Ka Lun Leung | Search arXiv

arXiv2026

A Structure-Preserving Scheme for Computing the Gradient of the Eikonal Equation

We propose a first-order, structure-preserving method for directly computing the normalized traveltime gradient of the eikonal equation in smooth first-arrival regions. The method combines analytical derivatives of spherical linear interpolation (SLERP) in a Godunov-type upwind discretization with spherical forward Euler (SFE) pseudo-time integration. This construction gives a tangent residual and keeps every iterate on $\mathbb{S}^2$ without projection or normalization. The SLERP discretization exactly preserves a family of curved orientation equilibria. Removing the background discretization bias improves absolute orientation accuracy near these equilibria, allowing a prescribed error tolerance to be reached on a coarser grid. We prove the equilibrium-preservation property and, separately, a smaller angular error for SFE than normalized Euler in a one-cell relaxation under the stated step-size condition. Two- and three-dimensional experiments examine convergence, geometric preservation, and computational cost. Comparisons with gradient recovery from scalar traveltimes show more accurate directions on the same grid in the curved media considered. The method provides direction fields for wave propagation and geometric optics, with potential use in adjoint-based inverse problems.

math.NA↗

arXiv2014

Consistency Techniques for Flow-Based Projection-Safe Global Cost Functions in Weighted Constraint Satisfaction

Many combinatorial problems deal with preferences and violations, the goal of which is to find solutions with the minimum cost. Weighted constraint satisfaction is a framework for modeling such problems, which consists of a set of cost functions to measure the degree of violation or preferences of different combinations of variable assignments. Typical solution methods for weighted constraint satisfaction problems (WCSPs) are based on branch-and-bound search, which are made practical through the use of powerful consistency techniques such as AC*, FDAC*, EDAC* to deduce hidden cost information and value pruning during search. These techniques, however, are designed to be efficient only on binary and ternary cost functions which are represented in table form. In tackling many real-life problems, high arity (or global) cost functions are required. We investigate efficient representation scheme and algorithms to bring the benefits of the consistency techniques to also high arity cost functions, which are often derived from hard global constraints from classical constraint satisfaction. The literature suggests some global cost functions can be represented as flow networks, and the minimum cost flow algorithm can be used to compute the minimum costs of such networks in polynomial time. We show that naive adoption of this flow-based algorithmic method for global cost functions can result in a stronger form of null-inverse consistency. We further show how the method can be modified to handle cost projections and extensions to maintain generalized versions of AC* and FDAC* for cost functions with more than two variables. Similar generalization for the stronger EDAC* is less straightforward. We reveal the oscillation problem when enforcing EDAC* on cost functions sharing more than one variable. To avoid oscillation, we propose a weak version of EDAC* and generalize it to weak EDGAC* for non-binary cost functions. Using various benchmarks involving the soft variants of hard global constraints ALLDIFFERENT, GCC, SAME, and REGULAR, empirical results demonstrate that our proposal gives improvements of up to an order of magnitude when compared with the traditional constraint optimization approach, both in terms of time and pruning.

cs.AI↗