Efficient Parallel Implementation Research Articles

Improved algorithm for generating evenly-spaced streamlines from an orientation field on a triangulated surface

Background:Vector fields such as cardiac fiber orientation can be visualized on a surface using streamlines. The application of evenly-spaced streamline generation to the construction of interconnected cable structure for cardiac propagation models has more stringent requirements imperfectly fulfilled by current algorithms. Method:We developed an open-source C++/python package for the placement of evenly-spaced streamlines on a triangulated surface. The new algorithm improves upon previous works by more accurately handling streamline extremities, U-turns and limit cycles, by providing stronger geometrical guarantees on inter-streamline minimal distance, particularly when a high density of streamlines (up to 10μm spacing) is desired, and by making a more efficient parallel implementation available. The approach requires finding intersections between geometrical capsules and triangles to update an occupancy mask defined on the triangles. This enables fast streamline integration from thousands of seed points to identify optimal streamline placement. Results:The algorithm was assessed qualitatively on different left atrial models of fiber orientation with varying mesh resolutions (up to 375k triangles) and quantitatively by measuring streamline lengths and distribution of inter-streamline minimal distance. The complexity and the computational performance of the algorithm were studied as a function of streamline spacing in relation to triangular mesh resolution. Conclusion:More accurate geometrical computations, attention to details and fine-tuning led to an algorithm more amenable to applications that require precise positioning of streamlines.

GPU Algorithms for Structured Sparse Matrix Multiplication with Diagonal Storage Schemes

Matrix–matrix multiplication is of singular importance in linear algebra operations with a multitude of applications in scientific and engineering computing. Data structures for storing matrix elements are designed to minimize overhead information as well as to optimize the operation count. In this study, we utilize the notion of the compact diagonal storage method (CDM), which builds upon the previously developed diagonal storage—an orientation-independent uniform scheme to store the nonzero elements of a range of matrices. This study exploits both these storage schemes and presents efficient GPU-accelerated parallel implementations of matrix multiplication when the input matrices are banded and/or structured sparse. We exploit the data layouts in the diagonal storage schemes to expose a substantial amount of fine-grained parallelism and effectively utilize the GPU shared memory to improve the locality of data access for numerical calculations. Results from an extensive set of numerical experiments with the aforementioned types of matrices demonstrate orders-of-magnitude speedups compared with the sequential performance.

A Computational Model for All-Solid-State Batteries Coupling Electrochemistry and Solid Mechanics on Resolved Microstructures Enabling Optimization of Battery Electrode Design

All-Solid-State Batteries (ASSB) are a promising technology to replace conventional Lithium-Ion Batteries (LIB) since the energy density and fast charging capability of LIBs are projected to reach a physical limit soon. In this regard, the ASSB technology is considered superior, especially if the incorporation of the lithium metal anode, or other high-capacity anode materials like silicon is achieved. However, these systems still require intensive research as many physical effects, especially those concerning the interaction of electrochemistry and solid mechanics, are to date not well understood [1]. Most experimental investigations on ASSB cells are performed on laboratory scale and are usually operated at elevated mechanical pre-stress to obtain satisfactory system performance. The delamination of active materials and solid electrolyte during cycling is expected to be one of the key effects for the required mechanical pre-stress. Furthermore, a lot of experimental research focuses on the development of new material classes and their stability in the cell compound, while a systematic optimization of the cell setup is missing.Comprehensive computational models can help to generate insights in these regards as they allow for detailed investigations of local effects, which is often not possible in experimental setups. Another advantage of computational models compared to experiments is the ability to change input parameters in a fast and systematic way to quantify their influence on the battery cell performance and thereby help to speed up the optimization of battery designs. Since the multitude of physical effects that are relevant in ASSB are computationally challenging, a lot of modeling approaches rely on numerous simplifications like spatial homogenization or neglect of mechanical effects to name just two. However, such assumptions can significantly limit the validity of the model.We propose a novel computational model for ASSB [2] to thoroughly investigate the interdependence of electrochemistry and solid mechanics on resolved microstructures that vastly influence the behavior and the performance of an ASSB cell. It is based on nonlinear continuum mechanics and accounts for large deformations due to lithiation dependent volume changes of the active materials. The mass and charge conservations are consistently formulated to also ensure the conservation property in large deformation scenarios. Furthermore, the model is extended by interaction effects of electrochemistry and solid mechanics like a nonlinear contact formulation at the electrode-electrolyte interfaces to account for delamination phenomena and the subsequent absence of charge transfer reaction (see Fig. 1). We thereby show how delamination phenomena drastically change available percolation paths in the microstructure. The efficient parallel implementation of the computational model allows to investigate realistic three-dimensionally resolved microstructures of ASSB cells while accounting for the coupling effects of electrochemistry and solid mechanics. A combination of elaborate physical models and probabilistic methods enables to gain further understanding and to quantify main influencing factors on the cell performance. To showcase these capabilities and how they can be exploited to increase system understanding, we analyze how statistical values like the porosity, or the composition of the electrode microstructure influence the capacity of an ASSB cell and thereby contribute to find optimal electrode designs.

