Abstract
For many systems of linear equations that arise from the discretization of partial differential equations, the construction of an efficient multigrid solver is challenging. Here we present EvoStencils, a novel approach for optimizing geometric multigrid methods with grammar-guided genetic programming, a stochastic program optimization technique inspired by the principle of natural evolution. A multigrid solver is represented as a tree of mathematical expressions that we generate based on a formal grammar. The quality of each solver is evaluated in terms of convergence and compute performance by automatically generating an optimized implementation using code generation that is then executed on the target platform to measure all relevant performance metrics. Based on this, a multi-objective optimization is performed using a non-dominated sorting-based selection. To evaluate a large number of solvers in parallel, they are distributed to multiple compute nodes. We demonstrate the effectiveness of our implementation by constructing geometric multigrid solvers that are able to outperform hand-crafted methods for Poisson’s equation and a linear elastic boundary value problem with up to 16 million unknowns on multi-core processors with Ivy Bridge and Broadwell microarchitecture.
Highlights
Solving the linear or nonlinear systems that arise from the discretization of partial differential equations efficiently is an unprecedented challenge
We have been able to overcome this limitation through a distributed code generation-based solver evaluation and, could demonstrate the construction of multigrid solvers that are able to outperform efficient reference methods both in the previously considered linear elastic boundary value problem, as well as a two-dimensional Poisson problem
While we could not achieve the same degree of efficiency for a three-dimensional Poisson problem, the constructed solvers still represent functioning and efficient multigrid methods
Summary
Solving the linear or nonlinear systems that arise from the discretization of partial differential equations efficiently is an unprecedented challenge. Geometric multigrid methods are a class of asymptotically optimal multilevel solution algorithms for (non-)linear systems, which were first formulated by Fedorenko in 1961 [12] and have been later pioneered by Brandt [4] and Hackbusch [16]. These methods are based on accelerating the convergence of stationary iterative methods by applying corrections obtained on a lower resolution of the original problem. Since the invention of multigrid, significant effort has been put into the design of efficient solvers for many important cases, such as Helmholtz [11] and saddle point problems [2], this task is still an open challenge
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
