Dimensional Reduction of Direct Statistical Simulation
Brad Marston, Brown University
Direct Statistical Simulation (DSS) solves the equations of motion for the statistics of dynamical systems in place of the traditional route of accumulating statistics by Direct Numerical Simulation (DNS). I discuss two forms of DSS and illustrate their application to the Lorenz attractor. For higher dimensional systems, however, DSS is usually more expensive computationally than DNS because even low order statistics typically have higher dimension than the underlying fields (depending on the symmetry of the problem and the choice of averaging operation). That low-order statistics usually evolve slowly compared with instantaneous dynamics is one important advantage of DSS. I will show that it is possible to go further by using Proper Orthogonal Decomposition (POD) to address the “curse of dimensionality.” POD is applied to DSS in the form of expansions in the equal-time cumulants to second order (CE2). I discuss two averaging operations (zonal and ensemble) and test the approach on an idealized barotropic models on a rotating sphere (a stochastically-driven flow that spontaneously organizes into jets). Order-of-magnitude savings in computational cost are obtained in the reduced basis, enabling access to parameter regimes beyond the reach of DNS.