A Multi-GPU, Multi-CPU Implementation of RBF-FD for PDE Solutions

Abstract

During the last few decades, numerical methods that use collections of radially symmetric, univariate func- tions for approximation have surfaced for PDE solutions. Such functions are referred to as Radial Basis Functions (RBFs) and have been successfully employed in spectral, pseudo-spectral and localized modes. RBF methods are ideal for unstructured or scattered nodes; e.g., node sets generated with centroidal Voronoi tessellation.

One of the newest areas of research leverages RBFs in the calculation of weights for generalized finite difference stencils in a scheme referred to as RBF-FD. RBF-FD ad- dresses a major concern with other RBF methods—namely, ill-conditioning.

The RBF-FD method exhibits a large amount of parallelism, which we target in a multi-CPU, multi-GPU framework for solving 2D and 3D PDEs. To span multiple CPUs and multiple GPUs, an Additive Schwarz domain decomposi- tion method is used to partition the physical domain and distribute work across processors. Each processor then works with a GPU to offload computationally demanding tasks. Since individual RBF-FD stencils may span multiple subdomains, inter-processor communication is required at each iteration.

Details of our implementation, along with case studies solv- ing elliptic and parabolic PDEs using implicit and explicit schemes, are provided. Our end goal is to use this code for Tsunami and other geophysical simulations.