A novel joint diagonalization approach for linear stochastic systems and reliability analysis

Engineering Computations (Swansea, Wales)

Volumn 29, Issue 2, 2012, Pages 221-244

  Wang, Feng ^a   Li, Chenfeng ^a   Feng, Jianwen ^a   Cen, Song ^b   Owen, D R J ^a  

^a SWANSEA UNIVERSITY   (United Kingdom)

^b TSINGHUA UNIVERSITY   (China)

Author keywords

Conjugate gradient method; Least squares; Mathematics; Programming and algorithm theory; Real symmetric matrices; Reliability analysis; Steepest descent method; Stochastic processes

Indexed keywords

LEAST SQUARE; PROGRAMMING AND ALGORITHM THEORY; STEEPEST DESCENT; STOCHASTIC PROCESS; SYMMETRIC MATRICES;

COMPUTATIONAL EFFICIENCY; CONJUGATE GRADIENT METHOD; COST FUNCTIONS; FINITE ELEMENT METHOD; ITERATIVE METHODS; LINEAR TRANSFORMATIONS; MATHEMATICAL TECHNIQUES; MONTE CARLO METHODS; RANDOM PROCESSES; RELIABILITY ANALYSIS;

ALGORITHMS;

EID: 84863245672     PISSN: 02644401     EISSN: None     Source Type: Journal    
DOI: 10.1108/02644401211206061     Document Type: Article

References (55)

