An E cient Graph Coloring Algorithm for Stencil-based Jacobian Computations

Numerical methods for the solution of partial di↵erential equations constitute an important class of techniques in scientific computing. Often, the discretization is based on approximating the partial derivatives by finite di↵erences on a regular Cartesian grid. The resulting computations are structured in the sense of updating a large, multidimensional array by a stencil operation. A stencil defines the update of the value at a grid point based on values at neighboring grid points. We consider the problem of computing the Jacobian matrix of some functions that is given in the form of a computer program involving stencil operations on a regular Cartesian grid. Due to the stencil operations this Jacobian matrix is sparse. The exploitation of the stencil-type to compute this sparse Jacobian matrix using automatic di↵erentiation with minimal computational e↵ort can be modeled as a graph coloring problem [1]. By definition, exact solutions of this combinatorial optimization problem use the minimal number of colors. Exact solutions in terms of explicit formuae are known for various stencil types [2, 4, 5]. However, a formula for the exact solution is not readily available for an arbitrary stencil type. So, by ignoring any structure implied by a given stencil, it is not uncommon to use coloring heuristics for general graphs. These heuristics try to approximate the exact solution and will, most likely, not attain the minimal number of colors. Recently, Lülfesmann and Kawarabayashi [3] introduced a graph coloring algorithm for an arbitrary stencil on a regular multidimensional Cartesian grid that computes the exact solution. This algorithm eliminates the need for deriving an explicit formula for the exact solution. It is based on a divide-and-conquer approach that establishes a hierarchy of vertex separators that recursively decomposes the grid into smaller and smaller subgrids. The main advantage of this algorithm is that it always computes a coloring with the minimal number of colors. However, the disadvantage of this algorithm is its high computational complexity. In fact, there are problem instances reported in [3] where, compared to the running time of a traditional graph coloring heuristic, the running time of this algorithm is larger than a factor of more than 800. So, there is urgent need to look for alternative ways to compute exact solutions of this structured graph coloring problem while reducing the resulting running time. In this extended abstract we propose a novel graph coloring algorithm for stencil-based Jacobian computations on a regular …