Communication-avoiding Algorithms Research Articles

Parallel Fast Multipole Method accelerated FFT on HPC clusters

With increasing sizes of distributed systems, there comes an increased risk of communication bottlenecks. In the past decade there has been a growing interest in communication-avoiding algorithms. The distributed memory Fast Fourier Transform is an important algorithm which suffers from major communication bottlenecks. In this work, we take a look at an existing communication-avoiding algorithm FMM-FFT, an alternative to FFT which utilizes the Fast Multipole Method (FMM) to reduce communications to a single all-to-all communication. We present a detailed implementation of FMM-FFT relying on modern libraries and demonstrate it on two distinct distributed memory architectures notably a traditional Intel Xeon based HPC cluster and then a Beowulf cluster. We show that while the FMM-FFT is significantly slower than FFT on the traditional HPC cluster, on the Beowulf cluster it outperforms standard FFT, consistently getting speedups of 1.5x or more against FFTW. We then proceed to show how the communication to computation cost metric is important and useful in explaining the performance results of FMM-FFT against standard FFT. The source code pertaining to this work is being made publicly available under a permissive open source licence at Github.

Time-stepping DPG formulations for the heat equation

For a wide range of PDEs, the discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan provides discrete stability starting from a coarse mesh and minimization of the residual in a user-controlled norm, among other appealing features. Research on DPG for transient problems has mainly focused on spacetime discretizations, which has theoretical advantages, but practical costs for computations and software implementations. The sole examination of time-stepping DPG formulations was performed by Führer, Heuer, and Gupta, who applied Rothe’s method to an ultraweak formulation of the heat equation to develop an implicit time-stepping scheme; their work emphasized theoretical results, including error estimates in time and space.In the present work, we follow Führer, Heuer, and Gupta in examining the heat equation; our focus is on numerical experiments, examining the stability and accuracy of several formulations, including primal as well as ultraweak, and explicit as well as implicit and Crank–Nicolson time-stepping schemes. We are additionally interested in communication-avoiding algorithms, and we therefore include a highly experimental formulation that places all the trace terms on the right-hand side of the equation.

Varying the s in your s-step GMRES

Krylov subspace methods are commonly used iterative methods for solving large sparse linear systems, however they suffer from communication bottlenecks on parallel computers. Therefore, $s$-step methods have been developed where the Krylov subspace is built block by block, so that $s$ matrix-vector multiplications can be done before orthonormalizing the block. Then Communication-Avoiding algorithms can be used for both kernels. This paper introduces a new variation on $s$-step GMRES in order to reduce the number of iterations necessary to ensure convergence, with a small overhead in the number of communications. Namely, we develop a $s$-step GMRES algorithm, where the block size is variable and increases gradually. Our numerical experiments show a good agreement with our analysis of condition numbers and demonstrate the efficiency of our variable $s$-step approach.

A Class of Communication-avoiding Algorithms for Solving General Dense Linear Systems on CPU/GPU Parallel Machines

We study several solvers for the solution of general linear systems where the main objective is to reduce the communication overhead due to pivoting. We first describe two existing algorithms for the LU factorization on hybrid CPU/GPU architectures. The first one is based on partial pivoting and the second uses a random preconditioning of the original matrix to avoid pivoting. Then we introduce a solver where the panel factorization is performed using a communication-avoiding pivoting heuristic while the update of the trailing submatrix is performed by the GPU. We provide performance comparisons and tests on accuracy for these solvers on current hybrid multicore-GPU parallel machines.