Droplet splitting on chemically heterogeneous surface: A 3D lattice Boltzmann study

The uniform splitting of droplets on heterogeneous surfaces without external stimuli has gained increasing attention due to its wide and flexible applications. To precisely control and predict the droplet splitting, the 3D multiphase lattice Boltzmann model (LBM) is used to study the mechanism of the impact splitting on a hydrophilic surface with a superhydrophobic strip. A parallel implementation of the 3D multiphase LBM is presented to improve the performance by using an effective kernel fusion to reduce the off-chip memory data traffic. Two splitting modes with single or double liquid bridge are contrasted and analyzed. The results show that the size of the air bubble trapped during the impacting process leads to various splitting modes. By calculating the unbalanced Young's force in the three-phase contact lines, unbalanced surface tension at the superhydrophobic/hydrophilic surface is observed. In addition, we find that the increase in the initial velocity of the drop can not only shorten the splitting time but also lead to a change in splitting mode. The increase in the wettability contrast of the surface can stably shorten the splitting time. However the splitting time did not linearly shorten with the increase in strip width. Our findings would advance the understanding of drop impact splitting and possibly inspire new guidance for the design of functional interfaces. The present study also provides an efficient parallel implementation to simulate the 3D droplet motion on heterogeneous surfaces.

Complex Linear Response Functions for a Multiconfigurational Self-Consistent Field Wave Function in a High Performance Computing Environment.

We present novel developments for the highly efficient evaluation of complex linear response functions of a multiconfigurational self-consistent field (MCSCF) wave function as implemented in MultiPsi. Specifically, expressions for the direct evaluation of linear response properties at given frequencies using the complex polarization propagator (CPP) approach have been implemented, within both the Tamm-Dancoff approximation (TDA) and the random phase approximation (RPA). Purely real algebra with symmetric and antisymmetric trial vectors in a shared subspace is used wherein the linear response equations are solved. Two bottlenecks of large scale MC-CPP calculations, namely, the memory footprint and computational time, are addressed. The former is addressed by limiting the size of the subspace of trial vectors by using singular value decomposition (SVD) on either orbital or CI subspaces. The latter is addressed using an efficient parallel implementation as well as the strategy of dynamically adding linear response equations at near-convergence to neighboring roots. Furthermore, a novel methodology for decomposing MC-CPP spectra in terms of intuitive orbital excitations in an approximate fashion is presented. The performance of the code is illustrated with several numerical examples, including the X-ray spectrum of a molecule with nearly one hundred atoms. Additionally, for X-ray spectroscopy, the effect of including or excluding the core orbital in the active space on small covalent metal complexes is discussed.

Hybrid GPU–CPU Efficient Implementation of a Parallel Numerical Algorithm for Solving the Cauchy Problem for a Nonlinear Differential Riccati Equation of Fractional Variable Order

