A novel stochastic locally transversal linearization (LTL) technique for engineering dynamical systems: Strong solutions

Abstract

Most available integration techniques for stochastically driven engineering dynamical systems are based on stochastic Taylor expansions of the response variables and thus require numerical modelling of multiple stochastic integrals (MSI-s). Since the latter is an extremely involved numerical task and becomes inaccurate for higher level MSI-s, these methods fail to achieve an accuracy beyond a limited order. Recently, the first author has proposed a locally transversal linearization (LTL) technique that completely avoids the use of Taylor-like expansions in the construction of the integration map [Proc. Roy. Soc. Ser. A, 457 (2001) 539; Int. J. Numer. Methods Eng. 61 (2004) 764]. A crucial step in the implementation of the LTL method is to arrive at a conditionally linearized solution which is then made to transversally intersect the non-linear solution manifold in the associated phase space. The present paper is the first part of an investigation consisting of two parts to considerably simplify and numerically expedite the generation of the conditionally linearized solution without affecting the local and global error orders of the original LTL method. In particular, the derivation of the conditionally linear form of the stochastic differential equations is done in such a way that the corresponding fundamental solution matrix (and consequently its inverse) remains unchanged during the entire integration process. In this part of the work, only strong (path wise) stochastic solutions are constructed. Through formal error estimates, it is verified that the present version of the LTL method has the same error orders as its older counterpart. Further, a host of numerical examples on stochastically driven non-linear oscillators are presented to illustrate its superior computational speed and ease 0307-904X/$ -see front matter Ó

Higher order weak linearizations of stochastically driven nonlinear oscillators

“…In order to obtain the linearized vector field forX using the older version of SLTL, we replace the nonlinear part of the drift field and the diffusion field corresponding to the multiplicative noise by using the still-unknown state vector X iC1 at tZt iC1 . Thus, the WSLTL-based SDEs take the (implicit) form (Roy & Dash 2005a) dX 1 ZX 2 dt;…”

A pseudo-dynamical systems approach to a class of inverse problems in engineering

“…Finally, we consider the implicit LTL-based linearization (Roy & Dash 2005) of the pseudo-dynamical system corresponding to the normal equation (2.2),…”

“…This is because the O(h 2 ) term in a Taylor expansion may not be smaller than the O(h) term (even for small h), unless the linearized operator is Lipschitz and the domain of the nonlinear minimizer is strictly non-empty. Finally, we consider the implicit LTL-based linearization (Roy & Dash 2005) of the pseudo-dynamical system corresponding to the normal equation (2.2),…”