Corner Cases, Singularities, and Dynamic Factoring

Abstract

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.

Computing the quasipotential for nongradient SDEs in 3D

“…We give an explanation of this phenomenon on a 2D model in Section 4.5. Effects of local factoring near asymptotically stable equilibria [29] adjusted for OLIMs are studied in Section 4.2. It is found that local factoring reduces computational errors for linear SDEs but may or may not be beneficial for nonlinear ones.…”

“…An additive factoring serving the same purpose was proposed in [20]. The idea of factoring was adapted for the fast marching method (FMM) in [29]. Furthermore, the fact that the FMM propagates the solution throughout the domain from smaller values to larger values without iteration allows for factoring locally; i.e., the eikonal equation only needs to be factored around point sources and rarefaction fans.…”

“…where The effect of local factoring in the case of the quasipotential solver is less dramatic than that in the case of the eikonal equation [29,27]. Numerical errors committed by the OLIMs in computing the quasipotential near asymptotically stable equilibria are mainly due to (i) the approximation of U in the triangle and simplex bases using linear interpolation, and (ii) the use of line segments to approximate segments of the MAPs.…”

Riemannian Fast-Marching on Cartesian Grids, Using Voronoi's First Reduction of Quadratic Forms

“…This produces a non-differentiable singularity with dominant term q → p * − q M(p * ) at the source point p * . Similar effects are encountered at the corners of obstacles [59]. Factoring techniques, additive or multiplicative, incorporate corrective terms in the numerical scheme based on the analytic expression of the singularity [46].…”

Ordered Line Integral Methods for Solving the Eikonal Equation

“…The logarithmic factor only appears when rarefaction fans are present: e.g., point source boundary data, or if the wavefront diffracts around a singular corner or edge. In these cases, full O(h) accuracy can be recovered by proper initialization near rarefaction fans, or by employing a variety of factoring schemes [16,23,30].…”