The numerical solution for fractional dynamics problems can create a high computational load, which makes it necessary to implement efficient algorithms for their solution. The main contribution to the computational load of such computations is created by heredity (memory), which is determined by the dependence of the current value of the solution function on previous values in the time interval. In terms of mathematics, the heredity here is described using a fractional differentiation operator in the Gerasimov–Caputo sense of variable order. As an example, we consider the Cauchy problem for the non-linear fractional Riccati equation with non-constant coefficients. An efficient parallel implementation algorithm has been proposed for the known sequential non-local explicit finite-difference numerical solution scheme. This implementation of the algorithm is a hybrid one, since it uses both GPU and CPU computational nodes. The program code of the parallel implementation of the algorithm is described in C and CUDA C languages, and is developed using OpenMP and CUDA hardware, as well as software architectures. This paper presents a study on the computational efficiency of the proposed parallel algorithm based on data from a series of computational experiments that were obtained using a computing server NVIDIA DGX STATION. The average computation time is analyzed in terms of the following: running time, acceleration, efficiency, and the cost of the algorithm. As a result, it is shown on test examples that the hybrid version of the numerical algorithm can give a significant performance increase of 3–5 times in comparison with both the sequential version of the algorithm and OpenMP implementation.

A dynamical core based on a discontinuous Galerkin method for higher-order finite-element sea ice modeling

Abstract. The ability of numerical sea ice models to reproduce localized deformation features associated with fracture processes is key for an accurate representation of the ice dynamics and of dynamically coupled physical processes in the Arctic and Antarctic. Equally key is the capacity of these models to minimize the numerical diffusion stemming from the advection of these features to ensure that the associated strong gradients persist in time, without the need to unphysically re-inject energy for re-localization. To control diffusion and improve the approximation quality, we present a new numerical core for the dynamics of sea ice that is based on higher-order finite-element discretizations for the momentum equation and higher-order discontinuous Galerkin methods for the advection. The mathematical properties of this core are discussed, and a detailed description of an efficient shared-memory parallel implementation is given. In addition, we present different numerical tests and apply the new framework to a benchmark problem to quantify the advantages of the higher-order discretization. These tests are based on Hibler's viscous–plastic sea ice model, but the implementation of the developed framework in the context of other physical models reproducing a strong localization of the deformation is possible.

Generalizing Downsampling from Regular Data to Graphs

Downsampling produces coarsened, multi-resolution representations of data and it is used, for example, to produce lossy compression and visualization of large images, reduce computational costs, and boost deep neural representation learning. Unfortunately, due to their lack of a regular structure, there is still no consensus on how downsampling should apply to graphs and linked data. Indeed reductions in graph data are still needed for the goals described above, but reduction mechanisms do not have the same focus on preserving topological structures and properties, while allowing for resolution-tuning, as is the case in regular data downsampling. In this paper, we take a step in this direction, introducing a unifying interpretation of downsampling in regular and graph data. In particular, we define a graph coarsening mechanism which is a graph-structured counterpart of controllable equispaced coarsening mechanisms in regular data. We prove theoretical guarantees for distortion bounds on path lengths, as well as the ability to preserve key topological properties in the coarsened graphs. We leverage these concepts to define a graph pooling mechanism that we empirically assess in graph classification tasks, providing a greedy algorithm that allows efficient parallel implementation on GPUs, and showing that it compares favorably against pooling methods in literature.

Efficient parallel implementation of Gaussian Mixture Model background subtraction algorithm on an embedded multi-core Digital Signal Processor

Scene segmentation is crucial in video processing applications, such as vehicle detection. The Gaussian Mixture Model (GMM) based background subtraction (BS) is a widely employed technique for scene segmentation. However, due to its high computational demands, an efficient parallel implementation is necessary. This paper proposes a parallelized implementation of the GMM algorithm on the C6678 digital signal processor (DSP), utilizing its eight cores. Three different parallelization approaches are presented: Open Multiprocessing (OpenMP), manually optimized method, and manually optimized method involving Direct Memory Access (DMA) engine. To evaluate their effectiveness, a metrics comparison is conducted using background truth images provided by the literature dataset references. The experimental results demonstrate that the proposed manually optimized method surpasses the current state-of-the-art. It achieves a remarkable parallel efficiency of approximately 99% for low and medium frame sizes and 83% and 79% for high-definition (HD) and full-HD resolution frames, respectively, when all eight cores are enabled. In contrast, the parallel efficiency obtained using OpenMP is only 45%.