1. Al-Baali, M. (1985), "Descent property and global convergence of the Fletcher-Reeves method with inexact line search", IMA J. Numer. Anal., Vol. 5, pp. 121-4.
2. Al-Baali, M. and Fletcher, R. (1996), "On the order of convergence of preconditioned nonlinear conjugate gradient methods", SIAM J. Sci. Comput., Vol. 17, pp. 658-65. (Pubitemid 126621173)
3. Askey, R. and Wilson, J.A. (1985), "Some basic hypergeometric polynomials that generalize Jacobi polunomials", American Mathematical Society, Vol. 319, pp. 1-55.
4. Battiti, R. (1992), "1st-order and 2nd-order methods for learning-between steepest descent and Newton method", Neural Computation, Vol. 4, pp. 141-66.
5. Bayer, V., Roos, D. and Adam, U. (2007), "Structural reliability analysis by random field, modeling with robustness methods and adaptive response surfaces", paper presented at 11 th International Conference on Civil, Structural and Enviromental Engineering Computing, St Julians, Malta.
6. Bjorck, A. (1996), Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA.
7. British Standard Institution (1990a), EN 1990: Eurocode: Basis of Structural Design, Annex B, British Standard Institution, London.
8. British Standard Institution (1990b), EN 1990: Eurocode: Basis of Structural Design, Annex C, British Standard Institution, London.
9. Cornell, C.A. (1967), "Bounds on the reliability of structural systems", J. Struct. Div., ASCE, Vol. 30.
10. Cornell, C.A. (1969), "A probability based structural code", J. Amer. Concrete Inst., Vol. 66, pp. 974-85.
11. Cropper, W.H. (2004), Great Physicists: The Life and Times of Leading Physicists from Galileo to Hawking, Oxford University Press, Oxford.
12. Dai, Y.H. and Yuan, Y. (1995), "Further studies on the Polak-Ribiere-Polyak method", Research Report ICM-95-040, Chinese Academy of Science.
13. Dai, Y.H. and Yuan, Y. (1996), "Convergence of the Fletcher-Reeves method under a generalized Wolfe search", Journal Computational Mathematics of Chinese Universities, Vol. 2, p. 7.
14. Dai, Y.H. and Yuan, Y. (1998), "A class of globally convergent conjugate gradient methods", Research Report ICM-98-030, Institue of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Science.
15. Evans, D.H. (1992), Probability and Its Applications for Engineers, ASQC Quality Press, New York, NY.
16. Fishman, G.S. (1995), Monte Carlo: Concepts, Algorithms, and Applications, Springer, New York, NY.
17. Fletcher, R. (1987), Practical Methods of Optimization, Wiley, New York, NY.
18. Fletcher, R. and Powell, M.J.D. (1963), "A rapidly convergent descent method for minimization", The Computational Journal, Vol. 6, pp. 163-8.
19. Fletcher, R. and Reeves, C.M. (1964), "Function minimization by conjugate gradients", The Computational Journal, Vol. 7, pp. 149-54.
20. Ghanem, R.G. and Spanos, P.D. (2003), Stochastic Finite Elemens: A Spectral Approach, Springer, New York, NY.
21. Gilbert, J.C. and Nocedal, J. (1992), "Global convergence properties of conjugate gradient methods for optimization", SIAM Journal of Optimization, Vol. 2, pp. 21-42.
22. Gill, P.E., Murray, W. and Wright, M.H. (1981), Practical Optimization, Academic Press, New York, NY.
23. Golub, G.H. and Pereyra, V. (1973), "The differentiation of pseudoinverses and nonlinear least squares problems whose variables separate", SIAM J. Numer., pp. 413-32.
24. Grippo, L. and Lucidi, S. (1997), "A globally convergent version of the Polak-Ribiere conjugate gradient method", Mathematics Programming, Vol. 78, pp. 375-91. (Pubitemid 127398232)
25. Gulvanessian, H. and Holicky, M. (2005), "Eurocodes: using reliability analysis to combine action effects", Proceedings of the Institution of Civil Engineers, pp. 243-52. (Pubitemid 41186660)
26. Hairer, E. (1993), Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin.
27. Hasofer, A.M. and Lind, N.C. (1974), "Exact and invariant second-moment code format", Journal of the Engineering Mechanics Division, Vol. 100, pp. 111-21.
28. Huh, J., Lee, S.Y. and Haldar, A. (2003), "Reliability evaluation using finite element method", Proceedings of the Fourth International Symposium on Uncertainty Modelling and Analysis.
29. Iserles, A. (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, Cambridge.
30. Khoda, K.M., Lin, Y. and Storey, C. (1992), "Generalized Polak-Ribiere algorithm", Journal of Optimization Theory and Applications, Vol. 75, pp. 345-54. (Pubitemid 23581289)
31. Li, C.F., Feng, Y.T. and Owen, D.R.J. (2006a), "Explicit solution to stochastic system of linear algebraic equations (a1A1 + ⋯ a1A1)x=b", Computer Methods in Applied Mechanics and Engineering, Vol. 195 Nos 44-47, pp. 6560-76. (Pubitemid 44057615)
32. Li, C.F., Adhikari, S., Cen, S., Feng, Y.T. and Owen, D.R.J. (2009), "A joint diagonalisation approach for linear stochastic systems", Computers & Structures, Vol. 88, pp. 1137-48.
33. Li, C.F., Feng, Y.T., Owen, D.R.J., Li, D.F. and Davies, I.M. (2006b), "Fourier representation of random media fields in stochastic finite element modelling", Engineering Computations, Vol. 23, pp. 794-817. (Pubitemid 44450666)
34. Li, C.F., Feng, Y.T., Owen, D.R.J., Li, D.F. and Davies, I.M. (2008), "A Fourier-Karhunen-Loeve discretization scheme for stationary random material properties in SFEM", International Journal for Numerical Methods in Engineering, Vol. 73, pp. 1942-65. (Pubitemid 351454756)
35. Melchers, R.E. (1999), Structural Reliability Analysis and Prediction, Wiley, Chichester.
36. Nakagiri, S. (1989), "Reliability enhancement of portal frame structure by finite element synthesis", Transactions of the International Conference on Structural Mechanics in Reactor Technology, Vol. 10, pp. 155-60.
37. Neville, A.M. and Kennedy, J.B. (1964), Basic Statistical Methods for Engineers and Scientists, Harper & Row, London.
38. Nocedal, J. (1991), "Theory of algorithm for unconstrained optimization", Acta Numerical, Vol. 1, pp. 199-242.
39. Nocedal, J. (1995), "Conjugate gradient methods and nonlinear optimization", Proceedings of the AMS-IMS-SIAM Summer Research Conference on Linear and Nonlinear Conjugate Gradient-related Methods.
40. Nocedal, J. (1996), Large Scale Unconstrained Optimization, Northwestern University, Chicago, IL.
41. Nocedal, J. and Wright, S.J. (1999), Numerical Optimization, Springer, New York, NY.
42. Oldenburg, D.W., McGillivray, P.R. and Ellis, R.G. (1993), "Generalized subspace methods for large-scale inverse problems", Geophysical Journal International, Vol. 114, pp. 12-20. (Pubitemid 23394825)
43. Rosenbrock, H.H. (1960), "An automatic method for finding the greatest or least value of a function", The Computer Journal, Vol. 3, pp. 175-84.
44. Sachdeva, S.K., Nair, P.B. and Keane, A.J. (2006), "Comparative study of projection schemes for stochastic finite element analysis", Computer Methods in Applied Mechanics and Engineering, Vol. 195, pp. 2371-92.
45. Shewchuk, J.R. (1994), "An introduction to the conjugate gradient method without the agonizing pain", Technical Report, August.
46. Sudret, B. and Der Kiureghian, A. (2000), Stochastic Finite Element Methods and Reliability, University of California, Berkeley, CA.
47. Sudret, B. and Der Kiureghian, A. (2002), "Comparison of finite element reliability methods", Probabilistic Engineering Mechanics, Vol. 17, pp. 337-48. (Pubitemid 35460256)
48. Sudret, B., Berveiller, M. and Lemaire, M. (2003), "Application of a stochastic finite element procedure to reliability analysis", Proceedings of the 11th IFIP WG7.5 Conference on Reliability and Optimization of Structural Systems, Banff, Canada.
49. Thoft-Christensen, P. and Baker, M.J. (1982), Structural Reliability Theory and Its Applications, Springer, Berlin.
50. Wang, X. (2008), "Method of steepest descent and its application", IEEE Microwave and Wireless Components Letters, Vol. 12, pp. 24-6.
51. Wiener, N. (1938), "The homogeneous chaos", American Journal of Mathematics, Vol. 60, pp. 897-936.
52. Wolfe, P. (1969), "Convergence conditions for ascent methods", SIAM Review, Vol. 11, pp. 226-35.
53. Xiu, D. (2010), Numerical Methods for Stochastic Computations, University Press, Princeton, NJ.
54. Xu, H. and Rahman, S. (2005), "Decomposition methods for structural reliability analysis", Probabilistic Engineering Mechanics, Vol. 20, pp. 239-50. (Pubitemid 41145456)
55. Cameron, R. and Martin, W. (1947), "The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals", Ann. Math, Vol. 48, pp. 385-92.