Dynamical Polynomial Chaos Expansions and Long Time Evolution of Differential Equations with Random Forcing

Ozen, H. Cagan; Bal, Guillaume

doi:10.1137/15m1019167

Cited by 17 publications

(29 citation statements)

References 22 publications

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“…In both cases, the long-time simulation of SPDEs proves to be quite expensive [6,7,8]. In this paper, we extend the PC-based method developed for stochastic differential equations in [1] to the setting of SPDEs.…”

Section: Introductionmentioning

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%

“…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. The presence of a temporal random forcing is a serious challenge to the PC method as the number of stochastic variables increases linearly with time, which hinders the capability of the PC method for long-time computations [1,6]. Moreover, the optimality of the initial basis typically degrades as time evolves due to the presence of nonlinearities in the equation.…”

Section: Introductionmentioning

confidence: 99%

“…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. The novelty of the DgPC algorithm is that it utilizes a restart procedure and constructs polynomial chaos expansions (PCEs) dynamically in time by using orthogonal polynomials of the projections of the solution at the current steps and the future random forcing [1].…”

Section: Introductionmentioning

confidence: 99%

“…This allows the algorithm to keep reasonably sparse random bases, and to mitigate the aforementioned curse of dimensionality and loss of optimality. Numerical experiments in that reference illustrate that such a restart procedure has the ability to accurately estimate long-time solutions of SDEs; see the relevant computational and theoretical details in [1].…”

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: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

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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Dynamic stochastic finite element method using time‐dependent generalized polynomial chaos

Lacour

Bal

Abrahamson

2020

Num Anal Meth Geomechanics

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Summary We present an intrusive formulation for the dynamic stochastic finite‐element method to propagate the epistemic uncertainty in material properties into a finite‐element system over time. The stochastic finite‐element method, originally developed for the static case, uses generalized polynomial chaos (gPC) expansions to represent the uncertainty in both material/load fields and displacement fields and solves for the unknown PC coefficients of displacement at each degree of freedom of the finite‐element system. In this case, the gPC basis used to represent the solution is optimal and can be kept the same throughout the static simulation; however, when integrating a stochastic system over time, it is proven that using gPC tends to break down for integrations over long times. The reason is that the solution's complexity increases in time, and the set of polynomials used in the gPC expansion to represent the solution, therefore, does not stay optimal. In this formulation, new stochastic variables and orthogonal polynomials are constructed as time progresses. These variables are obtained as the Kahrunen‐Loeve expansion of the finite‐element solution at the times of update, thus, optimally representing the distribution of the solution using a minimal set of orthogonal polynomials. The result from the method is a time‐domain polynomial chaos representation of the entire finite‐element solution. A fast post‐processing phase can be used to (a) obtain the probability distribution of the solution at each degree of freedom and (b) generate any number of time series realizations of the solution, which correspond to the same time series that would be obtained from a set of simulations based on direct Monte‐Carlo simulations of the uncertain material properties.

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Polynomial stochastic dynamical indicators

Vasile

Manzi

2023

Celest Mech Dyn Astron

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This paper introduces three types of dynamical indicators that capture the effect of uncertainty on the time evolution of dynamical systems. Two indicators are derived from the definition of finite-time Lyapunov exponents, while a third indicator directly exploits the property of the polynomial expansion of the dynamics with respect to the uncertain quantities. The paper presents the derivation of the indicators and a number of numerical experiments that illustrates the use of these indicators to depict a cartography of the phase space under parametric uncertainty and to identify robust initial conditions and regions of practical stability in the restricted three-body problem.

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Dynamical Polynomial Chaos Expansions and Long Time Evolution of Differential Equations with Random Forcing

Cited by 17 publications

References 22 publications

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

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

Dynamic stochastic finite element method using time‐dependent generalized polynomial chaos

Polynomial stochastic dynamical indicators

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