Adaptive density-based robust topology optimization under uncertain loads using parallel computing

This work presents an efficient parallel implementation of density-based robust topology optimization (RTO) using adaptive mesh refinement (AMR) schemes permitting us to address the problem with modest computational resources. We use sparse grid stochastic collocation methods (SCMs) for transforming the RTO problem into a weighted multiple-loading deterministic problem at the collocation points. The calculation of these deterministic problems and the functional sensitivity is computationally expensive. We combine distributed-memory parallel computing and AMR techniques to address the problem efficiently. The former allows us to exploit the computational resources available, whereas the latter permits us to increase performance significantly. We propose the parallel incremental calculation of the deterministic problems and the contribution to the functional sensitivity maintaining a similar memory allocation to the one used in the deterministic counterpart. The cumulative computing uses buffers to adapt the evaluation at the collocation points to the parallel computing resources permitting the exploitation of the embarrassing parallelism of SCMs. We evaluate the deterministic problems in a coarse mesh generated for each topology optimization iteration to increase the performance. We perform the regularization and design variable update in a fine mesh to obtain an equivalent design to the one generated in such a mesh. We evaluate the proposal in two- and three-dimensional problems to test its feasibility and scalability. We also check the performance improvement using computational buffers in parallel computing nodes. Finally, we compare the proposal to the same approach using different preconditioners without AMR schemes showing significant performance improvements.

Global Optimization Approach for Parameter Estimation in Stochastic Dynamic Models of Biosystems.

Mechanistic dynamic models have become an essential tool for understanding biomolecular networks and other biological systems. Biochemical stochasticity can be extremely important in some situations, e.g., at the single-cell level where there is a low copy number of the species involved. In these scenarios, deterministic models are not suitable to characterize the dynamics, so stochastic dynamic models should be considered. Here, we address the challenging problem of parameter estimation in stochastic dynamic models. Despite recent advances, this area is considerably less mature than its deterministic counterpart. We present a novel strategy based on two components: (i) global optimization via a hybrid stochastic-deterministic approach, and (ii) stochastic simulation techniques tailored to the sparsity of the available experimental data. Regarding the latter, for cases of dense population data we make use of a novel approach using a Partial Integro-Differential Equation (PIDE) model solved using a semilagrangian method. In order to further speed up the simulations, we also present efficient parallel implementations for multi-core CPUs and also for graphical processing units (GPUs). Importantly, whereas SDE and Fokker Planck approximations of the Chemical Master Equation (CME) apply when the reactant populations are sufficiently large, the PIDE approximation to the CME is valid for very low copy numbers, and therefore they enable us to tackle parameter estimation for systems with large intrinsic molecular noise, (highly stochastic regimes far from the thermodynamic limit). We test our strategy with four challenging problems: a Lotka-Volterra system, a polarization system in S. cerevisiae, a genetic toggle switch, and a genetic circadian oscillator. Our method could successfully solve these problems in very reasonable computation times (often a few minutes for the first two problems) using standard low-cost hardware, showing very significant speedups with respect to recent alternative methods. The code used to obtain the results reported here is available at https://doi.org/10.5281/zenodo.5195408.

A distributed computing model for big data anonymization in the networks.

Recently big data and its applications had sharp growth in various fields such as IoT, bioinformatics, eCommerce, and social media. The huge volume of data incurred enormous challenges to the architecture, infrastructure, and computing capacity of IT systems. Therefore, the compelling need of the scientific and industrial community is large-scale and robust computing systems. Since one of the characteristics of big data is value, data should be published for analysts to extract useful patterns from them. However, data publishing may lead to the disclosure of individuals' private information. Among the modern parallel computing platforms, Apache Spark is a fast and in-memory computing framework for large-scale data processing that provides high scalability by introducing the resilient distributed dataset (RDDs). In terms of performance, Due to in-memory computations, it is 100 times faster than Hadoop. Therefore, Apache Spark is one of the essential frameworks to implement distributed methods for privacy-preserving in big data publishing (PPBDP). This paper uses the RDD programming of Apache Spark to propose an efficient parallel implementation of a new computing model for big data anonymization. This computing model has three-phase of in-memory computations to address the runtime, scalability, and performance of large-scale data anonymization. The model supports partition-based data clustering algorithms to preserve the λ-diversity privacy model by using transformation and actions on RDDs. Therefore, the authors have investigated Spark-based implementation for preserving the λ-diversity privacy model by two designed City block and Pearson distance functions. The results of the paper provide a comprehensive guideline allowing the researchers to apply Apache Spark in their own researches.

