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Results for “Stochastic collocation method”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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DOE OSTI2011

Investigation of advanced UQ for CRUD prediction with VIPRE.

This document summarizes the results from a level 3 milestone study within the CASL VUQ effort. It demonstrates the application of 'advanced UQ,' in particular dimension-adaptive p-refinement for polynomial chaos and stochastic collocation. The study calculates statistics for several quantities of interest that are indicators for the formation of CRUD (Chalk River unidentified deposit), which can lead to CIPS (CRUD induced power shift). Stochastic expansion methods are attractive methods for uncertainty quantification due to their fast convergence properties. For smooth functions (i.e., analytic, infinitely-differentiable) in L{sup 2} (i.e., possessing finite variance), exponential convergence rates can be obtained under order refinement for integrated statistical quantities of interest such as mean, variance, and probability. Two stochastic expansion methods are of interest: nonintrusive polynomial chaos expansion (PCE), which computes coefficients for a known basis of multivariate orthogonal polynomials, and stochastic collocation (SC), which forms multivariate interpolation polynomials for known coefficients. Within the DAKOTA project, recent research in stochastic expansion methods has focused on automated polynomial order refinement ('p-refinement') of expansions to support scalability to higher dimensional random input spaces [4, 3]. By preferentially refining only in the most important dimensions of the input space, the applicability of these methods can be extended from O(10{sup 0})-O(10{sup 1}) random variables to O(10{sup 2}) and beyond, depending on the degree of anisotropy (i.e., the extent to which randominput variables have differing degrees of influence on the statistical quantities of interest (QOIs)). Thus, the purpose of this study is to investigate the application of these adaptive stochastic expansion methods to the analysis of CRUD using the VIPRE simulation tools for two different plant models of differing random dimension, anisotropy, and smoothness.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2013

Improved statistical models for limited datasets in uncertainty quantification using stochastic collocation

This paper presents a data-driven framework for performing uncertainty quantification (UQ) by choosing a stochastic model that accurately describes the sources of uncertainty in a system. This model is propagated through an appropriate response surface function that approximates the behavior of this system using stochastic collocation. Given a sample of data describing the uncertainty in the inputs, our goal is to estimate a probability density function (PDF) using the kernel moment matching (KMM) method so that this PDF can be used to accurately reproduce statistics like mean and variance of the response surface function. Instead of constraining the PDF to be optimal for a particular response function, we show that we can use the properties of stochastic collocation to make the estimated PDF optimal for a wide variety of response functions. We contrast this method with other traditional procedures that rely on the Maximum Likelihood approach, like kernel density estimation (KDE) and its adaptive modification (AKDE). We argue that this modified KMM method tries to preserve what is known from the given data and is the better approach when the available data is limited in quantity. We test the performance of these methods for both univariate and multivariate density estimation by sampling random datasets from known PDFs and then measuring the accuracy of the estimated PDFs, using the known PDF as a reference. Comparing the output mean and variance estimated with the empirical moments using the raw data sample as well as the actual moments using the known PDF, we show that the KMM method performs better than KDE and AKDE in predicting these moments with greater accuracy. This improvement in accuracy is also demonstrated for the case of UQ in electrostatic and electrothermomechanical microactuators. We show how our framework results in the accurate computation of statistics in micromechanical systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Stochastic multiscale flux basis for Stokes-Darcy flows

Three algorithms are developed for uncertainty quantification in modeling coupled Stokes and Darcy flows. The porous media may consist of multiple regions with different properties. The permeability is modeled as a non-stationary stochastic variable, with its log represented as a sum of local Karhunen-Loève (KL) expansions. The problem is approximated by stochastic collocation on either tensor-product or sparse grids, coupled with a multiscale mortar mixed finite element method for the spatial discretization. A non-overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem, which is solved by an iterative solver, for each stochastic realization. In the traditional domain decomposition implementation, each subdomain solves a local Dirichlet or Neumann problem in every interface iteration. To reduce this cost, two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced. The basis consists of the local flux (or velocity trace) responses from each mortar degree of freedom. It is computed for each subdomain independently before the interface iteration begins. The use of the multiscale flux basis avoids the need for subdomain solves on each iteration. The deterministic basis is computed at each stochastic collocation and used only at this realization. The stochastic basis is formed by further looping over all local realizations of a subdomain's KL region before the stochastic collocation begins. It is used over multiple realizations. Numerical tests are presented to illustrate the performance of the three algorithms, with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes

