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Research Articles

A Nearly Exact Reformulation of the Girsanov Linearization for Stochastically Driven Nonlinear Oscillators

[+] Author and Article Information
Tara Raveendran, R. M. Vasu

Department of Instrumentation and
Applied Physics,
Indian Institute of Science,
Bangalore, Karnataka 560012 India

D. Roy

Computational Mechanics Lab,
Department of Civil Engineering,
Indian Institute of Science,
Bangalore, Karnataka 560012 India
e-mail: royd@civil.iisc.ernet.in

1Corresponding author.

Manuscript received March 18, 2011; final manuscript received September 10, 2012; accepted manuscript posted October 8, 2012; published online January 25, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 80(2), 021020 (Jan 25, 2013) (11 pages) Paper No: JAM-11-1087; doi: 10.1115/1.4007779 History: Received March 18, 2011; Revised September 10, 2012; Accepted October 08, 2012

The Girsanov linearization method (GLM), proposed earlier in Saha, N., and Roy, D., 2007, “The Girsanov Linearisation Method for Stochastically Driven Nonlinear Oscillators,” J. Appl. Mech.,74, pp. 885–897, is reformulated to arrive at a nearly exact, semianalytical, weak and explicit scheme for nonlinear mechanical oscillators under additive stochastic excitations. At the heart of the reformulated linearization is a temporally localized rejection sampling strategy that, combined with a resampling scheme, enables selecting from and appropriately modifying an ensemble of locally linearized trajectories while weakly applying the Girsanov correction (the Radon–Nikodym derivative) for the linearization errors. The semianalyticity is due to an explicit linearization of the nonlinear drift terms and it plays a crucial role in keeping the Radon–Nikodym derivative “nearly bounded” above by the inverse of the linearization time step (which means that only a subset of linearized trajectories with low, yet finite, probability exceeds this bound). Drift linearization is conveniently accomplished via the first few (lower order) terms in the associated stochastic (Ito) Taylor expansion to exclude (multiple) stochastic integrals from the numerical treatment. Similarly, the Radon–Nikodym derivative, which is a strictly positive, exponential (super-) martingale, is converted to a canonical form and evaluated over each time step without directly computing the stochastic integrals appearing in its argument. Through their numeric implementations for a few low-dimensional nonlinear oscillators, the proposed variants of the scheme, presently referred to as the Girsanov corrected linearization method (GCLM), are shown to exhibit remarkably higher numerical accuracy over a much larger range of the time step size than is possible with the local drift-linearization schemes on their own.

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References

Socha, L., 2005, “Linearization in Analysis of Nonlinear Stochastic Systems: Recent Results—Part I: Theory,” ASME Appl. Mech. Rev., 58(3), pp. 178–205. [CrossRef]
Kallianpur, G., 1980, Stochastic Filtering Theory, Springer-Verlag, New York.
Kloeden, P. E., and Platen, E., 1999, Numerical Solution of Stochastic Differential Equations, Springer, New York.
Maruyama, G., 1955, “Continuous Markov Processes and Stochastic Equations,” Rend. Circ. Mat. Palermo, 4, pp. 48–90. [CrossRef]
Gard, T. C., 1988, Introduction to Stochastic Differential Equations, Marcel Dekker Inc., New York.
Burrage, K., BurrageP. M., and Tian, T., 2004, “Numerical Methods for Strong Solutions of Stochastic Differential Equations: An Overview,” Proc. R. Soc. London, Ser. A, 460(2041), pp. 373–402. [CrossRef]
Rumelin, W., 1982, “Numerical Treatment of Stochastic Differential Equations,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 19(3), pp. 604–613. [CrossRef]
Roy, D., and Dash, M. K., 2005, “Explorations of a Family of Stochastic Newmark Methods in Engineering Dynamics,” Comput. Methods Appl. Mech. Eng., 194(45–47), pp. 4758–4796. [CrossRef]
Roy, D., 2006, “A Family of Weak Stochastic Newmark Methods for Simplified and Efficient Monte Carlo Simulations of Oscillators,” Int. J. Numer. Methods Eng., 67(3), pp. 364–399. [CrossRef]
Roy, D., 2001, “A Numeric-Analytic Technique For Non-Linear Deterministic and Stochastic Dynamical Systems,” Proc. R. Soc. A, 457, pp. 539–566. [CrossRef]
Bernard, P., and Wu, L., 1998, “Stochastic Linearization: The Theory,” J. Appl. Probab., 35(3), pp. 718–730. [CrossRef]
Elishakoff, I., and Falsone, G., 1993, “Some Recent Developments in Stochastic Linearization Technique,” Computational Stochastic Mechanics, A.Cheng and C. Y.Yang, eds., Elsevier Applied Science, London, pp. 175–194.
Bouc, R., 1994, “The Power Spectral Density of Response for a Strongly Nonlinear Random Oscillator,” J. Sound Vib., 175(3), pp. 317–331. [CrossRef]
Socha, L., 1995, “Application of Probability Metrics to the Linearization and Sensitivity Analysis of Stochastic Dynamic Systems,” Proceedings of the International Conference on Nonlinear Stochastic Dynamics, Hanoi, Vietnam, December, pp. 193–202.
Socha, L., 2005, “Linearization in Analysis of Nonlinear Stochastic Systems: Recent Results—Part II: Applications,” ASME Appl. Mech. Rev., 58(5), pp. 303–314. [CrossRef]
Jimenez, J. C., 2002, “A Simple Algebraic Expression to Evaluate the Local Linearization Schemes for Stochastic Differential Equations,” Appl. Math. Lett., 15, pp. 775–780. [CrossRef]
Saha, N., and Roy, D., 2007, “The Girsanov Linearisation Method for Stochastically Driven Nonlinear Oscillators,” ASME J. Appl. Mech., 74, pp. 885–897. [CrossRef]
Rubinstein, R. Y., 1981, Simulation and the Monte Carlo Method, Wiley, New York.
Oksendal, B. K., 2003, Stochastic Differential Equations—An Introduction With Applications, 6th ed., Springer, New York.
Roy, D., 2000, “Exploration of the Phase-Space Linearization Method for Deterministic and Stochastic Nonlinear Dynamical Systems,” Nonlinear Dyn., 23(3), pp. 225–258. [CrossRef]
Robert, C. P., and Casella, G., 2004, Monte Carlo Statistical Methods, Springer, New York.
Handschin, J. E., and Mayne, D. Q., 1969, “Monte Carlo Techniques to Estimate the Conditional Expectation in Multi-State, Nonlinear Filtering,” Int. J. Control, 9, pp. 547–559. [CrossRef]
Beskos, A., and Roberts, G. O., 2005, “Exact Simulation of Diffusions,” Ann. Appl. Probab., 15(4), pp. 2422–2444. [CrossRef]
Wang, R., and Zhang, Z., 2000, “Exact Stationary Solutions of the Fokker-Planck Equation for Nonlinear Oscillators under Stochastic Parametric and External Excitations,” Nonlinearity, 13(3), pp. 907–920. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

