T. Forrest Kieffer | Searcharxiv

arXiv2025

Robust series linearization of nonlinear advection-diffusion equations

We consider nonlinear partial differential equations (PDEs) for advection-diffusion processes which are augmented by an auxiliary parameter $δ$ such that $δ=0$ corresponds to linear advection-diffusion. We derive potentially non-perturbative series expansions in $δ$ that provide a process to obtain the solution of the nonlinear PDE through solving a hierarchical system of linear, forced PDEs with the forcing terms dependent on solutions at lower orders in the hierarchy. We rigorously detail our approach for a particular deformation that interpolates between linear advection-diffusion and the canonical Burgers' equation modeling nonlinear advection. In this case, we prove that the series has infinite radius of convergence for arbitrary integrable initial data, analyze the cases of a Dirac-delta initial condition (IC) (i.e., the fundamental solution) in an infinite domain and arbitrary IC in a periodic domain, and demonstrate the approach to turbulent behavior in a scenario with periodic forcing. We then treat models of nonlinear diffusion involving the $p$-Laplacian operator, including generalizations of the Poisson equation in $1$ and $2$ dimensions, and the heat equation in $1+1$ dimensions. We detail series expansions for two different deformations of these equations about their linear (ordinary Laplacian) counterparts, providing numerical evidence for the convergence of the series outside of a perturbative regime and demonstrating that the rate and radius of convergence are affected by choice of deformation. Our results provide a rigorous foundation for using series expansion techniques to study nonlinear advection-diffusion PDEs, opening new pathways for analysis and potential applications for quantum-assisted computational fluid dynamics.

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arXiv2025

Dynamics on Lie groups with applications to attitude estimation

The problem of filtering - propagation of states through stochastic differential equations (SDEs) and association of measurement data using Bayesian inference - in a state space which forms a Lie group is considered. Particular emphasis is given to concentrated Gaussians (CGs) as a parametric family of probability distributions to capture the uncertainty associated with an estimated state. The so-called group-affine property of the state evolution is shown to be necessary and sufficient for linearity of the dynamics on the associated Lie algebra, in turn implying CGs are invariant under such evolution. A putative SDE on the group is then reformulated as an SDE on the associated Lie algebra. The vector space structure of the Lie algebra together with the notion of a CG enables the leveraging of techniques from conventional Gaussian-based Kalman filtering in an approach called the tangent space filter (TSF). We provide example calculations for several Lie groups that arise in the problem of estimating position, velocity, and orientation of a rigid body from a noisy, potentially biased inertial measurement unit (IMU). For the specific problem of attitude estimation, numerical experiments demonstrate that TSF-based approaches are more accurate and robust than another widely used attitude filtering technique.

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