Ab initio study of water dissociation on a charged Pd(111) surface.

The interactions between molecules and electrode surfaces play a key role in electrochemical processes and are a subject of extensive research, both experimental and theoretical. In this paper, we address the water dissociation reaction on a Pd(111) electrode surface, modeled as a slab embedded in an external electric field. We aim at unraveling the relationship between surface charge and zero-point energy in aiding or hindering this reaction. We calculate the energy barriers with dispersion-corrected density-functional theory and an efficient parallel implementation of the nudged-elastic-band method. We show that the lowest dissociation barrier and consequently the highest reaction rate take place when the field reaches a strength where two different geometries of the water molecule in the reactant state are equally stable. The zero-point energy contributions to this reaction, on the other hand, remain nearly constant across a wide range of electric field strengths, despite significant changes in the reactant state. Interestingly, we show that the application of electric fields that induce a negative charge on the surface can make nuclear tunneling more significant for these reactions.

Hybrid CPU–GPU implementation of the transformed spatial domain channel estimation algorithm for mmWave MIMO systems

Hybrid platforms combining multicore central processing units (CPU) with many-core hardware accelerators such as graphic processing units (GPU) can be smartly exploited to provide efficient parallel implementations of wireless communication algorithms for Fifth Generation (5G) and beyond systems. Massive multiple-input multiple-output (MIMO) systems are a key element of the 5G standard, involving several tens or hundreds of antenna elements for communication. Such a high number of antennas has a direct impact on the computational complexity of some MIMO signal processing algorithms. In this work, we focus on the channel estimation stage. In particular, we develop a parallel implementation of a recently proposed MIMO channel estimation algorithm. Its performance in terms of execution time is evaluated both in a multicore CPU and in a GPU. The results show that some computation blocks of the algorithm are more suitable for multicore implementation, whereas other parts are more efficiently implemented in the GPU, indicating that a hybrid CPU–GPU implementation would achieve the best performance in practical applications based on the tested platform.

Efficient parallel implementation of crowd simulation using a hybrid CPU+GPU high performance computing system

In the paper we present a modern efficient parallel OpenMP+CUDA implementation of crowd simulation for hybrid CPU+GPU systems and demonstrate its higher performance over CPU-only and GPU-only implementations for several problem sizes including 10 000, 50 000, 100 000, 500 000 and 1 000 000 agents. We show how performance varies for various tile sizes and what CPU–GPU load balancing settings shall be preferred for various domain sizes among CPUs and GPUs of a high performance system with 2 Intel Xeon Silver multicore CPUs and 8 NVIDIA Quadro RTX 5000 GPUs. We then present how execution time depends on the number of agents as well as the number of CUDA streams used for parallel execution of several CUDA kernels. We discuss the design and implementation of an algorithm with CPU computational threads, GPU management threads, assignment of particular tasks to threads as well as usage of pinned memory and CUDA shared memory for maximizing performance.

An efficient geometric method for incompressible hydrodynamics on the sphere

We present an efficient and highly scalable geometric numerical method for two-dimensional ideal fluid dynamics on the sphere. The starting point is Zeitlin's finite-dimensional model of hydrodynamics. The efficiency stems from exploiting a tridiagonal splitting of the discrete spherical Laplacian combined with highly optimized, scalable numerical algorithms. For time-stepping, we adopt a recently developed isospectral integrator able to preserve the geometric structure of Euler's equations, in particular conservation of the Casimir functions. To overcome previous computational bottlenecks, we formulate the matrix Lie algebra basis through a sequence of tridiagonal eigenvalue problems, efficiently solved by well-established linear algebra libraries. The same tridiagonal splitting allows for computation of the stream matrix, involving the inverse Laplacian, for which we design an efficient parallel implementation on distributed memory systems. The resulting overall computational complexity is O(N3) per time-step for N2 spatial degrees of freedom. The dominating computational cost is matrix-matrix multiplication, carried out via the parallel library ScaLAPACK. Scaling tests show approximately linear scaling up to around 2500 cores for the matrix size N=4096 with a computational time per time-step of about 0.55 seconds. These results allow for long-time simulations and the gathering of statistical quantities while simultaneously conserving the Casimir functions. We illustrate the developed algorithm for Euler's equations at the resolution N=2048.

