Probabilistic equivalence and stochastic model reduction in multiscale analysis

This paper presents a probabilistic upscaling of mechanics models. A reduced-order probabilistic model is constructed as a coarse-scale representation of a specified fine-scale model whose probabilistic structure can be accurately determined. Equivalence of the fine- and coarse-scale representations is identified such that a reduction in the requisite degrees of freedom can be achieved while accuracy in certain quantities of interest is maintained. A significant stochastic model reduction can a priori be expected if a separation of spatial and temporal scales exists between the fine- and coarse-scale representations. The upscaling of probabilistic models is subsequently formulated as an optimization problem suitable for practical computations. An illustration in stochastic structural dynamics is provided to demonstrate the proposed framework.