Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities

Branicki, Michał; Majda, Andrew J.

doi:10.4310/cms.2013.v11.n1.a3

Cited by 64 publications

(64 citation statements)

References 38 publications

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“…Therefore, the computational cost increases rapidly with dimension, which decreases the efficiency of the PC method. The second related problem of PC is that initial predetermined bases may lose their optimality and even fail to converge for long-time evolutions [19,20,6,17,38,22,1].…”

Section: Given a Complete Countable Orthonormal Systemmentioning

confidence: 99%

“…Once these deterministic equations are solved, the statistical properties of the solution can be readily inferred from the coefficients of the expansion, which facilitates uncertainty quantification. In some cases, the PC method can propagate uncertainties with a substantially lower cost than MC methods; especially for low dimensional uncertainties [4,16,7,6,17,1]; see also [18]. However, in cases of high dimensional random parameters, the efficiency of the PC method is reduced because of the large number of terms that appear in the expansion.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the optimality of the initial basis typically degrades as time evolves due to the presence of nonlinearities in the equation. These drawbacks were addressed in [19,20,6,21,22,23,17]. In our previous work [1], we proposed the Dynamical generalized Polynomial Chaos (DgPC) method to solve low dimensional stochastic differential equations (SDEs) driven by white noise.…”

Section: Introductionmentioning

confidence: 99%

“…Equilibrium statistics and asymptotic properties of the solutions are relevant in many applications and have been extensively studied in the literature [12,11,36,6,37,17]. Based on these motivations, we provide numerical experiments for a 1D randomly forced Burgers equation and a 2D stochastic Navier-Stokes (SNS) system.…”

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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Section: Given a Complete Countable Orthonormal Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…For n = 2, the condition in (3.2) is only satisfied in regimes I and III. In regime I, whereγ decays much faster than u, this condition implies bounded first-and second-order statistics (see appendix D of [49]). Next, we will find a reduced stochastic prior model corresponding to the nonlinear system in (3.1).…”

Section: Nonlinear Modelmentioning

confidence: 99%

The role of additive and multiplicative noise in filtering complex dynamical systems

Gottwald

Harlim

2013

Proc. R. Soc. A.

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Covariance inflation is an ad hoc treatment that is widely used in practical real-time data assimilation algorithms to mitigate covariance underestimation owing to model errors, nonlinearity, or/and, in the context of ensemble filters, insufficient ensemble size. In this paper, we systematically derive an effective 'statistical' inflation for filtering multi-scale dynamical systems with moderate scale gap, = O(10 −1 ), to the case of no scale gap with = O(1), in the presence of model errors through reduced dynamics from rigorous stochastic subgridscale parametrizations. We will demonstrate that for linear problems, an effective covariance inflation is achieved by a systematically derived additive noise in the forecast model, producing superior filtering skill. For nonlinear problems, we will study an analytically solvable stochastic test model, mimicking turbulent signals in regimes ranging from a turbulent energy transfer range to a dissipative range to a laminar regime. In this context, we will show that multiplicative noise naturally arises in addition to additive noise in a reduced stochastic forecast model. Subsequently, we will show that a 'statistical' inflation factor that involves mean correction in addition to covariance inflation is necessary to achieve accurate filtering in the presence of intermittent instability in both the turbulent energy transfer range and the dissipative range.

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A comparison of efficient uncertainty quantification techniques for stochastic multiscale systems

Kimaev

Ricardez‐Sandoval

2017

AIChE Journal

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The aim of this article is to compare the performance of efficient uncertainty propagation techniques (Polynomial Chaos [PCE] and Power Series [PSE] expansions) for uncertainty quantification in multiscale systems where discrete (molecular) scale is modeled without closed‐form expressions. A multiscale model of thin film formation by chemical vapor deposition was used to study the effects of single parameter and multivariate uncertainty. For the single parameter uncertainty, 2nd order PSE approximations were the most accurate and computationally attractive. For the multivariate uncertainty, PSE performance deteriorated, while 2nd order PCE provided the highest accuracy when its expansion coefficients were calculated using the Least Squares method. However, comparable accuracy was achieved at half the computational cost when the coefficients were calculated using Nonintrusive Spectral Projection (NISP). The response variables were subsequently controlled using robust optimization, and the results obtained using PCE NISP satisfied the optimization constraints more closely than other methods. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3361–3373, 2017

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Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities

Cited by 64 publications

References 38 publications

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

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

The role of additive and multiplicative noise in filtering complex dynamical systems

A comparison of efficient uncertainty quantification techniques for stochastic multiscale systems

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