Parallelization Strategies for Density Matrix Renormalization Group Algorithms on Shared-Memory Systems

Shared-memory parallelization (SMP) strategies for density matrix renormalization group (DMRG) algorithms enable the treatment of complex systems in solid state physics. We present two different approaches by which parallelization of the standard DMRG algorithm can be accomplished in an efficient way. The methods are illustrated with DMRG calculations of the two-dimensional Hubbard model and the one-dimensional Holstein-Hubbard model on contemporary SMP architectures. The parallelized code shows good scalability up to at least eight processors and allows us to solve problems which exceed the capability of sequential DMRG calculations.Comment: 18 pages, 9 figure

OpenACC Based GPU Parallelization of Plane Sweep Algorithm for Geometric Intersection

Line segment intersection is one of the elementary operations in computational geometry. Complex problems in Geographic Information Systems (GIS) like finding map overlays or spatial joins using polygonal data require solving segment intersections. Plane sweep paradigm is used for finding geometric intersection in an efficient manner. However, it is difficult to parallelize due to its in-order processing of spatial events. We present a new fine-grained parallel algorithm for geometric intersection and its CPU and GPU implementation using OpenMP and OpenACC. To the best of our knowledge, this is the first work demonstrating an effective parallelization of plane sweep on GPUs. We chose compiler directive based approach for implementation because of its simplicity to parallelize sequential code. Using Nvidia Tesla P100 GPU, our implementation achieves around 40X speedup for line segment intersection problem on 40K and 80K data sets compared to sequential CGAL library

Towards an Achievable Performance for the Loop Nests

Numerous code optimization techniques, including loop nest optimizations, have been developed over the last four decades. Loop optimization techniques transform loop nests to improve the performance of the code on a target architecture, including exposing parallelism. Finding and evaluating an optimal, semantic-preserving sequence of transformations is a complex problem. The sequence is guided using heuristics and/or analytical models and there is no way of knowing how close it gets to optimal performance or if there is any headroom for improvement. This paper makes two contributions. First, it uses a comparative analysis of loop optimizations/transformations across multiple compilers to determine how much headroom may exist for each compiler. And second, it presents an approach to characterize the loop nests based on their hardware performance counter values and a Machine Learning approach that predicts which compiler will generate the fastest code for a loop nest. The prediction is made for both auto-vectorized, serial compilation and for auto-parallelization. The results show that the headroom for state-of-the-art compilers ranges from 1.10x to 1.42x for the serial code and from 1.30x to 1.71x for the auto-parallelized code. These results are based on the Machine Learning predictions.Comment: Accepted at the 31st International Workshop on Languages and Compilers for Parallel Computing (LCPC 2018

Tackling Exascale Software Challenges in Molecular Dynamics Simulations with GROMACS

GROMACS is a widely used package for biomolecular simulation, and over the last two decades it has evolved from small-scale efficiency to advanced heterogeneous acceleration and multi-level parallelism targeting some of the largest supercomputers in the world. Here, we describe some of the ways we have been able to realize this through the use of parallelization on all levels, combined with a constant focus on absolute performance. Release 4.6 of GROMACS uses SIMD acceleration on a wide range of architectures, GPU offloading acceleration, and both OpenMP and MPI parallelism within and between nodes, respectively. The recent work on acceleration made it necessary to revisit the fundamental algorithms of molecular simulation, including the concept of neighborsearching, and we discuss the present and future challenges we see for exascale simulation - in particular a very fine-grained task parallelism. We also discuss the software management, code peer review and continuous integration testing required for a project of this complexity.Comment: EASC 2014 conference proceedin

Parallelization of Modular Algorithms

In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gr\"obner bases (resp. standard bases) and the modular computation of the associated primes of a zero-dimensional ideal and describe their parallel implementation in SINGULAR. Our modular algorithms to solve problems over Q mainly consist of three parts, solving the problem modulo p for several primes p, lifting the result to Q by applying Chinese remainder resp. rational reconstruction, and a part of verification. Arnold proved using the Hilbert function that the verification part in the modular algorithm to compute Gr\"obner bases can be simplified for homogeneous ideals (cf. \cite{A03}). The idea of the proof could easily be adapted to the local case, i.e. for local orderings and not necessarily homogeneous ideals, using the Hilbert-Samuel function (cf. \cite{Pf07}). In this paper we prove the corresponding theorem for non-homogeneous ideals in case of a global ordering.Comment: 16 page

BioEM: GPU-accelerated computing of Bayesian inference of electron microscopy images

In cryo-electron microscopy (EM), molecular structures are determined from large numbers of projection images of individual particles. To harness the full power of this single-molecule information, we use the Bayesian inference of EM (BioEM) formalism. By ranking structural models using posterior probabilities calculated for individual images, BioEM in principle addresses the challenge of working with highly dynamic or heterogeneous systems not easily handled in traditional EM reconstruction. However, the calculation of these posteriors for large numbers of particles and models is computationally demanding. Here we present highly parallelized, GPU-accelerated computer software that performs this task efficiently. Our flexible formulation employs CUDA, OpenMP, and MPI parallelization combined with both CPU and GPU computing. The resulting BioEM software scales nearly ideally both on pure CPU and on CPU+GPU architectures, thus enabling Bayesian analysis of tens of thousands of images in a reasonable time. The general mathematical framework and robust algorithms are not limited to cryo-electron microscopy but can be generalized for electron tomography and other imaging experiments