We study the convergence properties of the recently developed Dynamically Orthogonal (DO) field equations [1] in comparison with the Polynomial Chaos (PC) method. To this end, we consider a series of one-dimensional prototype SPDEs, whose solution can be expressed analytically, and which are associated with both linear (advection equation) and nonlinear (Burgers equation) problems with excitations that lead to unimodal and strongly bi-modal distributions. We also propose a hybrid approach to tackle the singular limit of the DO equations for the case of deterministic initial conditions. The results reveal that the DO method converges exponentially fast with respect to the number of modes (for the problems considered) giving same levels of computational accuracy comparable with the PC method but (in many cases) with substantially smaller computational cost compared to stochastic collocation, especially when the involved parametric space is high-dimensional.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2012

Characterizing the elements of Earth's radiative budget: Applying uncertainty quantification to the CESM

Understanding and characterizing sources of uncertainty in climate modeling is an important task. Because of the ever increasing sophistication and resolution of climate modeling it is increasing important to develop uncertainty quantification methods that minimize the computational cost that occurs when these methods are added to climate modeling. This research explores the application of sparse stochastic collocation with polynomial edge detection to characterize portions of the probability space associated with the Earth's radiative budget in the Community Earth System Model (CESM). Specifically, we develop surrogate models with error estimates for a range of acceptable input parameters that predict statistical values of the Earth's radiative budget as derived from the CESM simulation.

42 ENGINEERING↗
DOE OSTI2012

Characterizing the elements of Earth s radiative budget: Applying uncertainty quantification to the CESM

Understanding and characterizing sources of uncertainty in climate modeling is an important task. Because of the ever increasing sophistication and resolution of climate modeling it is increasing important to develop uncertainty quantification methods that minimize the computational cost that occurs when these methods are added to climate modeling. This research explores the application of sparse stochastic collocation with polynomial edge detection to characterize portions of the probability space associated with the Earth s radiative budget in the Community Earth System Model (CESM). Specifically, we develop surrogate models with error estimates for a range of acceptable input parameters that predict statistical values of the Earth s radiative budget as derived from the CESM simulation. We extend these results in resolution from T31 to T42 and in parameter space increasing the degrees of freedom from two to three.

Archibald, Richard K↗
DOE OSTI2010

Stochastic Response Surface Based Simulation Of Ground Water Modeling

Estimation of Uncertainty associated with groundwater modeling is an important aspect in nuclear industries. Traditional methods for propagating the uncertainty, such as Monte Carlo require prohibitive number of model simulation for complex model. Stochastic response surface method which is more computationally efficient is based on polynomial expansion of model input and output in terms of well behaved random variables that are identically and independently distributed. The model output at collocation point is evaluated to know the unknown coefficients in the series expansion of output. The uncertainty in concentration profile and degree of uncertainty is evaluated for model. The present paper describes the novel approach of stochastic response surface technique to estimate the uncertainty in the groundwater modeling.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

