An Invariant Preserving Sparse Spectral Discretization of the Continuum Equation

Parker Ewen, Henry O. Jacobs, Ram Vasudevan

Abstract: Uncertainty propagation is a critical component in various applications such as stochastic optimal control, optimal transport, probabilistic inference, and filtering. This paper frames uncertainty propagation of a dynamical system whose domain is the torus through the lens of advecting a probability density through a vector field via the continuum equation. This equation exhibits two conservation laws of interest: positivity and mass conservation. This article constructs a novel sparse spectral discretization technique that conserves these invariants while simultaneously qualitatively improving spectral convergence rates. Specifically, densities are transformed into their corresponding half-densities and it is shown that the advection of these half-densities corresponds to a unitary evolution. A hyperbolic cross approximation is implemented to provide sub-exponential complexity with respect to system dimension. The performance of the proposed method is illustrated in several numerical experiments and outperforms a standard spectral scheme. The convergence rate and error bounds of the proposed approach are likewise derived.