HD oscillator: plot of Ψmax(k); (a) GCLM1, (b) GCLM2: C=5.0, K=100.0, α=100.0, σ = 5.0, hi = 0.01, X0¯ = {0,0}T, N = 2000

Grahic Jump Location
Fig. 2

HD oscillator: second moment histories through GCLM1 and GCLM2; (a) E [x12], (b) E [x22], and (c) acceptance rate; C = 5.0, K = 100.0, α = 100.0, σ = 5.0, hi = 0.01, X0¯ = {0,0}T, N = 10,000; solid black lines indicate exact stationary limits

Grahic Jump Location
Fig. 3

HD oscillator: second-moment histories of second moments through GCLM1 and GCLM2; (a) E [x12], (b) E [x22]; C = 1.0, K = 10.0, α = 1.0,σ = 0.5, hi = 0.01, X0¯ = {0,0}T, N = 5000; solid black lines indicate exact stationary limits

Grahic Jump Location
Fig. 4

HD oscillator: second moment histories via LM1 and GCLM1: (a) E [x12] and (b) E [x22];C=5.0,K=100.0,α=100.0,σ=5.0,hi=0.01,X¯0={0,0}T,N=5000; solid black lines indicate exact stationary limits

Grahic Jump Location
Fig. 5

HD oscillator: second-moment histories via GCLM1 and GCLM2: (a) E [x12], (b) E [x22], (c) Ψmax(k) for GCLM1 and (d) Ψmax(k) for GCLM2; C=1.0,K=10.0,α=100.0,σ=5.0,hi=0.005,X¯0={0,0}T,N=1000; solid black lines indicate exact stationary limits

Grahic Jump Location
Fig. 6

HD oscillator: second-moment history of E [x12] via LM1 and GCLM1; C=1.0,K=10.0,α=10.0,σ=0.5,hi=0.25,X¯0={0,0}T,N=10,000; solid black line indicate exact stationary limits

Grahic Jump Location
Fig. 7

The 2-DOF oscillator: second moment histories via GCLM1 and GCLM2: (a) E [(x2(1))2] and (b) E [(x1(2))2]; C1=C2=5.0, K1=K2=K3=100.0, α=100.0,σ1=5.0,σ2=5.0, hi=0.01,X¯0={0,0,0,0}T;N=10,000

Grahic Jump Location
Fig. 8

The 2-DOF oscillator: second moment histories via LM1 and GCLM1: E [(x2(1))2]; C1=C2=1.0, K1=K2=K3=10.0,σ1=0.5, α=10.0, σ2=0.5,hi=0.25,X¯0={0,0,0,0}T;N=10,000

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