Numerical methods and mathematical aspects for simulation of homogeneous and non homogeneous gaussian vector fields

Poirion, Fabrice; Soize, Christian

doi:10.1007/3-540-60214-3_50

Cited by 30 publications

(32 citation statements)

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“…This is true in particular if the random field depends only on one physical length scale. For such "single-scale" Gaussian homogenous random fields, a wide variety of simulation techniques beyond the Fourierwavelet and Randomization methods may well be adequate [36]. Our concern is with simulating multiscale random fields, meaning that ℓ s ≪ ℓ c .…”

Section: Length Scales Of the Random Fieldmentioning

confidence: 99%

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Comparative analysis of multiscale Gaussian random field simulation algorithms

Kramer

Kurbanmuradov

Sabelfeld

2007

Journal of Computational Physics

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We analyze and compare the efficiency and accuracy of two simulation methods for homogeneous random fields with multiscale resolution. We consider in particular the Fourier-wavelet method and three variants of the Randomization method: (A) without any stratified sampling of wavenumber space, (B) with stratified sampling of wavenumbers with equal energy subdivision, (C) with stratified sampling with a logarithmically uniform subdivision. We focus primarily on fractal Gaussian random fields with Kolmogorov-type spectra. Previous work has shown that variants (A) and (B) of the Randomization method are only able to generate a self-similar structure function over three to four decades with reasonable computational effort. By contrast, variant (C), along with the Fourier-wavelet method, is able to reproduce accurate self-similar scaling of the structure function over a number of decades increasing linearly with computational effort (for our examples we will show that nine decades can be reproduced). We provide some conceptual and numerical comparison of the various cost contributions to each random field simulation method.We find that when evaluating ensemble averaged quantities like the correlation and structure functions, as well as some multi-point statistical characteristics, the Randomization method can provide good accuracy with less cost than the Fourierwavelet method. The cost of the Randomization method relative to the Fourierwavelet method, however, appears to increase with the complexity of the random field statistics which are to be calculated accurately. Moreover, the Fourier-wavelet method has better ergodic properties, and hence becomes more efficient for the computation of spatial (rather than ensemble) averages which may be important in simulating the solutions to partial differential equations with random field coefficients.

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Section: Length Scales Of the Random Fieldmentioning

confidence: 99%

“…A Riemann sum discretization of the stochastic integral is easy to implement [44,47,48,36,16], and following [8,20,30], we shall refer to it as the stan-dard Fourier method. As documented in [8], this method can suffer from false periodicity artifacts of the discretization, particularly if the wavenumbers are chosen with uniform spacing.…”

Section: Introductionmentioning

confidence: 99%

Comparative analysis of multiscale Gaussian random field simulation algorithms

Kramer

Kurbanmuradov

Sabelfeld

2007

Journal of Computational Physics

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“…The details concerning the generation of realizations of random field U can be found in [82,83]. One possible method is based on the usual numerical simulation of homogeneous Gaussian vector-valued random field constructed with the stochastic integral representation of homogeneous stochastic fields (see [69,79]). …”

Section: A Construction Of the Non-gaussian Random Field [G 0 ] And Imentioning

confidence: 99%

Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm

Soize

2017

Comput Mech

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This paper deals with the optimal design of a titanium mesoscale implant in a cortical bone for which the apparent elasticity tensor is modeled by a non-Gaussian random field at mesoscale, which has been experimentally identified. The external applied forces are also random. The design parameters are geometrical dimensions related to the geometry of the implant. The stochastic elastostatic boundary value problem is discretized by the finite element method. The objective function and the constraints are related to normal, shear, and von Mises stresses inside the cortical bone. The constrained nonconvex optimization problem in presence of uncertainties is solved by using a probabilistic learning algorithm that allows for considerably reducing the numerical cost with respect to the classical approaches.

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“…The numerical simulation of homogeneous Gaussian vector-valued random field was introduced by Shinozuka [66][67][68]. A detailed development with additional mathematical properties related to convergence properties can be found in [54] and…”

Section: Representation Of the Random Field U Adapted To Its Numericamentioning

confidence: 99%

“…We then have the following vectorized ν-order approximation For any ν = 2p fixed, V ν is a Gaussian vector and taking into account the properties of the random field U given in Section 3.2.1, it can be proved [54] that:…”

Section: Soize -Cmame -Revised Version December 2004mentioning

confidence: 99%

Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators

Soize¹

2006

Computer Methods in Applied Mechanics and Engineering

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167

235

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To cite this version:Christian Soize. Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2006, 195 (1-3) AbstractThis paper deals with the construction of a class of non Gaussian positive-definite matrix-valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled. The properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given. Such a matrix-valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters. The class of random fields which can be approximated is presented and their experimental identification is analyzed. An example is given in three-dimensional linear elasticity for which the fourth-order elasticity tensor-valued random field is constructed for a random non homogeneous anisotropic elastic material.

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Numerical methods and mathematical aspects for simulation of homogeneous and non homogeneous gaussian vector fields

Cited by 30 publications

References 32 publications

Comparative analysis of multiscale Gaussian random field simulation algorithms

Comparative analysis of multiscale Gaussian random field simulation algorithms

Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm

Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators

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