Fast computation of fractal dimension for 2D, 3D and 4D data

The box-counting (BC) algorithm is one of the most popular methods for calculating the fractal dimension (FD) of binary data. FD analysis has many important applications in the biomedical field, such as cancer detection from 2D computed axial tomography images, Alzheimer’s disease diagnosis from magnetic resonance 3D volumetric data, and consciousness states characterization based on 4D data extracted from electroencephalography (EEG) signals, among many others. Currently, these kinds of applications use data whose size and amount can be very large, with high computation times needed to calculate the BC of the whole datasets. In this study we present a very efficient parallel implementation of the BC algorithm for its execution on Graphics Processing Units (GPU). Our algorithm can process 2D, 3D and 4D data and we tested it on two platforms with different hardware configurations. The results showed speedups of up to 92.38 × (2D), 57.27 × (3D) and 75.73 × (4D) with respect to the corresponding CPU single-thread implementations of the same algorithm. Against an OpenMP multi-thread CPU implementation, our GPU algorithm achieved speedups of up to 16.12 × (2D), 6.86 × (3D) and 7.49 × (4D). We have also compared our algorithm to a previous GPU implementation of the BC algorithm in 3D, achieving a speedup of up to 4.79 × . Finally, as a practical application of our GPU BC algorithm a study comparing the FD of 4D data extracted from the EEGs of a schizophrenia patient and a healthy subject was performed. The computation time for processing 40 4D matrices was reduced from three hours (sequential CPU) to less than three minutes with our GPU algorithm.

Recent advances in domain decomposition methods for large-scale saddle point problems

Scalability of parallel solvers for problems with high heterogeneities relies on adaptive coarse spaces built from generalized eigenvalue problems in the subdomains. The corresponding theory is powerful and flexible but the development of an efficient parallel implementation is challenging. We report here on recent advances in adaptive coarse spaces and on their open source implementations in domain specific languages such as FreeFem, focusing on a new domain decomposition for saddle point formulations with some numerical tests.

Parallel Implementation of Sparse Distributed Memory for Semantic Storage

Introduction. Sparse Distributed Memory (SDM) and Binary Sparse Distributed Representations (Binary Sparse Distributed Representations, BSDR), as two phenomenological approaches to biological memory modelling, have many similarities. The idea of ??their integration into a hybrid semantic storage model with SDM as a low-level cleaning memory (brain cells) for BSDR, which is used as an encoder of high-level symbolic information, is natural. A hybrid semantic store should be able to store holistic data (for example, structures of interconnected and sequential key-value pairs) in a neural network. A similar design has been proposed several times since the 1990s. However, the previously proposed models are impractical due to insufficient scalability and/or low storage density. The gap between SDM and BSDR can be bridged by the results of a third theory related to sparse signals: Compressive Sensing or Sampling (CS). In this article, we focus on the highly efficient parallel implementation of the CS-SDM hybrid memory model for graphics processing units on the NVIDIA CUDA platform, analyze the computational complexity of CS-SDM operations for the case of parallel implementation, and offer optimization techniques for conducting experiments with big sequential batches of vectors. The purpose of the paper is to propose an efficient software implementation of sparse-distributed memory for preserving semantics on modern graphics processing units. Results. Parallel algorithms for CS-SDM operations are proposed, their computational complexity is estimated, and a parallel implementation of the CS-SDM hybrid semantic store is given. Optimization of vector reconstruction for experiments with sequential data batches is proposed. Conclusions. The obtained results show that the design of CS-SDM is naturally parallel and that its algorithms are by design compatible with the architecture of systems with massive parallelism. The conducted experiments showed high performance of the developed implementation of the SDM memory block. Keywords: GPU, CUDA, neural network, Sparse Distributed Memory, associative memory, Compressive Sensing.

Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries

Scalability of parallel solvers for problems with high heterogeneities relies on adaptive coarse spaces built from generalized eigenvalue problems in the subdomains. The corresponding theory is powerful and flexible but the development of an efficient parallel implementation is challenging. We report here on recent advances in adaptive coarse spaces and on their open source implementations.