ANALYSIS OF VARIANCE-BASED MIXED MULTISCALE FINITE ELEMENT METHOD AND APPLICATIONS IN STOCHASTIC TWO-PHASE FLOWS

The stochastic partial differential systems have been widely used to model physical processes, where the inputs involve large uncertainties. Flows in random and heterogeneous porous media is one of the cases where the random inputs (e.g., permeability) are often modeled as a stochastic field with high-dimensional random parameters. To treat the high dimensionality and heterogeneity efficiently, model reduction is employed in both stochastic space and physical space. An analysis of variance (ANOVA)-based mixed multiscale finite element method MsFEM) is developed to decompose the high-dimensional stochastic problem into a set of lower-dimensional stochastic subproblems, which require much less computational complexity and significantly reduce the computational cost in stochastic space, and the mixed MsFEM can capture the heterogeneities on a coarse grid to greatly reduce the computational cost in the spatial domain. In addition, to enhance the efficiency of the traditional ANOVA method, an adaptive ANOVA method based on a new adaptive criterion is developed, where the most active dimensions can be selected to greatly reduce the computational cost before conducting ANOVA decomposition. This novel adaptive criterion is based on variance-decomposition method coupled with sparse-grid probabilistic collocation method or multilevel Monte Carlo method. The advantage of this adaptive criterion lies in its much lower computational overhead for identifying the active dimensions and interactions. A number of numerical examples in two-phase stochastic flows are presented and demonstrate the accuracy and performance of the adaptive ANOVA-based mixed Ms-FEM.

Wei, Jia↗
DOE OSTI2016

Numerical methods for high-dimensional probability density function equations

In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker–Planck and Dostupov–Pugachev equations), random wave theory (Malakhov–Saichev equations) and coarse-grained stochastic systems (Mori–Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Staggered scheme for the compressible fluctuating hydrodynamics of multispecies fluid mixtures

Here, we present a numerical formulation for the solution of nonisothermal, compressible Navier-Stokes equations with thermal fluctuations to describe mesoscale transport phenomena in multispecies fluid mixtures. The novelty of our numerical method is the use of staggered grid momenta along with a finite volume discretization of the thermodynamic variables to solve the resulting stochastic partial differential equations. The key advantages of the numerical scheme are that it significantly simplifies the discretization of diffusive and stochastic momentum fluxes into a more compact form, and it provides an unambiguous prescription of boundary conditions involving pressure. The staggered grid scheme more accurately reproduces the equilibrium static structure factor of hydrodynamic fluctuations in gas mixtures compared to a collocated scheme described previously by Balakrishnan et al. [Phys. Rev. E 89, 013017 (2014)1539-375510.1103/PhysRevE.89.013017]. The numerical method is tested for ideal noble gases mixtures under various nonequilibrium conditions, such as applied thermal and concentration gradients, to assess the role of cross-diffusion effects, such as Soret and Dufour, on the long-ranged correlations of hydrodynamic fluctuations, which are also more accurately reproduced compared to the collocated scheme. We numerically study giant nonequilibrium fluctuations driven by concentration gradients and fluctuation-driven Rayleigh-Taylor instability in gas mixtures. Wherever applicable, excellent agreement is observed with theory and measurements from the direct simulation Monte Carlo method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Systematic parameter inference in stochastic mesoscopic modeling

We propose a method to efficiently determine the optimal coarse-grained force field in mesoscopic stochastic simulations of Newtonian fluid and polymer melt systems modeled by dissipative particle dynamics (DPD) and energy conserving dissipative particle dynamics (eDPD). The response surfaces of various target properties (viscosity, diffusivity, pressure, etc.) with respect to model parameters are constructed based on the generalized polynomial chaos (gPC) expansion using simulation results on sampling points (e.g., individual parameter sets). To alleviate the computational cost to evaluate the target properties, we employ the compressive sensing method to compute the coefficients of the dominant gPC terms given the prior knowledge that the coefficients are “sparse”. The proposed method shows comparable accuracy with the standard probabilistic collocation method (PCM) while it imposes a much weaker restriction on the number of the simulation samples especially for systems with high dimensional parametric space. Fully access to the response surfaces within the confidence range enables us to infer the optimal force parameters given the desirable values of target properties at the macroscopic scale. Moreover, it enables us to investigate the intrinsic relationship between the model parameters, identify possible degeneracies in the parameter space, and optimize the model by eliminating model redundancies. The proposed method provides an efficient alternative approach for constructing mesoscopic models by inferring model parameters to recover target properties of the physics systems (e.g., from experimental measurements), where those force field parameters and formulation cannot be derived from the microscopic level in a straight forward way.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

