A Multilevel Stochastic Collocation Method for Partial Differential Equations with Random Input Data

Teckentrup, Aretha L.; Jantsch, Peter; Webster, Clayton G.; Gunzburger, Max

doi:10.1137/140969002

Cited by 106 publications

(113 citation statements)

References 29 publications

Supporting

Mentioning

110

Contrasting

Order By: Relevance

“…Introduction. Multilevel techniques have a long and successful history in computational science and engineering, e.g., multigrid for solving systems of equations [8,25,9], multilevel discretizations for representing functions [50,18,10], and multilevel Monte Carlo and multilevel stochastic collocation for estimating mean solutions of partial differential equations (PDEs) with stochastic parameters [27,22,45]. These multilevel techniques typically start with a fine-grid discretization-a high-fidelity model-of the underlying PDE or function.…”

mentioning

confidence: 99%

Optimal Model Management for Multifidelity Monte Carlo Estimation

Peherstorfer¹,

Willcox²,

Gunzburger³

2016

SIAM J. Sci. Comput.

218

233

View full text Add to dashboard Cite

Abstract. This work presents an optimal model management strategy that exploits multifidelity surrogate models to accelerate the estimation of statistics of outputs of computationally expensive high-fidelity models. Existing acceleration methods typically exploit a multilevel hierarchy of surrogate models that follow a known rate of error decay and computational costs; however, a general collection of surrogate models, which may include projection-based reduced models, data-fit models, support vector machines, and simplified-physics models, does not necessarily give rise to such a hierarchy. Our multifidelity approach provides a framework to combine an arbitrary number of surrogate models of any type. Instead of relying on error and cost rates, an optimization problem balances the number of model evaluations across the high-fidelity and surrogate models with respect to error and costs. We show that a unique analytic solution of the model management optimization problem exists under mild conditions on the models. Our multifidelity method makes occasional recourse to the high-fidelity model; in doing so it provides an unbiased estimator of the statistics of the high-fidelity model, even in the absence of error bounds and error estimators for the surrogate models. Numerical experiments with linear and nonlinear examples show that speedups by orders of magnitude are obtained compared to Monte Carlo estimation that invokes a single model only. 1. Introduction. Multilevel techniques have a long and successful history in computational science and engineering, e.g., multigrid for solving systems of equations [8,25,9], multilevel discretizations for representing functions [50,18,10], and multilevel Monte Carlo and multilevel stochastic collocation for estimating mean solutions of partial differential equations (PDEs) with stochastic parameters [27,22,45]. These multilevel techniques typically start with a fine-grid discretization-a high-fidelity model-of the underlying PDE or function. The fine-grid discretization is chosen to guarantee an approximation of the output of interest with the accuracy required by the current problem at hand. Additionally, a hierarchy of coarser discretizations-lowerfidelity surrogate models-is constructed, where a parameter (e.g., mesh width) controls the trade-off between error and computational costs. Changing this parameter gives rise to a multilevel hierarchy of discretizations with known error and cost rates. Multilevel techniques use these error and cost rates to distribute the computational work among the discretizations in the hierarchy, shifting most of the work onto the

show abstract

mentioning

confidence: 99%

Optimal Model Management for Multifidelity Monte Carlo Estimation

Peherstorfer¹,

Willcox²,

Gunzburger³

2016

SIAM J. Sci. Comput.

218

233

View full text Add to dashboard Cite

show abstract

“…A recent development for alleviating such complexity and accelerating the convergence of parameterized PDE solutions is to utilize multilevel methods (see e.g., multilevel Monte Carlo (MLMC) methods [4,5,11,20,40] and the multilevel stochastic collocation (MLSC) approach [41]). The main ingredient to multilevel methods is the exploitation of a hierarchical sequence of spatial approximations to the underlying PDE, which are then combined with discretizations in parameter space in such a way as to minimize the overall computational cost.…”

mentioning

confidence: 99%

Accelerating Stochastic Collocation Methods for Partial Differential Equations with Random Input Data

Galindo¹,

Jantsch²,

Webster³

et al. 2016

SIAM/ASA J. Uncertainty Quantification

Self Cite

View full text Add to dashboard Cite

Abstract. This work proposes and analyzes a generalized acceleration technique for decreasing the computational complexity of using stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. The SC approaches considered in this effort consist of sequentially constructed multi-dimensional Lagrange interpolant in the random parametric domain, formulated by collocating on a set of points so that the resulting approximation is defined in a hierarchical sequence of polynomial spaces of increasing fidelity. Our acceleration approach exploits the construction of the SC interpolant to accelerate the underlying ensemble of deterministic solutions. Specifically, we predict the solution of the parametrized PDE at each collocation point on the current level of the SC approximation by evaluating each sample with a previously assembled lower fidelity interpolant, and then use such predictions to provide deterministic (linear or nonlinear) iterative solvers with improved initial approximations. As a concrete example, we develop our approach in the context of SC approaches that employ sparse tensor products of globally defined Lagrange polynomials on nested one-dimensional Clenshaw-Curtis abscissas. This work also provides a rigorous computational complexity analysis of the resulting fully discrete sparse grid SC approximation, with and without acceleration, which demonstrates the effectiveness of our proposed methodology in reducing the total number of iterations of a conjugate gradient solution of the finite element systems at each collocation point. Numerical examples include both linear and nonlinear parametrized PDEs, which are used to illustrate the theoretical results and the improved efficiency of this technique compared with several others.

show abstract

“…For example, when the time step is to large, refining the spatial mesh has no effect on error in the mean E[f ]. Analogously (22) implies that refining the mesh when the stochastic error is larger will have no effect on overall error ||f −f , || L p (Γ) .…”

Section: Multi-index Collocationmentioning

confidence: 99%

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Jakeman

Eldred

Geraci

et al. 2019

Numerical Meth Engineering

View full text Add to dashboard Cite

Summary In this paper, we present an adaptive algorithm to construct response surface approximations of high‐fidelity models using a hierarchy of lower fidelity models. Our algorithm is based on multi‐index stochastic collocation and automatically balances physical discretization error and response surface error to construct an approximation of model outputs. This surrogate can be used for uncertainty quantification (UQ) and sensitivity analysis (SA) at a fraction of the cost of a purely high‐fidelity approach. We demonstrate the effectiveness of our algorithm on a canonical test problem from the UQ literature and a complex multiphysics model that simulates the performance of an integrated nozzle for an unmanned aerospace vehicle. We find that, when the input‐output response is sufficiently smooth, our algorithm produces approximations that can be over two orders of magnitude more accurate than single fidelity approximations for a fixed computational budget.

show abstract

A Multilevel Stochastic Collocation Method for Partial Differential Equations with Random Input Data

Cited by 106 publications

References 29 publications

Optimal Model Management for Multifidelity Monte Carlo Estimation

Optimal Model Management for Multifidelity Monte Carlo Estimation

Accelerating Stochastic Collocation Methods for Partial Differential Equations with Random Input Data

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Contact Info

Product

Resources

About