A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

Abstract

SUMMARYEngineers are often more concerned with the computation of statistical moments (or mathematical expectations) of the response of stochastically driven dynamical systems than with the determination of path-wise response histories. With this in perspective, weak stochastic solutions of dynamical systems, modelled as n degrees-of-freedom (DOF) mechanical oscillators and driven by additive and/or multiplicative white noise (or, filtered white noise) processes, are considered in this study. While weak stochastic solutions are simpler and quicker to compute than strong (sample path-wise) solutions, it must be emphasized that sample realizations of weak solutions have no path-wise similarity with strong solutions. However, the statistical moment of any continuous and sufficiently differentiable deterministic function of the weak stochastic response is 'close' to that of the true response (if it exists) within a certain order of a given time step size. Computation of weak response therefore assumes great significance in the context of simulations of stochastically driven dynamical systems (oscillators) of engineering interest. To efficiently generate such weak responses, a novel class of weak stochastic Newmark methods (WSNMs), based on implicit Ito-Taylor expansions of displacement and velocity fields, is proposed. The resulting multiple stochastic integrals (MSIs) in these expansions are replaced by a set of random variables with considerably simpler and discrete probability densities. In fact, yet another simplifying feature of the present strategy is that there is no need to model and compute some of the higher-level MSIs. Estimates of error orders of these methods in terms of a given time step size are derived and a proof of global convergence provided. Numerical illustrations are provided and compared with exact solutions whenever available to demonstrate the accuracy, simplicity and higher computational speed of WSNMs vis-à-vis a few other popularly used stochastic integration schemes as well as the path-wise versions of the stochastic Newmark scheme.

Higher order weak linearizations of stochastically driven nonlinear oscillators

“…Such backward expansions enable the linearized fields to remain identical with the original ones at the forward time point (of a given time-step) and also ensure a higher closeness between the two vector fields. This forms the basis of arguments that indicate higher formal orders achievable through the present schemes vis-à-vis SLTL-1 and, indeed, most existing schemes in the literature Roy 2006;Komori 2007). Apart from the derivative-free nature of this class of linearizations, the other source of simplicity is in the usage of not-too-many stochastic integrals, especially for the weak formulations.…”

“…Substituting (3.23) into (3.22) and noting that EkXðsÞk 2z is bounded by some quantity Milstein 1995), the first inequality of (3.18) follows. In a similar way, the other inequalities of (3.18)-(3.21) may be shown (Roy 2006).& Now, introduce the following notations for n-dimensional exact (or true), strong and weak response increment vectors as:…”

“…From the first principles of Ito calculus, it may be shown that EðI r $I l0 ÞZ0:5h 2 d rl . Thus, we may write We refer to Milstein (1995) or Roy (2006) for the derivation of the above expression.…”

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

“…The Radon-Nikodym derivative is an exponential martingale (strictly positive) computable as 1 f " ? (9) subject to M(0) = l.With increasing order of linearization and decreasing time step size, |e^' | becomes smaller and hence M(i) approaches 1, in principle. While this highlights the importance of linearization (and smaller time steps) in the present setup, an exact evaluation of M{t) remains infeasible.…”

“…More specifically, owing to the computational intractability of the multiple stochastic integrals (MSIs), the strongest schemes for SDEs, generally derived using variants of the stochastic Taylor expansion, have considerably lower orders of accuracy. Strong explicit schemes include the Euler-Maruyama [4], stochastic Heun [5], stochastic Runge-Kutta [6,7], and stochastic Newmark [8,9]. For stiff SDEs, where explicit numerical schemes could lose stability and demand inordinately low step sizes, implicit methods [3] often offer higher numerical stability.…”