Matthias Messner

Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation

Many applications lead to large systems of linear equations with dense matrices. Direct matrix-vector products become prohibitive, since the computational cost increases quadratically with the size of the problem. By exploiting specific kernel properties fast algorithms can be constructed. A directional multilevel algorithm for translation-invariant oscillatory kernels of the type K(x,y)=G(x-y)e^i^k^|^x^-^y^|, with G(x-y) being any smooth kernel, will be presented. We will first present a general approach to build fast multipole methods (FMMs) based on Chebyshev interpolation and the adaptive cross approximation (ACA) for smooth kernels. The Chebyshev interpolation is used to transfer information up and down the levels of the FMM. The scheme is further accelerated by compressing the information stored at Chebyshev interpolation points using ACA and QR decompositions. This leads to a nearly optimal computational cost with a small pre-processing time due to the low computational cost of ACA. This approach is in particular faster than performing singular value decompositions. This does not address the difficulties associated with the oscillatory nature of K. For that purpose, we consider the following modification of the kernel K^u=K(x,y)e^-^i^k^u^.^(^x^-^y^), where u is a unit vector (see Brandt [1]). We proved that the kernel K^u can be interpolated efficiently when x-y lies in a cone of direction u. This result is used to construct an FMM for the kernel K. Theoretical error bounds will be presented to control the error in the computation as well as the computational cost of the method. The paper ends with the presentation of 2D and 3D numerical convergence studies, and computational cost benchmarks.

Task-Based FMM for Multicore Architectures

Fast multipole methods (FMM) are a fundamental operation for the simulation of many physical problems. The high-performance design of such methods usually requires to carefully tune the algorithm for both the targeted physics and the hardware. In this paper, we propose a new approach that achieves high performance across architectures. Our method consists of expressing the FMM algorithm as a task flow and employing a state-of-the-art runtime system, StarPU, to process the tasks on the different computing units. We carefully design the task flow, the mathematical operators, their implementations, and scheduling schemes. Potentials and forces on 200 million particles are computed in 42.3 seconds on a homogeneous 160-core SGI Altix UV 100 and good scalability is shown.

Task-based FMM for heterogeneous architectures

High performance fast multipole method is crucial for the numerical simulation of many physical problems. In a previous study, we have shown that task‐based fast multipole method provides the flexibility required to process a wide spectrum of particle distributions efficiently on multicore architectures. In this paper, we now show how such an approach can be extended to fully exploit heterogeneous platforms. For that, we design highly tuned graphics processing unit (GPU) versions of the two dominant operators P2P and M2L) as well as a scheduling strategy that dynamically decides which proportion of subsequent tasks is processed on regular CPU cores and on GPU accelerators. We assess our method with the StarPU runtime system for executing the resulting task flow on an Intel X5650 Nehalem multicore processor possibly enhanced with one, two, or three Nvidia Fermi M2070 or M2090 GPUs (Santa Clara, CA, USA). A detailed experimental study on two 30 million particle distributions (a cube and an ellipsoid) shows that the resulting software consistently achieves high performance across architectures. Copyright

Poster: Matrices over Runtime Systems at Exascale

The goal of Matrices Over Runtime Systems at Exascale (MORSE) project is to design dense and sparse linear algebra methods that achieve the fastest possible time to an accurate solution on large-scale multicore systems with GPU accelerators, using all the processing power that future high end systems can make available. In this poster, we propose a framework for describing linear algebra algorithms at a high level of abstraction and delegating the actual execution to a runtime system in order to design software whose performance is portable accross architectures. We illustrate our methodology on three classes of problems: dense linear algebra, sparse direct methods and fast multipole methods. The resulting codes have been incorporated into Magma, Pastix and ScalFMM solvers, respectively.

Abstract: Matrices Over Runtime Systems at Exascale

The goal of Matrices Over Runtime Systems at Exascale (MORSE) project is to design dense and sparse linear algebra methods that achieve the fastest possible time to an accurate solution on large-scale multicore systems with GPU accelerators, using all the processing power that future high end systems can make available. In this poster, we propose a framework for describing linear algebra algorithms at a high level of abstraction and delegating the actual execution to a runtime system in order to design software whose performance is portable accross architectures. We illustrate our methodology on three classes of problems: dense linear algebra, sparse direct methods and fast multipole methods. The resulting codes have been incorporated into Magma, Pastix and ScalFMM solvers, respectively.