Parallel resolution of the 3D Helmholtz equation based on multi‐graphics processing unit clusters

Abstract

SummaryThe resolution of the 3D Helmholtz equation is required in the development of models related to a wide range of scientific and technological applications. For solving this equation in complex arithmetic, the biconjugate gradient (BCG) method is one of the most relevant solvers. However, this iterative method has a high computational cost because of the large sparse matrix and the vector operations involved. In this paper, a specific BCG method, adapted for the regularities of the Helmholtz equation is presented. This BCG is based on the implementation of a novel format (named ‘Regular Format’) that allows the storage of the large sparse matrix involved in the sparse matrix vector product in a compact form. The contribution of this work is twofold: (1) decreasing the memory requirements of the 3D Helmholtz equation using the ‘Regular Format’ and (2) speeding up the resolution of the equation using high performance computing resources. A hybrid Message Passing Interface (MPI)‐graphics processing unit CUDA GPU parallelization that is capable of solving complex problems in short time has carried out (Fast‐Helmholtz). Fast‐Helmholtz combines optimizations at Message Passing Interface and GPU levels to reduce communications costs and to improve the exploitation of GPU architecture. This strategy makes it possible to extend the dimension of the Helmholtz problem to be solved, thanks to the relevant reduction of memory requirements and runtime. Copyright © 2014 John Wiley & Sons, Ltd.

High‐performance computing: to boldly go where no human has gone before

“…The work by Ortega et al [146] presents a novel and compact form to storage of large sparse matrices coupled with the adoption of a hybrid MPI-GPU parallelization. This combined approach allows systems to extend the dimension of the Helmholtz problem they are able to solve.…”

High performance computing for a 3-D optical diffraction tomographic application in fluid velocimetry

“…Let us note that a similar matrix will be obtained with a Finite Difference Method (FDM) approach with a second order approximation and the same regular mesh. The matrix M is very large and its size depends on the number of spatial discretization points or voxels into the volume (Vol) [14]. To avoid instability effects the voxels should be located such that the nearest neighbors are not further away than one tenth of the wavelength [19].…”

“…Most of the modern supercomputers consist of clusters of multi-core nodes which include accelerator devices such as GPUs [12]. Earlier works have shown the parallel computation capability of GPUs in performing ODT models [13,14]. In this paper, we will focus our attention on the combination of MATLAB with GPU devices in order to offload part of the computation and improving the global performance.…”

Non-linear Iterative Optimization Method for Locating Particles Using HPC Techniques

“…Thus, when the discretization process is based on FEM and a spatial regular 3D mesh, the linear system of equations resulting from Equation 1 is described by a matrix with only seven non-zero diagonals. Therefore, FEM transforms the 3D Helmholtz equation into the linear system (Ax = b), where the independent term, b, depends on the illumination field (E r ), the unknown vector, x, identifies the scattered field and the matrix A, related to the refractive index (n(r)), is sparse and exhibits a strong regularity in both the pattern and the values of its non-zero elements and has a very large size which depends on the number of spatial discretization points or voxels into the volume (V ol) [17]. It means that the Forward solver actually consists on the resolution of a large size linear system of equation composed by complex number.…”