Measure transformation and efficient quadrature in reduced‐dimensional stochastic modeling of coupled problems

Arnst, Maarten; Ghanem, Roger; Phipps, Eric Todd; Red-Horse, John

doi:10.1002/nme.4368

Cited by 30 publications

(64 citation statements)

References 44 publications

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“…We note that the set (3.9) can be computed based on the knowledge of moments of the variables ξ j and η j . Following [56,29], we assemble the Gram matrix H j with the entries…”

Section: Moments and Orthogonal Polynomials By A Sampling Approachmentioning

confidence: 99%

“…Multi-index operations can still be used to construct the polynomial set (3.9) with respect to the joint distribution, although the estimation of multivariate moments of η j becomes necessary because of such a dependency. In this case, it is known that orthogonal polynomials are not unique and depend on the ordering imposed on the multi-index set; see [1,29,43]. In all computations, we use the graded lexicographic ordering for multi-indices.…”

Section: Moments and Orthogonal Polynomials By A Sampling Approachmentioning

confidence: 99%

“…In this work, we estimate the moments of the solution using its PCE through a sampling approach to greatly reduce their computational cost; see [29] for a similar sampling methodology. We also show that the method is robust with respect to re-sampling.…”

Section: Introductionmentioning

confidence: 99%

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A dynamical polynomial chaos approach for long-time evolution of SPDEs

Ozen

Bal

2017

Journal of Computational Physics

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We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time. Exploiting the Markovian property of white noise, DgPC [1] implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise. The dimension of the representation is kept minimal by application of a Karhunen-Loeve (KL) expansion. Using frequent restarts and low degree polynomials on sparse multi-index sets, the method allows us to perform long time simulations, including the calculation of invariant measures for systems which possess one. We apply the method to the numerical simulation of stochastic Burgers and Navier Stokes equations with white noise forcing. Our method also allows us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations and show that the algorithm compares favorably with standard Monte Carlo methods.

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“…We note that the set (3.9) can be computed based on the knowledge of moments of the variables ξ j and η j . Following [56,29], we assemble the Gram matrix H j with the entries…”

Section: Moments and Orthogonal Polynomials By A Sampling Approachmentioning

confidence: 99%

Section: Moments and Orthogonal Polynomials By A Sampling Approachmentioning

confidence: 99%

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A dynamical polynomial chaos approach for long-time evolution of SPDEs

Ozen

Bal

2017

Journal of Computational Physics

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“…Recent work on uncertainty quantification in coupled multiphysics models has focused on dimensionality reduction in the information shared at the interface between different model components [4,5]. More specific to coupled atomistic-to-continuum simulations, we have studied two-way coupling between uncertainties across the scale interface.…”

mentioning

confidence: 98%

Quantifying Sampling Noise and Parametric Uncertainty in Atomistic-to-Continuum Simulations Using Surrogate Models

Salloum¹,

Sargsyan²,

Jones³

et al. 2015

Multiscale Model. Simul.

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We present a methodology to assess the predictive fidelity of multiscale simulations by incorporating uncertainty in the information exchanged between the components of an atomisticto-continuum simulation. We account for both the uncertainty due to finite sampling in molecular dynamics (MD) simulations and the uncertainty in the physical parameters of the model. Using Bayesian inference, we represent the expensive atomistic component by a surrogate model that relates the long-term output of the atomistic simulation to its uncertain inputs. We then present algorithms to solve for the variables exchanged across the atomistic-continuum interface in terms of polynomial chaos expansions (PCEs). We consider a simple Couette flow where velocities are exchanged between the atomistic and continuum components, while accounting for uncertainty in the atomistic model parameters and the continuum boundary conditions. Results show convergence of the coupling algorithm at a reasonable number of iterations. The uncertainty in the obtained variables significantly depends on the amount of data sampled from the MD simulations and on the width of the time averaging window used in the MD simulations. Introduction.A wide variety of devices, such as physical nanodevices and microfluid systems, function through phenomena that operate on a range of time and length scales. To properly resolve the key phenomena on all relevant scales, multiscale methods are required [13,15,18,21,22,25,43], often coupling different physical models across scales, such as atomistic models (e.g., molecular dynamics (MD)) and continuum macroscale formulations [1,32,37].While recent years have brought tremendous progress in the field of multiscale modeling [13,15,18,21,22,25,43], a key requirement for predictive simulations is to also quantify the uncertainty in the multiscale simulation results. Uncertainty in multiscale simulations stems from input parameter, initial condition, and boundary condition uncertainties; from modelling assumptions on each resolved scale level; and also from modelling assumptions in the coupling between the different scales [12,20,35,40]. Uncertainty quantification (UQ) in multiscale simulations is an active research topic, with many approaches being developed for the various types of multiscale coupling [12,16,20,35,46]. Most of the current research appears to focus on UQ on individual scale levels, or on one-way propagation of uncertainty across scales.

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“…We construct the reduced-dimensional representation through the truncation of a KL decomposition [1,2]. First, we compute the sample mean and covariance matrix:…”

Section: Dimension Reductionmentioning

confidence: 99%

Hybrid Sampling/Spectral Method for Solving Stochastic Coupled Problems

Arnst¹,

Soize²,

Ghanem³

2013

SIAM/ASA J. Uncertainty Quantification

Self Cite

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Abstract. In this paper, we present a hybrid method that combines Monte Carlo sampling and spectral methods for solving stochastic coupled problems. After partitioning the stochastic coupled problem into subsidiary subproblems, the proposed hybrid method entails iterating between these subproblems in a way that enables the use of the Monte Carlo sampling method for subproblems that depend on a very large number of uncertain parameters and the use of spectral methods for subproblems that depend on only a small or moderate number of uncertain parameters. To facilitate communication between the subproblems, the proposed hybrid method shares between the subproblems a reference representation of all the solution random variables in the form of an ensemble of samples; for each subproblem solved by a spectral method, it uses a dimension-reduction technique to transform this reference representation into a subproblem-specific reduced-dimensional representation to facilitate a computationally efficient solution in a reduced-dimensional space. After laying out the theoretical framework, we provide an example relevant to microelectomechanical systems. 1. Introduction. Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering: Multiphysics models can take the form of a single equation that tightly couples different types of physical behavior, or they can take the form of a system of equations wherein the solution to certain equations is passed to other equations to determine physical properties or loadings or both. Multiscale models couple different types of behavior with fundamentally different descriptions at different scales. In multidomain models, physical behavior in different regions of space is coupled through a shared interface.Partitioned methods are widely used for solving coupled problems. These methods most often split a coupled problem into subproblems that represent single-physics, single-scale, or single-domain behavior, and they then seek a global solution by iterating between subproblem solutions. The attraction of partitioned methods is that many coupled problems afford a natural decomposition into subproblems for which computational expertise and dedicated

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Measure transformation and efficient quadrature in reduced‐dimensional stochastic modeling of coupled problems

Cited by 30 publications

References 44 publications

A dynamical polynomial chaos approach for long-time evolution of SPDEs

A dynamical polynomial chaos approach for long-time evolution of SPDEs

Quantifying Sampling Noise and Parametric Uncertainty in Atomistic-to-Continuum Simulations Using Surrogate Models

Hybrid Sampling/Spectral Method for Solving Stochastic Coupled Problems

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