What Is the Fractional Laplacian? A Comparative Review with New Results

The fractional Laplacian in $\mathbb{R}^{d}$, which we write as (–Δ) α/2 with, α ϵ (0, 2) has multiple equivalent characterizations. Furthermore, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. Yet, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: “What is the fractional Laplacian?” Beginning from first principles, we compare several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties. Next, we present quantitative comparisons using a sample of state-of-the-art methods. We discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems: radial basis function collocation (for the Riesz fractional Laplacian) and nonharmonic lifting (for the spectral fractional Laplacian). In our numerical studies, we aim to compare different definitions on bounded domains using a collection of benchmark problems. We consider the fractional Poisson equation with both zero and nonzero boundary conditions, where the fractional Laplacian is defined according to the Riesz definition, the spectral definition, the directional definition, and the horizon-based nonlocal definition. We verify the accuracy of the numerical methods used in the approximations for each operator, and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

A GENERALIZED STOCHASTIC COLLOCATION APPROACH TO CONSTRAINED OPTIMIZATION FOR RANDOM DATA IDENTIFICATION PROBLEMS

We present a scalable, parallel mechanism for stochastic identification/control for problems constrained by partial differential equations with random input data. Several identification objectives will be discussed that either minimize the expectation of a tracking cost functional or minimize the difference of desired statistical quantities in the appropriate $L^p$ norm, and the distributed parameters/control can both deterministic or stochastic. Given an objective we prove the existence of an optimal solution, establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations. The modeling process may describe the solution in terms of high dimensional spaces, particularly in the case when the input data (coefficients, forcing terms, boundary conditions, geometry, etc) are affected by a large amount of uncertainty. For higher accuracy, the computer simulation must increase the number of random variables (dimensions), and expend more effort approximating the quantity of interest in each individual dimension. Hence, we introduce a novel stochastic parameter identification algorithm that integrates an adjoint-based deterministic algorithm with the sparse grid stochastic collocation FEM approach. This allows for decoupled, moderately high dimensional, parameterized computations of the stochastic optimality system, where at each collocation point, deterministic analysis and techniques can be utilized. The advantage of our approach is that it allows for the optimal identification of statistical moments (mean value, variance, covariance, etc.) or even the whole probability distribution of the input random fields, given the probability distribution of some responses of the system (quantities of physical interest). Our rigorously derived error estimates, for the fully discrete problems, will be described and used to compare the efficiency of the method with several other techniques. Numerical examples illustrate the theoretical results and demonstrate the distinctions between the various stochastic identification objectives.

Webster, Clayton G↗
DOE OSTI2017

Error Convergence Characterization for Stochastic Transport Methods

There has been increased interest in reliably and efficiently solving engineering systems involving uncertainty in recent years. Characterization of the convergence of errors within the problem, including statistical and bias errors, increases confidence that chosen solution methods are appropriate for a particular problem and quantity of interest (QoI), that implementation errors have not been made, and can aid users or algorithms in solving problems within a tolerance efficiently. We demonstrate a method for characterizing error convergence in a simple stochastic transport problem. Though the approach is applicable to problems with more complicated physics, our simple problem enables analytic QoI solutions so that error convergence behavior and the identification of limiting errors can be presented with high precision. A one-dimensional, two-material, neutral-particle, and absorption-only transport problem involving uniformly distributed uncertain total cross sections is presented. Random sampling (RS) and stochastic collocation (SC) are used as stochastic solvers which sample from the uncertain space and Monte Carlo (MC) transport is employed to solve transport results on the resulting realizations. Probability density functions (PDFs) are created using either RS solutions or samples taken from a polynomial chaos expansion (PCE) constructed with SC solutions. The convergence of each solver is characterized with respect to the QoI against analytic solutions, convergence behavior is used to efficiently solve a QoI within a tolerance, and convergence plots are used to identify dominant solution errors.

