particle methods to a variety of test problems and the numerical results indicate high accuracy, efficiency and robustness of both the deterministic and stochastic methods. In addition, our numerical experiments clearly demonstrate that the deterministic particle method outperforms its stochastic counterpart.     

Transient analysis of thermo-elastic waves in thick hollow cylinders using a stochastic hybrid numerical method,considering Gaussian mechanical properties

This article attempts to study the stochastic coupled thermo-elasticity of thick hollow cylinders subjected to thermal shock loading considering uncertainty in mechanical properties. The thermo-elastic governing equations based on Green–Naghdi theory (without energy dissipation) are stochastically solved using a hybrid numerical method (combined Galerkin finite element and Newmark finite difference methods). The mechanical properties are considered as random variables with Gaussian distribution, which are generated using Monte Carlo simulation method with various coefficients of variations (COVs). The effects of uncertainty in mechanical properties with various coefficients of variations on thermo-elastic wave propagation are studied in detail. Also, the maximum, mean and variance of temperature, displacement and stresses are illustrated across thickness of cylinder in various times.     

随机平面线弹性问题的一类弱Galerkin方法

本文针对带有随机杨氏模量和荷载的平面线弹性问题,提出了一类随机弱Galerkin有限元方法.先利用Karhunen-Loève展开把随机项参数化,将方程转化为一个确定性问题;再采用弱Galerkin有限元法和$k$-/$p$-型方法分别离散空间区域和随机场.在弱Galerkin离散中,用分片$s（s\geqslant 1$）和$s+1$次多项式逼近单元内部的应力和位移,用分片$s$次多项式逼近位移在单元边界上的迹.证明了该方法关于空间网格尺度最优且与Lamé常数$\lambda$一致无关的误差估计.最后通过数值算例验证了理论结果.     

The boundary element method for stochastic potential problems

Stochastic Dirichlet and Neumann boundary value problems and stochastic mixed problems have been formulated. As a result the stochastic singular integral equations have been obtained. A way of solving these equations by means of discretization of a boundary using stochastic boundary elements has been presented, resulting in a set of random algebraic equations. It has been proved that for Dirichlet and Neumann problems probabilistic characteristics (i.e. moments: expected value and correlation function) fulfilled deterministic singular integral equations. A numerical method of evaluation of moments on a boundary and inside a domain has been presented.     

Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients. These problems model the conversion of starch into sugars in growing apples. The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization. This method leads to high-order accurate stochastic solutions. A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method. After Newton linearization, a point-based algebraic multigrid solution method is applied. In order to decrease the computational cost, alternative multigrid preconditioners are presented. Numerical results demonstrate the convergence properties, robustness and efficiency of the proposed multigrid 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     

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.     

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.     

On spline collocation methods for boundary integral equations in the plane

Nowadays boundary elemen; methods belong to the most popular numerical methods for solving elliptic boundary value problems. They consist in the reduction of the problem to equivalent integral equations (or certain generalizations) on the boundary Γ of the given domain and the approximate solution of these boundary equations. For the numerical treatment the boundary surface is decomposed into a finite number of segments and the unknown functions are approximated by corresponding finite elements and usually determined by collocation and Galerkin procedures. One finds the least difficulties in the theoretical foundation of the convergence of Galerkin methods for certain classes of equations, whereas the convergence of collocation methods, which are mostly used in numerical computations, has yet been proved only for special equations and methods. In the present paper we analyse spline collocation methods on uniform meshes with variable collocation points for one-dimensional pseudodifferential equations on a closed curve with convolutional principal parts, which encompass many classes of boundary integral equations in the plane. We give necessary and sufficient conditions for convergence and prove asymptotic error estimates. In particular we generalize some results on nodal and midpoint collocation obtained in [2], [7] and [8]. The paper is organized as follows. In Section 1 we formulate the problems and the results, Section 2 deals with spline interpolation in periodic Sobolev spaces, and in Section 3 we prove the convergence theorems for the considered collocation methods.     

Adaptive sparse grid algorithms with applications to electromagnetic scattering under uncertainty

We discuss adaptive sparse grid algorithms for stochastic differential equations with a particular focus on applications to electromagnetic scattering by structures with holes of uncertain size, location, and quantity. Stochastic collocation (SC) methods are used in combination with an adaptive sparse grid approach based on nested Gauss-Patterson grids. As an error estimator we demonstrate how the nested structure allows an effective error estimation through Richardson extrapolation. This is shown to allow excellent error estimation and it also provides an efficient means by which to estimate the solution at the next level of the refinement. We introduce an adaptive approach for the computation of problems with discrete random variables and demonstrate its efficiency for scattering problems with a random number of holes. The results are compared with results based on Monte Carlo methods and with Stroud based integration, confirming the accuracy and efficiency of the proposed techniques.     

Estimating Orthant Probabilities of High-Dimensional Gaussian Vectors with An Application to Set Estimation

The computation of Gaussian orthant probabilities has been extensively studied for low-dimensional vectors. Here, we focus on the high-dimensional case and we present a two-step procedure relying on both deterministic and stochastic techniques. The proposed estimator relies indeed on splitting the probability into a low-dimensional term and a remainder. While the low-dimensional probability can be estimated by fast and accurate quadrature, the remainder requires Monte Carlo sampling. We further refine the estimation by using a novel asymmetric nested Monte Carlo (anMC) algorithm for the remainder and we highlight cases where this approximation brings substantial efficiency gains. The proposed methods are compared against state-of-the-art techniques in a numerical study, which also calls attention to the advantages and drawbacks of the procedure. Finally, the proposed method is applied to derive conservative estimates of excursion sets of expensive to evaluate deterministic functions under a Gaussian random field prior, without requiring a Markov assumption. Supplementary material for this article is available online.     

Wavelets collocation methods for the numerical solution of elliptic BV problems

Based on collocation with Haar and Legendre wavelets, two efficient and new numerical methods are being proposed for the numerical solution of elliptic partial differential equations having oscillatory and non-oscillatory behavior. The present methods are developed in two stages. In the initial stage, they are developed for Haar wavelets. In order to obtain higher accuracy, Haar wavelets are replaced by Legendre wavelets at the second stage. A comparative analysis of the performance of Haar wavelets collocation method and Legendre wavelets collocation method is carried out. In addition to this, comparative studies of performance of Legendre wavelets collocation method and quadratic spline collocation method, and meshless methods and Sinc–Galerkin method are also done. The analysis indicates that there is a higher accuracy obtained by Legendre wavelets decomposition, which is in the form of a multi-resolution analysis of the function. The solution is first found on the coarse grid points, and then it is refined by obtaining higher accuracy with help of increasing the level of wavelets. The accurate implementation of the classical numerical methods on Neumann’s boundary conditions has been found to involve some difficulty. It has been shown here that the present methods can be easily implemented on Neumann’s boundary conditions and the results obtained are accurate; the present methods, thus, have a clear advantage over the classical numerical methods. A distinct feature of the proposed methods is their simple applicability for a variety of boundary conditions. Numerical order of convergence of the proposed methods is calculated. The results of numerical tests show better accuracy of the proposed method based on Legendre wavelets for a variety of benchmark problems.