  A fundamental limitation of various Equivalent Linearization Methods (ELMs)
in nonlinear random vibration analysis is that they are approximate by their
nature. A quantity of interest estimated from an ELM has no guarantee to be the
same as the solution of the original nonlinear system. In this study, we tackle
this fundamental limitation. We sequentially address the following two
questions: i) given an equivalent linear system obtained from any ELM, how do
we construct an estimator so that, as the linear system simulations are guided
by a limited number of nonlinear system simulations, the estimator converges on
the nonlinear system solution? ii) how to construct an optimized equivalent
linear system such that the estimator approaches the nonlinear system solution
as quickly as possible? The first question is theoretically straightforward
since classical Monte Carlo techniques such as the control variates and
importance sampling can improve upon the solution of any surrogate model. We
adapt the well-known Monte Carlo theories into the specific context of
equivalent linearization. The second question is challenging, especially when
rare event probabilities are of interest. We develop specialized methods to
construct and optimize linear systems. In the context of uncertainty
quantification (UQ), the proposed optimized ELM can be viewed as a physical
surrogate model-based UQ method. The embedded physical equations endow the
surrogate model with the capability to handle high-dimensional random vectors
in stochastic dynamics analysis.
