Probabilistic equivalence and stochastic model reduction in multiscale analysis

Arnst, Maarten; Ghanem, Roger

doi:10.1016/j.cma.2008.03.016

Cited by 54 publications

(43 citation statements)

References 29 publications

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“…Moreover, from an information theory perspective 8 , the relative entropy measures loss/change of information. Relative entropy for highdimensional systems was used as measure of loss of information in coarse-graining 2,20,24 , and sensitivity analysis for climate modeling problems 28 .…”

Section: Introductionmentioning

confidence: 99%

Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems

Katsoulakis

Plecháč

2013

The Journal of Chemical Physics

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In this paper we focus on the development of new methods suitable for efficient and reliable coarse-graining of non-equilibrium molecular systems. In this context, we propose error estimation and controlled-fidelity model reduction methods based on Path-Space Information Theory, combined with statistical parametric estimation of rates for non-equilibrium stationary processes. The approach we propose extends the applicability of existing information-based methods for deriving parametrized coarse-grained models to Non-Equilibrium systems with Stationary States (NESS). In the context of coarse-graining it allows for constructing optimal parametrized Markovian coarse-grained dynamics within a parametric family, by minimizing information loss (due to coarse-graining) on the path space. Furthermore, we propose an asymptotically equivalent methodrelated to maximum likelihood estimators for stochastic processes-where the coarse-graining is obtained by optimizing the information content in path space of the coarse variables, with respect to the projected computational data from a fine-scale simulation. Finally, the associated path-space Fisher Information Matrix can provide confidence intervals for the corresponding parameter estimators. We demonstrate the proposed coarse-graining method in (a) non-equilibrium systems with diffusing interacting particles, driven by out-ofequilibrium boundary conditions, as well as (b) multi-scale diffusions and their well-studied corresponding stochastic averaging limits, comparing them to our proposed methodologies.

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Section: Introductionmentioning

confidence: 99%

Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems

Katsoulakis

Plecháč

2013

The Journal of Chemical Physics

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“…, where k i and l i are uniformly distributed over the interval [0,4], while φ i and ϕ i are uniformly distributed over the interval [0,1] as the function class of the right-hand side in the preconditioning of the MsDSM method. We use this random training strategy to reduce the computational cost.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…However, when the dimension in stochastic direction is large, this method is inefficient due to the exponential growth of the number of the gPC basis elements. We also point out that in [1], Arnst and Ghanem considered the probabilistic equivalence and stochastic model reduction in multiscale analysis. In [27,37], Kevrekidis et al applied the equation-free idea to study stochastic incompressible flows.…”

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confidence: 99%

A Multiscale Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients

Zhang¹,

Ci²,

Hou³

2015

Multiscale Model. Simul.

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Abstract. In this paper, we propose a multiscale data-driven stochastic method (MsDSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. This method combines the advantages of the recently developed multiscale model reduction method [M. L. Ci, T. Y. Hou, and Z. Shi, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 449-474] and the datadriven stochastic method (DSM) [M. L. Cheng et al., SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 452-493]. Our method consists of offline and online stages. In the offline stage, we decompose the harmonic coordinate into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Based on the Karhunen-Loève (KL) expansion of the smooth parts and oscillatory parts of the harmonic coordinates, we can derive an effective stochastic equation that can be well-resolved on a coarse grid. We then apply the DSM to the effective stochastic equation to construct a data-driven stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions. In the online stage, we expand the SPDE solution using the data-driven stochastic basis and solve a small number of coupled deterministic partial differential equations (PDEs) to obtain the expansion coefficients. The MsDSM reduces both the stochastic and the physical dimensions of the solution. We have performed complexity analysis which shows that the MsDSM offers considerable savings over not only traditional methods but also DSM in solving multiscale SPDEs. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation. 1. Introduction. In recent years, there has been an increased interest in the simulation of systems with uncertainties. Many physical and engineering applications involving uncertainty quantification can be described by stochastic partial differential equations (SPDEs). Several numerical methods have been developed in the literature to solve SPDEs, such as the stochastic finite element method [24], Wiener chaos expansion or generalized polynomial chaos (gPC) method [31,25,40,41,42,39,33,32], and stochastic collocation method [43,5,30,6]. These methods can effectively solve the SPDEs when the dimension of stochastic input variables is low. However, their performance deteriorates dramatically when the dimension of stochastic input variables is high. This so-called curse of dimensionality is one of the essential challenges in uncertainty quantification. Recently, some progress has been made to alleviate this difficulty by exploring the sparse structure of the solutions and constructing a problem-dependent stochastic basis to solve these SPDEs; see, e.g., the data-driven stochastic method [11,44] and dynamically biorthogonal method [9,10].

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“…(ii.1) -The first one corresponds to the spectral methods such as the Polynomial Chaos representations (see [63,64] and also [65,66,67,68,69,70,71,72,73]) which can be applied in infinite dimension for stochastic processes and random fields, which allow the effective construction of mapping h to be carried out and which allow any random variable X in L 2 N , to be written as…”

Section: Types Of Representation For the Stochastic Modeling Of Uncermentioning

confidence: 99%

“…Today, many applications of such an approach have been carried out for direct and inverse problems. We refer the reader to [65,74,75,76,77,78,79,80,81] and in particular, to Section 3.7 for a short overview concerning the identification and inverse stochastic problems related to the parametric and nonparametric probabilistic approaches of uncertainties.…”

Section: Types Of Representation For the Stochastic Modeling Of Uncermentioning

confidence: 99%

Stochastic modeling of uncertainties in computational structural dynamics—Recent theoretical advances

Soize

2013

Journal of Sound and Vibration

137

106

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To cite this version:Christian Soize. Stochastic modeling of uncertainties in computational structural dynamics -Recent theoretical advances. Journal of Sound and Vibration, Elsevier, 2013, 332 (10) AbstractThis paper deals with a short overview on stochastic modeling of uncertainties. We introduce the types of uncertainties, the variability of real systems, the types of probabilistic approaches, the representations for the stochastic models of uncertainties, the construction of the stochastic models using the maximum entropy principle, the propagation of uncertainties, the methods to solve the stochastic dynamical equations, the identification of the prior and the posterior stochastic models, the robust updating of the computational models and the robust design with uncertain computational models. We present recent theoretical advances in this field concerning the parametric and the nonparametric probabilistic approaches of uncertainties in computational structural dynamics for the construction of the prior stochastic models of both the uncertainties on the computational model parameters and on the modeling uncertainties, and for their identification with experimental data. We also present the construction of the posterior stochastic model of uncertainties using the Bayesian method when experimental data are available.

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Probabilistic equivalence and stochastic model reduction in multiscale analysis

Cited by 54 publications

References 29 publications

Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems

Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems

A Multiscale Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients

Stochastic modeling of uncertainties in computational structural dynamics—Recent theoretical advances

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