Dynamically orthogonal field equations for continuous stochastic dynamical systems

By Themistoklis P. Sapsis, Pierre FJ Lermusiaux. 2009

290 citations recorded by the source corpus.

Abstract In this work we derive an exact, closed set of evolution equations for general continuous stochastic fields described by a Stochastic Partial Differential Equation (SPDE). By hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component, we derive a system of field equations consisting of a Partial Differential Equation (PDE) for the mean field, a family of PDEs for the orthonormal basis that describe the stochastic subspace where the stochasticity ‘lives’ as well as a system of Stochastic Differential Equations that defines how the stochasticity evolves in the time varying stochastic subspace. These new evolution equations are derived directly from the original SPDE, using nothing more than a dynamically orthogonal condition on the representation of the solution. If additional restrictions are assumed on the form of the representation, we recover both the Proper Orthogonal Decomposition equations and the generalized Polynomial Chaos equations. We apply this novel methodology to two cases of two-dimensional viscous fluid flows described by the Navier–Stokes equations and we compare our results with Monte Carlo simulations.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f762cc28cb17ddeab009cf580ba8f578e35c14ee

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks (2019, methodology): “For example, this can be achieved by imposing dynamical constraints on the spatial basis, which is the so-called “dynamically orthogonal" (DO) methodology first proposed in [15, 16].”
Time-optimal path planning in dynamic flows using level set equations: theory and schemes (2014, background): “Planning paths in predicted probabilistic flows (Sapsis and Lermusiaux 2009; Ueckermann et al. 2013) are reported by Lermusiaux et al.”