42 ENGINEERING↗
DOE OSTI2022

Adaptive experimental design for multi-fidelity surrogate modeling of multi-disciplinary systems

We present an adaptive algorithm for constructing surrogate models of multi-disciplinary systems composed of a set of coupled components. With this goal we introduce “coupling” variables with a priori unknown distributions that allow surrogates of each component to be built independently. Once built, the surrogates of the components are combined to form an integrated-surrogate that can be used to predict system-level quantities of interest at a fraction of the cost of the original model. The error in the integrated-surrogate is greedily minimized using an experimental design procedure that allocates the amount of training data, used to construct each component-surrogate, based on the contribution of those surrogates to the error of the integrated-surrogate. Here, the multi-fidelity procedure presented is a generalization of multi-index stochastic collocation that can leverage ensembles of models of varying cost and accuracy, for one or more components, to reduce the computational cost of constructing the integrated-surrogate. Extensive numerical results demonstrate that, for a fixed computational budget, our algorithm is able to produce surrogates that are orders of magnitude more accurate than methods that treat the integrated system as a black-box.

42 ENGINEERING↗
DOE OSTI2016

Time-Dependent Sensitivity Analysis of OECD Benchmark using BISON and RAVEN

Developments in uncertainty quantification for nuclear simulations have decreased the computational cost required to perform accurate sensitivity analysis. This work demonstrates implementation of these methods in the RAVEN framework, allowing for time-dependent sensitivity analysis. By demonstration we consider an OECD benchmark case. We propagate uncertainties using RAVEN operating on the BISON fuels performance code. We then consider the time-evolution of output response sensitivity to uncertain inputs. We perform sensitivity analysis using time-based stochastic collocation for generalized polynomial chaos (SCgPC) and high-dimension model reduction (HDMR). While uncertainty quantification applied to end-of-cycle statistics is useful, we demonstrate that dominant sources of uncertainty for figures of merit may fluctuate throughout the simulation. Time-dependent sensitivity analysis allows more insight on key design parameters for fuel. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2012

New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs

By using functional integral methods we determine new evolution equations satisfied by the joint response-excitation probability density function (PDF) associated with the stochastic solution to first-order nonlinear partial differential equations (PDEs). The theory is presented for both fully nonlinear and for quasilinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are discussed for the classical linear and nonlinear advection equations and for the advection-reaction equation. By using a Fourier-Galerkin spectral method we obtain numerical solutions of the proposed response-excitation PDF equations. These numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi-element probabilistic collocation methods. It is found that the response-excitation approach yields accurate predictions of the statistical properties of the system. In addition, it allows to directly ascertain the tails of probabilistic distributions, thus facilitating the assessment of rare events and associated risks. The computational cost of the response-excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the PDE is subject to high-dimensional random boundary or initial conditions. The question of high-dimensionality for evolution equations involving multidimensional joint response-excitation PDFs is also addressed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Conditional Karhunen-Loève expansion for uncertainty quantification and active learning in partial differential equation models

We use a conditional Karhunen-Lo` eve (KL) model to quantify and reduce uncertainty in a stochastic partial differential equation (SPDE) problem with partially-known space-dependent coefficient, Y (x). We assume that a small number of Y (x) measurements are available and model Y (x) with a KL expansion. We achieve reduction in uncertainty by conditioning the KL expansion coefficients on measurements. We consider two approaches for conditioning the KL expansion: In Approach 1, we condition the KL model first and then truncate it. In Approach 2, we first truncate the KL expansion and then condition it. We employ the conditional KL expansion together with Monte Carlo and sparse grid collocation methods to compute the moments of the solution of the SPDE problem. Uncertainty of the problem is further reduced by adaptively selecting additional observation locations using two active learning methods. Method 1 minimizes the variance of the PDE coefficient, while Method 2 minimizes the variance of the solution of the PDE. We demonstrate that conditioning leads to dimension reduction of the KL representation of Y (x). For a linear diffusion SPDE with uncertain log-normal coefficient, we show that Approach 1 provides a more accurate approximation of the conditional log-normal coefficient and solution of the SPDE than Approach 2 for the same number of random dimensions in a conditional KL expansion. Furthermore, Approach 2 provides a good estimate for the number of terms of the truncated KL expansion of the conditional field of Approach 1. Finally, we demonstrate that active learning based on Method 2 is more efficient for uncertainty reduction in the SPDE’s states (i.e., it leads to a larger reduction of the variance) than active learning using Method 2.

Conditioned Karhunen-Lo` eve expanion, machine lea↗