A method for solving stochastic equations by reduced order models and local approximations

Grigoriu, Mircea

doi:10.1016/j.jcp.2012.06.013

Cited by 26 publications

(20 citation statements)

References 28 publications

(34 reference statements)

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“…We define these two steps in more detail in the following sections. A detailed discussion on the errors introduced by the SROM approach is provided in [9]. …”

Section: Stochastic Reduced Order Models (Srom)mentioning

confidence: 99%

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On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems

Field

Grigoriu

Emery

2015

Probabilistic Engineering Mechanics

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a b s t r a c tThe stochastic collocation (SC) and stochastic Galerkin (SG) methods are two well-established and successful approaches for solving general stochastic problems. A recently developed method based on stochastic reduced order models (SROMs) can also be used. Herein we provide a comparison of the three methods for some numerical examples; our evaluation only holds for the examples considered in the paper. The purpose of the comparisons is not to criticize the SC or SG methods, which have proven very useful for a broad range of applications, nor is it to provide overall ratings of these methods as compared to the SROM method. Rather, our objectives are to present the SROM method as an alternative approach to solving stochastic problems and provide information on the computational effort required by the implementation of each method, while simultaneously assessing their performance for a collection of specific problems.

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“…We define these two steps in more detail in the following sections. A detailed discussion on the errors introduced by the SROM approach is provided in [9]. …”

Section: Stochastic Reduced Order Models (Srom)mentioning

confidence: 99%

“…1 and 2. These methods include (1) stochastic collocation (SC) [3][4][5]; (2) stochastic Galerkin (SG) [5][6][7]; and (3) stochastic reduced order models (SROMs) [8,9]. The comparisons will be based on both accuracy and computational cost for a variety of examples.…”

Section: Introductionmentioning

confidence: 99%

On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems

Field

Grigoriu

Emery

2015

Probabilistic Engineering Mechanics

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“…We callZ a stochastic reduced order model of Z. Moreover, we denote by {Γ k , k = 1, ..., m 2 } the cells of a Voronoi partition of the range of Z, with centers {z k } [4]. A cell Γ k contains all samples z i of Z that are closer toz k than any other z l , i.e.…”

Section: Second-order Srom Solutionmentioning

confidence: 99%

“…The method has been originally developed for dynamic response [3]. Recently, it has been shown that for static problems the SROM-based method can be improved significantly [4]. Preliminary studies indicate that similar improvements will provide accurate solutions for random vibration problems [5].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Reduced-Order Models for Calculating Response Statistics of Structural Systems Subjected to Random Input

Radu¹,

Grigoriu²

2015

Proceedings of the 5th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. Probabilistic characterization of a system's response under random dynamic input is essential for performance-based engineering. Monte Carlo simulation is the only general method available for estimating response statistics. However, the method is computationally expensive for realistic structural systems since it involves repeated deterministic dynamic analyses for a large number of samples of the random input. The framework of random vibration is also inadequate for calculating response statistics for linear systems under non-Gaussian inputs and non-linear systems subjected to both Gaussian and non-Gaussian excitation.We propose a novel, highly efficient method for calculating response statistics. The method is based on stochastic reduced-order models (SROM), i.e., stochastic processes that have a finite number of samples selected in an optimal manner from samples of the input process. The SROM-based method can be viewed as a smart Monte-Carlo. Like Monte-Carlo simulation, the method uses random samples of the input to characterize structural response. Unlike Monte Carlo, which uses a large number of samples selected at random, SROM selects just a small number of samples in an optimal way. Numerical examples of the newly proposed method are presented for single-degree-of-freedom linear and non-linear Duffing and Bouc-Wen systems.1342

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“…The essential limitation of the method is the computational cost, which can be excessive when dealing with realistic microstructures. This study presents a practical method for characterizing material response at small scale that extends developments in [6,7] and [15,17]. The method can be viewed as a smart Monte Carlo simulation algorithm.…”

Section: Introductionmentioning

confidence: 99%

Response Statistics for Random Heterogeneous Microstructures

Grigoriu¹

2014

SIAM/ASA J. Uncertainty Quantification

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A practical method is developed for calculating statistics of material responses at small scale and functionals of these responses. The method is applied to characterize probabilistically potentials in specimens with conductivities that fluctuate randomly in space. Parametric models for microstructures, that is, deterministic functions of space coordinates that depend on random vectors Z, and SROMsZ of Z, that is, random vectors with finite numbers of samples matching statistics of Z, are the essential tools used in analysis. Two surrogate models are constructed for material responses U (x, Z) at small scale providing approximations for the mapping Z → U (x, Z), where x denotes the space coordinate. Samples of the surrogate models, which are generated by elementary calculations, are employed to characterize U (x, Z). Bounds are developed on the discrepancy between exact solutions and solutions by surrogate models. Numerical examples are presented to demonstrate the implementation of the proposed method and assess its accuracy by Monte Carlo simulation.

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A method for solving stochastic equations by reduced order models and local approximations

Cited by 26 publications

References 28 publications

On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems

On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems

Stochastic Reduced-Order Models for Calculating Response Statistics of Structural Systems Subjected to Random Input

Response Statistics for Random Heterogeneous Microstructures

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