arXiv subjects

Richard Sowers

Publications and source records attributed to Richard Sowers.

3 recordsLinked to original sources

arXiv2026

Koopman Representations for Non-Vanishing Time Intervals: An Optimization Approach and Sampling Effects

Koopman operator theory is a key tool in data assimilation of complex dynamical systems, with the potential to be applied to multimodal data. We formulate the problem of learning Koopman eigenfunctions from observations at arbitrary, possibly non-vanishing, time intervals as an optimization problem. Analysis of the formulation reveals aliasing induced by oscillatory dynamics and the sampling pattern, making an inherent identifiability limit explicit. The analysis also uncovers phase alignment near the true Koopman frequency, which creates a steep loss valley and demands careful optimization. We further show that irregular sampling can break aliasing and lead to phase cancellation. Numerical results demonstrate the efficacy of the proposed method under large regular time intervals compared to generator extended dynamic mode decomposition, and support the idea that irregular sampling can help recover the true Koopman spectrum.

eess.SY↗

arXiv2025

On the Enumeration of all Unique Paths of Recombining Trinomial Trees

Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree naively yields $\mathcal{O}(3^{D})$ trajectories, making exhaustive enumeration infeasible. Under time-homogeneous dynamics, however, the graph exhibits two exploitable symmetries: (i) translational invariance of nodes and (ii) a canonical bijection between admissible paths and ordered tuples encoding weak compositions. Leveraging these, we introduce a mass-shifting enumeration algorithm that slides integer "masses" through a cardinality tuple to generate exactly one representative per path-equivalence class while implicitly counting the associated weak compositions. This trims the search space by an exponential factor, enabling markedly deeper trees -- and therefore tighter numerical approximations of the underlying evolution -- to be processed in practice. We further derive an upper bound on the combinatorial counting expression that induces a theoretical lower bound on the algorithmic cost of approximately $\mathcal{O}\bigl(D^{1/2}1.612^{D}\bigr)$. This correspondence permits direct benchmarking while empirical tests, whose pseudo-code we provide, corroborate the bound, showing only a small constant overhead and substantial speedups over classical breadth-first traversal. Finally, we highlight structural links between our algorithmic/combinatorial framework and Motzkin paths with Narayana-type refinements, suggesting refined enumerative formulas and new potential analytic tools for path-dependent functionals.

cs.DS↗

arXiv2024

Side Boundary potentials for a Kolmogorov-type PDE

We solve a Kolmogorov-type hypoelliptic parabolic partial differential equation with a "side" boundary condition (in the direction of the weak Hörmander condition). We construct an approximate boundary potential which captures the effect of the boundary condition. Integrals against this approximate boundary potential have a novel discontinuity at the boundary. We introduce some polynomial corrections to this approximate boundary potential and then construct a boundary-domain Volterra equation to solve the original partial differential equation. This Volterra integral equation is iteratively solved, and the bounds contain a periodic behavior resulting from the boundary effects.

math.AP↗