| Multiscale modeling problems have become an active research area in recent years. There are many systems involving a large set of variables and these variables mostly behave in largely different time scales. It is necessary to derive proper effective models when one needs to obtain dynamical models that reproduce statistical properties of essential variables without wasting the computational time to compute non-essential variables in high dimensional systems.;For the second part of this dissertation, we propose effective models using a stochastic delay differential equation. The memory part in stochastic delay models is a simple linear combination of essential variables with finite number of delays. We apply this technique to the Truncated Burgers-Hopf equation and show that the effective model reproduces statistical behaviours of the full model.;In this dissertation, we develop two new approaches for stochastic effective models. The Markov chain stochastic parameterization technique is proposed for the effective models in the first part of this dissertation. This is a numerically oriented approach where sonic parts of the right hand side of essential variables are modeled by conditional Markov chains. It is shown that, under the proper conditioning scheme, statistical properties of essential variables from effective models have a good agreement with full models. Furthermore, we illustrate that the implementation of effective models including the conditioning scheme and the estimation of the t probability matrices is simple and straightforward. |
