A performance spectrum for parallel computational frameworks that solve PDEs

Abstract

Time to solution I n t e n s i t y R a t eO p e r a t i o n s p e r b y t e t r a n s f e r r e d Proposed performance spectrum that documents time, intensity and rate. Abstract. Important computational physics problems are often large-scale in nature, and it is highly desirable to have robust and high performing computational frameworks that can quickly address these problems. However, it is no trivial task to determine whether a computational framework is performing efficiently or is scalable. The aim of this paper is to present various strategies for better understanding the performance of any parallel computational frameworks for solving PDEs. Important performance issues that negatively impact time-to-solution are discussed, and we propose a performance spectrum analysis that can enhance one's understanding of critical aforementioned performance issues. As proof of concept, we examine commonly used finite element simulation packages and software and apply the performance spectrum to quickly analyze the performance and scalability across various hardware platforms, software implementations, and numerical discretizations. It is shown that the proposed performance spectrum is a versatile performance model that is not only extendable to more complex PDEs such as hydrostatic ice sheet flow equations, but also useful for understanding hardware performance in a massively parallel computing environment. Potential applications and future extensions of this work are also discussed.

Comparative Study of Finite Element Methods Using the Time-Accuracy-Size(TAS) Spectrum Analysis

“…• Arithmetic intensity: A logical extension of the TAS spectrum performance analysis would be to incorporate the Arithmetic Intensity (AI), used in the performance spectrum [16] and roofline performance model [40]. The AI of an algorithm or software is a measure that aids in estimating how efficiently the hardware resources and capabilities can be utilized.…”

“…By taking into account the total time-to-solution, numerical accuracy with respect to an error norm, and the computation rate, a cost-benefit analysis can be performed to determine which algorithm and discretization are particularly suited for an application. This work extends the performance spectrum model in [16] for interpretation of hardware and algorithmic tradeoffs in numerical PDE simulation. As a proof-of-concept, popular finite element software packages are used to illustrate this analysis for Poisson's equation.…”

A Segregated Approach for Modeling the Electrochemistry in the 3-D Microstructure of Li-Ion Batteries and Its Acceleration Using Block Preconditioners

“…In the final region, the problem size is a perfect match for the number of processes, which minimizes solve time and results in optimal scaling. More about static scaling can be found in [6,9,26,31]. Strong scaling plot generated using the 3.52M DOF mesh for the electrode domain and the 3.14M DOF mesh for the electrolyte domain and 16 processes per node.…”

“…Unlike in the solid domains, the electrolyte problem contains coupling on more than just the interface boundary. This could lead to a degradation in parallel performance because sparse matrix-vector multiplications have very low arithmetic intensities and can present as serial bottlenecks at the memory levels of each server node (see [9] for further discussion). Fig.…”

Evaluation of PETSc on a Heterogeneous Architecture, the OLCF Summit System: Part 1: Vector Node Performance

“…Note the use of the log scale and that, toward the right, the GPUs are performing significantly better than the 42 CPU cores, while towards the left the CPUs are faster. Figure 3 presents an alternative view of the same data, known as a static scaling or work-time spectrum plot [4,5]. It has the advantage that both the asymptotic bandwidth and the latency of the operations can be directly read from the figure.…”