Stochastic elliptic operators defined by non-Gaussian random fields with uncertain spectrum
Author:
C. Soize
Journal:
Theor. Probability and Math. Statist. 105 (2021), 113-136
MSC (2020):
Primary 60G60, 35J25; Secondary 74Q05, 74A40
DOI:
https://doi.org/10.1090/tpms/1159
Published electronically:
December 7, 2021
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Additional Information
Abstract: This paper presents a construction and the analysis of a class of non-Gaussian positive-definite matrix-valued homogeneous random fields with uncertain spectral measure for stochastic elliptic operators. Then the stochastic elliptic boundary value problem in a bounded domain of the 3D-space is introduced and analyzed for stochastic homogenization.
References
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References
- R. Adler, The geometry of random fields, SIAM, 2010. MR 3396215
- M. Bornert, T. Bretheau, and P. Gilormini, Homogenization in mechanics of materials, ISTE Ltd and John Wiley and Sons, New York, 2008.
- A. Bowman and A. Azzalini, Applied smoothing techniques for data analysis: The kernel approach with s-plus illustrations, vol. 18, Oxford University Press, Oxford: Clarendon Press, New York, 1997.
- R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals, Annals of Mathematics 48 (1947), no. 2, 385–392. MR 20230
- R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology, Springer Science & Business Media, Berlin, 2013. MR 1064315
- J. Doob, Stochastic processes, John Wiley & Sons, New York, 1953. MR 0058896
- A. D. Egorov, P. I. Sobolevsky, and L. A. Yanovich, Functional integrals: Approximate evaluation and applications, Kluwer Academinc Publishers, 1993. MR 1233132
- I. I. Gikhman and A. Skorokhod, Introduction à la théorie des processus aléatoires, Edition Mir, 1980.
- J. Guilleminot and C. Soize, On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties, J. Elasticity 111 (2013), no. 2, 109–130. MR 3036127
- J. Guilleminot and C. Soize, Stochastic model and generator for random fields with symmetry properties: application to the mesoscopic modeling of elastic random media, Multiscale Model. Simul. 11 (2013), no. 3, 840–870. MR 3091362
- T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin, Determination of the size of the representative volume element for random composites: statistical and numerical approach, International Journal of Solids and Structures 40 (2003), no. 13-14, 3647–3679.
- P. Krée and C. Soize, Mathematics of random phenomena, Reidel Pub. Co, 1986, (first published by Bordas in 1983 and also published by Springer Science & Business Media in 2012). MR 873731
- P. Lax and A. Milgram, Parabolic equations: Contributions to the theory of partial differential equations, Ann. of Math. Stud. (1954), no. 33, 167–190. MR 0067317
- N. Leonenko and A. Olenko, Tauberian and Abelian theorems for long-range dependent random fields, Methodol. Comput. Appl. Probab. 15 (2013), no. 4, 715–742. MR 3117624
- J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications: Vol. 1, vol. 181 of Springer Science & Business Media, 2012. MR 0350178
- A. Malyarenko and M. Ostoja-Starzewski, Statistically isotropic tensor random fields: correlation structures, Math. Mech. Complex Syst. 2 (2014), no. 2, 209–231. MR 3336288
- A. Malyarenko and M. Ostoja-Starzewski, Tensor-valued random fields for continuum physics, Cambridge University Press, 2018. MR 3930601
- A. Nouy and C. Soize, Random field representations for stochastic elliptic boundary value problems and statistical inverse problems, European J. Appl. Math. 25 (2014), no. 3, 339–373. MR 3197997
- M. Ostoja-Starzewski, Random field models of heterogeneous materials, International Journal of Solids and Structures 35 (1998), no. 19, 2429–2455.
- M. Ostoja-Starzewski, Material spatial randomness: From statistical to representative volume element, Probabilistic Engineering Mechanics 21 (2006), no. 2, 112–132.
- M. Ostoja-Starzewski, X. Du, Z.F. Khisaeva, and W. Li, Comparisons of the size of the representative volume element in elastic, plastic, thermoelastic, and permeable random microstructures, International Journal for Multiscale Computational Engineering 5 (2007), no. 2, 73–82.
- F. Poirion and C. Soize, Numerical methods and mathematical aspects for simulation of homogeneous and non homogeneous Gaussian vector fields, Probabilistic Methods in Applied Physics (Paul Krée and Walter Wedig, eds.), Springer-Verlag, Berlin, 1995, pp. 17–53.
- C. Robert and G. Casella, Monte Carlo statistical methods, Springer Science & Business Media, 2005. MR 2080278
- M. Rosenblatt, Stationary sequences and random fields, Springer Science & Business Media, 2012. MR 885090
- Yu. Rozanov, Random fields and stochastic partial differential equations, Kluwer Academic Publishers, 1998. MR 1629699
- R. Rubinstein and D. Kroese, Simulation and the Monte Carlo method, second edition, John Wiley & Sons, New York, 2008. MR 2365210
- A. V. Skorokhod and M. I. Yadrenko, On absolute continuity of measures corresponding to homogeneous Gaussian fields, Theory Probab. Appl. 18 (1973), no. 1, 27–40. MR 0312563
- C. Soize, A nonparametric model of random uncertainties on reduced matrix model in structural dynamics, Probabilistic Engineering Mechanics 15 (2000), no. 3, 277–294.
- C. Soize, Non Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators, Computer Methods in Applied Mechanics and Engineering 195 (2006), no. 1-3, 26–64. MR 2174357
- C. Soize, Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size, Probabilistic Engineering Mechanics 23 (2008), no. 2-3, 307–323.
- C. Soize, Random vectors and random fields in high dimension: parametric model-based representation, identification from data, and inverse problems, Handbook of Uncertainty Quantification (Roger Ghanem, David Higdon, and Houman Owhadi, eds.), vol. 2, Springer, Cham, Switzerland, 2017, pp. 883–936. MR 3821488
- C. Soize, Uncertainty quantification, Interdiscip. Appl. Math., vol. 47, Springer, New York, 2017. MR 3618803
- C. Soize, Computational stochastic homogenization of heterogeneous media from an elasticity random field having an uncertain spectral measure, Comput. Mech. published online (2021), 1–19. MR 4318420
- E. Vanmarcke, Random fields: Analysis and synthesis, World Scientific, Singapore, 2010. MR 2676223
- M. I. Yadrenko, Spectral theory of random fields, Optimization Software, 1983. MR 697386
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Additional Information
C. Soize
Affiliation:
MSME UMR 8208, Université Gustave Eiffel, 5 bd Descartes, 77454 Marne-la-Vallée, France
Email:
christian.soize@univ-eiffel.fr
Keywords:
Non-Gaussian random field,
uncertain spectral measure,
stochastic elliptic boundary value problem,
stochastic homogenization,
random effective elasticity tensor
Received by editor(s):
March 30, 2021
Published electronically:
December 7, 2021
Article copyright:
© Copyright 2021
Taras Shevchenko National University of Kyiv
