When designing concurrent coupling techniques in multiscale simulation, one often needs to obtain information at the microscopic (detailed) level of description, when the only available information is given at a macroscopic level of description (where only some averaged quantities are modeled). Examples are:
- the probability distributions generated by stochastic differential equations, whose evolution might be approximated at a macroscopic level of description by the evolution of some moments of this distribution (such as mean and variance) – this happens, for instance, in polymeric fluid flow);
- high-dimensional stochastic differential equations with slow and fast degrees of freedom, for which an approximate evolution can be conceived in terms of only the slow degrees of freedom;
- kinetic equations, in which a distribution function in position-velocity phase space may be approximated by some low-order moments over velocity space – this happens, for instance, in biological problems, nuclear fusion or rarified gas flow.
In all these situations (and more), missing information needs to be added at the microscopic scale. We develop and analyze algorithms to do this, based on either a quasi-equilibrium approximation or via minimization of the perturbation to a reference (prior) microscopic state.
- T. Lelièvre, G. Samaey, and P. Zielinski, Analysis of a micro-macro acceleration method with minimum relative entropy moment matching, 2018. Submitted.
- K. Debrabant, G. Samaey and P. Zielinski, A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations, SIAM Journal on Numerical Analysis 55:2745–2786, 2017.
- F. Legoll, T. Lelièvre, and G. Samaey, A micro/macro parareal algorithm for singularly perturbed ordinary differential equations, SIAM Journal on Scientific Computing 35:A1951-A1986, 2013.
- G. Mazzi, Y. De Decker and G. Samaey, Towards an efficient multiscale modeling of low-dimensional reactive systems: study of numerical closure procedures, Journal of Chemical Physics 137:204115, 2012.
- G. Samaey, T. Lelièvre and V. Legat, A numerical closure approach for kinetic models of polymeric fluids: exploring closure relations for FENE dumbbells, Computers and Fluids, 43:119-133, 2011.