arXiv subjects

Jonathan Valyou

Publications and source records attributed to Jonathan Valyou.

2 recordsLinked to original sources

arXiv2026

Optimal mean-time path planning for unmanned underwater vehicles: a Hamilton-Jacobi approach

Unmanned underwater vehicles (UUV) integrate ocean forecasts with path planning algorithms in order to identify energy- or time-minimizing paths that enable mission completion. Typically, a well-defined deterministic ocean forecast is assumed to be available for path planning; however, in practice, different ocean forecasts can disagree. In this paper, we extend previous work on deterministic optimal path planning to identify optimal mean-time paths when presented with an ensemble of possible ocean forecasts. In particular, we formulate a system of time-independent Hamilton-Jacobi partial differential equations that incorporates forecast uncertainty and yields the optimal mean reachability travel time and the necessary controls to find the associated optimal path. An efficient numerical solution of this system of PDEs is obtained through an extension of the Fast Sweeping Method; verification and benchmarking results are provided. Additional numerical examples illustrate the impact uncertainty can have on the optimal path; in particular, these results demonstrate that the vehicle's optimal path can deviate significantly from the deterministic optimal paths associated with the individual ensemble members.

math.OC↗

arXiv2026

A Conjugate Gradient Formulation of the EnKF Algorithm

Ensemble Kalman Filter (EnKF) based data assimilation algorithms synthesize predictive numerical forecast models with accumulated data as time evolves and account for model uncertainty and noisy measurements. The computational cost of these algorithms can be expensive, in particular for highly dimensional dynamical systems. Often, EnKF based algorithms have traded accuracy for reduced computational cost. In this paper, we present a novel parallelizable Conjugate Gradient-based Ensemble Kalman Filter (CGD-EnKF) algorithm that maintains comparable computational cost to efficient algorithms while realizing better state estimation accuracy in select cases. Here, we established the new approach by reformulating a matrix inverse calculation with a classical Conjugate Gradient (CGD) method. In addition, we discuss the upper error bound under CGD, error convergence to the classical EnKF result, and the computational complexity of the algorithm. We also showcase the CGD-EnKF-Reduced algorithm that is shown to be further computationally efficient for highly dimensional dynamical systems under small ensemble formulation. Numerical examples demonstrate the performance of our proposed algorithms and analytical properties, highlighting their comparability and advantages with respect to some benchmark EnKF algorithms.

math.NA↗