Construction of probability distributions in high dimension using the maximum entropy principle: Applications to stochastic processes, random fields and random matrices

International Journal for Numerical Methods in Engineering

Volumn 76, Issue 10, 2008, Pages 1583-1611

  Soize, Christian ^a  

^a Université Paris Est Créteil   (France)

Author keywords

High dimension; Karhunen Loeve expansion; Maximum entropy principle; Random fields; Random matrix; Stochastic process

Indexed keywords

APPLICATIONS; COMPUTATIONAL MECHANICS; DATA COMPRESSION; DIFFERENTIAL EQUATIONS; ENTROPY; INVERSE PROBLEMS; MAXIMUM PRINCIPLE; NANOELECTRONICS; RANDOM PROCESSES; RANDOM VARIABLES; STRUCTURES (BUILT OBJECTS); TIME SERIES;

HIGH DIMENSION; KARHUNEN-LOEVE EXPANSION; MAXIMUM ENTROPY PRINCIPLE; RANDOM FIELDS; RANDOM MATRIX; STOCHASTIC PROCESS;

PROBABILITY DISTRIBUTIONS;

EID: 60949108887     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2385     Document Type: Article

References (37)

1. Schueller GI. A state-of-the-art report on computational stochastic mechanics. Probabilistic Engineering Mechanics 1997; 12(4):197-321.
2. Schueller GI. Computational stochastic mechanics - recent advances. Computers and Structures 2001; 79(22-25):2225-2234.
3. Wiener N. The homogeneous chaos. American Journal of Mathematics 1938; 60:897-936.
4. Spanos PD, Ghanem R. Stochastic finite element expansion for random media. Journal of Engineering Mechanics (ASCE) 1989; 115(5):1035-1053.
5. Ghanem R, Spanos PD. Stochastic Finite Elements: A Spectral Approach. Springer: New York, 1991.
6. Ghanem R. Ingredients for a general purpose stochastic finite elements formulation. Computer Methods in Applied Mechanics and Engineering 1999; 168:19-34.
7. Soize C, Ghanem R. Physical systems with random uncertainties: chaos representation with arbitrary probability measure. SIAM Journal on Scientific Computing 2004; 26(2):395-410.
8. Soize C. A nonparametric model of random uncertainties for reduced matrix models in structural dynamics. Probabilistic Engineering Mechanics 2000; 15(3):277-294.
9. Soize C. Maximum entropy approach for modeling random uncertainties in transient elastodynamics. Journal of the Acoustical Society of America 2001; 109(5):1979-1996.
10. Soize C. Random matrix theory for modeling uncertainties in computational mechanics. Computer Methods in Applied Mechanics and Engineering 2005; 194(12-16):1333-1366.
11. Shannon CE. A mathematical theory of communication. Bell System Technical Journal 1948; 27:379-423, 623-659.
12. Jaynes ET. Information theory and statistical mechanics. Physical Review 1957; 106(4):620-630; 108(2):171-190.
13. Serfling RJ. Approximation Theorems of Mathematical Statistics. Wiley: New York, 1980.
14. Kapur JN, Kesavan HK. Entropy Optimization Principles with Applications. Academic Press: San Diego, 1992.
15. Jumarie G. Maximum Entropy, Information Without Probability and Complex Fractal. Kluwer Academic Publishers: Dordrecht, 2000.
16. Sobczyk K, Trebicki J. Maximum entropy principle in stochastic dynamics. Probabilistic Engineering Mechanics 1990; 5(3):102-110.
17. Mead LR, Papanicolaou N. Maximum entropy in the problem of moments. Journal of Mathematical Physics 1984; 25(8):2404-2417.
18. Ciulli S, Mounsif M. On the application of maximum entropy to the moments problem. Journal of Mathematical Physics 1991; 32(7):1717-1719.
19. Tagliani A. Maximum entropy in the Hamburger moments problem. Journal of Mathematical Physics 1994; 35(9):5087-5096.
20. Volpe EV, Baganoff D. Maximum entropy pdfs and the moment problem under near-Gaussian conditions. Probabilistic Engineering Mechanics 2003; 18(3):17-29.
21. Coleman TF, Li Y. On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to sounds. Mathematical Programming 1994; 67(2):189-224.
22. Coleman TF, Li Y. An interior, trust region approach for nonlinear minimization subject to bounds. SIAM Journal on Optimization 1996; 6:418-445.
23. Ergorov AD, Sobolevsky PI, Yanovich LA. Functional Integrals: Approximate Evaluation and Applications. Kluwer Academic Publisher: Dordrecht, 1993.
24. MacKeown PK. Stochastic Simulation in Physics. Springer: Singapore, 1997.
25. Hastings WK. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 1970; 109:57-97.
26. Kaipio J, Somersalo E. Statistical and Computational Inverse Problems. Springer: New York, 2005.
27. Spall JC. Introduction to Stochastic Search and Optimization. Wiley: Hoboken, NJ, 2003.
28. Geman S, Geman D. Stochastic relaxation, Gibbs distribution and the Bayesian distribution of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 1984; I-6:721-741.
29. Mehta ML. Random Matrices (2nd edn). Academic Press: New York, 1991.
30. Soize C. The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions. World Scientific: Singapore, 1994.
31. Caughey TK. Derivation and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white random excitation. The Journal of the Acoustical Society of America 1963; 35(11):1683-1692.
32. Yosida K. Functional Analysis. Springer: Berlin, 1995.
33. Talay D, Tubaro L. Expansion of the global error for numerical schemes solving stochastic differential equation. Stochastic Analysis and Applications 1990; 8(4):94-120.
34. Kloeden PE, Platen E. Numerical Solution of Stochastic Differential Equations. Springer: Berlin, 1992.
35. Talay D. Stochastic Hamiltonian system: exponential convergence to the invariant measure and discretization by the implicit Euler scheme. Markov Processes and Related Fields 2002; 8:163-198.
36. Clough RW, Penzien J. Dynamics of Structures. McGraw-Hill: New York, 1975.
37. Soize C. Non Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Computer Methods in Applied Mechanics and Engineering 2006; 195:26-64.