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 with compact support thru the lens of advecting a probability density through a vector field via the transport and continuum equations. These equations exhibit a number of conservation laws. For example, scalar multiplication is conserved by the transport equation, while positivity of probabilities is conserved by the continuum equation. Certain discretization techniques, such as particle-based methods, conserve these properties but converge slower than spectral discretization methods on smooth data. Standard spectral discretization methods, on the other hand, do not conserve the invariants of the transport equation and the continuum equation. This article constructs a novel sparse spectral discretization technique that conserves these important invariants, namely the positivity of probability measure, while simultaneously preserving spectral convergence rates. The performance of this proposed method is illustrated on several numerical experiments and the quantitative convergence rate of the proposed approach is derived.
Method #
Uncertainty propagation involves advecting an initial probability density along the flow of a dynamical system via the continuum equation. Ideally, a probability advection scheme should preserve the \(L^1\)-norm, ensure positivity, and offer spectral convergence with sub-exponential complexity with respect to system dimension.
This paper demonstrates that by considering the propagation of half-densities, the proposed advection scheme preserves \(L^1\)-norm and positivit. Furthermore, a sparse hyperbolic cross discretization is introduced to ensure sub-exponential complexity with respect to system dimension with minimal degradation in regards to approximation accuracy.
We note the proposed approach outperforms a classical spectral discretization scheme when ground truth propagated uncertainty is known. The proposed sparse spectral method requires fewer basis functions to achieve better performance and maintains accuracy for longer propagation times.
Simulation Results #
We validate our approach in simulation across several numerical experiments. The presented animations present the results for a 2D and 3D modified ABC flow, where the 3D experiment is marginalized along the third dimension. For a comparison against a Monte Carlo particle baseline, see the paper. We also demonstrate the proposed approach scales well with respect to system dimension. A 6D Kuramoto model is used and the proposed approach shows similar results to the Monte Carlo baseline. We note that in all experiments, the proposed sparse spectral approach requires orders of magnitude less memory compared to a standard spectral scheme.
Error Analysis #
We derive convergence rates for the proposed sparse spectral scheme. Upper bounds on the \(L^2\) error norm for the propogated half-densities are provided, and these translate directly into bounds on the \(L^1\) error norm of the propagated full densities. Lastly, an upper bound on the propagated probability measure is provided. All bounds are shown to improve as the number of sparse basis functions increases.
Citation #
This project was developed in the Robotics and Optimization for Analysis of Human Motion (ROAHM) Lab at the University of Michigan - Ann Arbor.
@article{ewen2024uncertainty,
author = "Ewen, Parker and Lymburner, Lucas and Jacobs, Henry and Vasudevan, Ram",
title = "An Invariant Preserving Sparse Spectral Discretization of the Continuum Equation",
journal = "In Submission",
year = 2024
}