In Eikonal equations, rarefaction is a common phenomenon known to degrade the rate of convergence of numerical methods. The "factoring" approach alleviates this difficulty by deriving a PDE for a new (locally smooth) variable while capturing the rarefaction-related singularity in a known (non-smooth) "factor". Previously this technique was successfully used to address rarefaction fans arising at point sources. In this paper we show how similar ideas can be used to factor the 2D rarefactions arising due to nonsmoothness of domain boundaries or discontinuities in PDE coefficients. Locations and orientations of such rarefaction fans are not known in advance and we construct a "just-in-time factoring" method that identifies them dynamically. The resulting algorithm is a generalization of the Fast Marching Method originally introduced for the regular (unfactored) Eikonal equations. We show that our approach restores the first-order convergence and illustrate it using a range of maze navigation examples with non-permeable and "slowly permeable" obstacles.

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Classical deterministic optimal control problems assume full information about the controlled process. The theory of control for general partially-observable processes is powerful, but the methods are computationally expensive and typically address the problems with stochastic dynamics and continuous (directly unobserved) stochastic perturbations. In this paper we focus on path planning problems which are in between -deterministic, but with an initial uncertainty on either the target or the running cost on parts of the domain. That uncertainty is later removed at some time T , and the goal is to choose the optimal trajectory until then. We address this challenge for three different models of information acquisition: with fixed T , discretely distributed and exponentially distributed random T . We develop models and numerical methods suitable for multiple notions of optimality: based on the average-case performance, the worst-case performance, the average constrained by the worst, the average performance with probabilistic constraints on the bad outcomes, risk-sensitivity, and distributional-robustness. We illustrate our approach using examples of pursuing random targets identified at a (possibly random) later time T .