arXiv subjects

Stefano Markidis

Publications and source records attributed to Stefano Markidis.

At least 73 records · Page 4Linked to original sources

arXiv2024

Supercomputers as a Continous Medium

As supercomputers' complexity has grown, the traditional boundaries between processor, memory, network, and accelerators have blurred, making a homogeneous computer model, in which the overall computer system is modeled as a continuous medium with homogeneously distributed computational power, memory, and data movement transfer capabilities, an intriguing and powerful abstraction. By applying a homogeneous computer model to algorithms with a given I/O complexity, we recover from first principles, other discrete computer models, such as the roofline model, parallel computing laws, such as Amdahl's and Gustafson's laws, and phenomenological observations, such as super-linear speedup. One of the homogeneous computer model's distinctive advantages is the capability of directly linking the performance limits of an application to the physical properties of a classical computer system. Applying the homogeneous computer model to supercomputers, such as Frontier, Fugaku, and the Nvidia DGX GH200, shows that applications, such as Conjugate Gradient (CG) and Fast Fourier Transforms (FFT), are rapidly approaching the fundamental classical computational limits, where the performance of even denser systems in terms of compute and memory are fundamentally limited by the speed of light.

cs.DC↗

arXiv2024

Experience and Analysis of Scalable High-Fidelity Computational Fluid Dynamics on Modular Supercomputing Architectures

The never-ending computational demand from simulations of turbulence makes computational fluid dynamics (CFD) a prime application use case for current and future exascale systems. High-order finite element methods, such as the spectral element method, have been gaining traction as they offer high performance on both multicore CPUs and modern GPU-based accelerators. In this work, we assess how high-fidelity CFD using the spectral element method can exploit the modular supercomputing architecture at scale through domain partitioning, where the computational domain is split between a Booster module powered by GPUs and a Cluster module with conventional CPU nodes. We investigate several different flow cases and computer systems based on the modular supercomputing architecture (MSA). We observe that for our simulations, the communication overhead and load balancing issues incurred by incorporating different computing architectures are seldom worthwhile, especially when I/O is also considered, but when the simulation at hand requires more than the combined global memory on the GPUs, utilizing additional CPUs to increase the available memory can be fruitful. We support our results with a simple performance model to assess when running across modules might be beneficial. As MSA is becoming more widespread and efforts to increase system utilization are growing more important our results give insight into when and how a monolithic application can utilize and spread out to more than one module and obtain a faster time to solution.

cs.DC↗

arXiv2024

A GPU-Accelerated Interior Point Method for Radiation Therapy Optimization

Optimization plays a central role in modern radiation therapy, where it is used to determine optimal treatment machine parameters in order to deliver precise doses adapted to each patient case. In general, solving the optimization problems that arise can present a computational bottleneck in the treatment planning process, as they can be large in terms of both variables and constraints. In this paper, we develop a GPU accelerated optimization solver for radiation therapy applications, based on an interior point method (IPM) utilizing iterative linear algebra to find search directions. The use of iterative linear algebra makes the solver suitable for porting to GPUs, as the core computational kernels become standard matrix-vector or vector-vector operations. Our solver is implemented in C++20 and uses CUDA for GPU acceleration. The problems we solve are from the commercial treatment planning system RayStation, developed by RaySearch Laboratories (Stockholm, Sweden), which is used clinically in hundreds of cancer clinics around the world. RayStation solves (in general) nonlinear optimization problems using a sequential quadratic programming (SQP) method, where the main computation lies in solving quadratic programming (QP) sub-problems in each iteration. GPU acceleration for the solution of such QP sub-problems is the focus of the interior point method of this work. We benchmark our solver against the existing QP-solver in RayStation and show that our GPU accelerated IPM can accelerate the aggregated time-to-solution for all QP sub-problems in one SQP solve by 1.4 and 4.4 times, respectively, for two real patient cases.

math.OC↗

arXiv2024

Brain-Inspired Physics-Informed Neural Networks: Bare-Minimum Neural Architectures for PDE Solvers

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving partial differential equations~(PDEs) in various scientific and engineering domains. However, traditional PINN architectures typically rely on large, fully connected multilayer perceptrons~(MLPs), lacking the sparsity and modularity inherent in many traditional numerical solvers. An unsolved and critical question for PINN is: What is the minimum PINN complexity regarding nodes, layers, and connections needed to provide acceptable performance? To address this question, this study investigates a novel approach by merging established PINN methodologies with brain-inspired neural network techniques. We use Brain-Inspired Modular Training~(BIMT), leveraging concepts such as locality, sparsity, and modularity inspired by the organization of the brain. With brain-inspired PINN, we demonstrate the evolution of PINN architectures from large, fully connected structures to bare-minimum, compact MLP architectures, often consisting of a few neural units! Moreover, using brain-inspired PINN, we showcase the spectral bias phenomenon occurring on the PINN architectures: bare-minimum architectures solving problems with high-frequency components require more neural units than PINN solving low-frequency problems. Finally, we derive basic PINN building blocks through BIMT training on simple problems akin to convolutional and attention modules in deep neural networks, enabling the construction of modular PINN architectures. Our experiments show that brain-inspired PINN training leads to PINN architectures that minimize the computing and memory resources yet provide accurate results.

cs.CE↗

arXiv2024

Krylov Solvers for Interior Point Methods with Applications in Radiation Therapy and Support Vector Machines

Interior point methods are widely used for different types of mathematical optimization problems. Many implementations of interior point methods in use today rely on direct linear solvers to solve systems of equations in each iteration. The need to solve ever larger optimization problems more efficiently and the rise of hardware accelerators for general purpose computing has led to a large interest in using iterative linear solvers instead, with the major issue being inevitable ill-conditioning of the linear systems arising as the optimization progresses. We investigate the use of Krylov solvers for interior point methods in solving optimization problems from radiation therapy and support vector machines. We implement a prototype interior point method using a so called doubly augmented formulation of the Karush-Kuhn-Tucker linear system of equations, originally proposed by Forsgren and Gill, and evaluate its performance on real optimization problems from radiation therapy and support vector machines. Crucially, our implementation uses a preconditioned conjugate gradient method with Jacobi preconditioning internally. Our measurements of the conditioning of the linear systems indicate that the Jacobi preconditioner improves the conditioning of the systems to a degree that they can be solved iteratively, but there is room for further improvement in that regard. Furthermore, profiling of our prototype code shows that it is suitable for GPU acceleration, which may further improve its performance in practice. Overall, our results indicate that our method can find solutions of acceptable accuracy in reasonable time, even with a simple Jacobi preconditioner.

math.OC↗

arXiv2023

Leveraging MLIR for Loop Vectorization and GPU Porting of FFT Libraries

FFTc is a Domain-Specific Language (DSL) for designing and generating Fast Fourier Transforms (FFT) libraries. The FFTc uniqueness is that it leverages and extend Multi-Level Intermediate Representation (MLIR) dialects to optimize FFT code generation. In this work, we present FFTc extensions and improvements such as the possibility of using different data layout for complex-value arrays, and sparsification to enable efficient vectorization, and a seamless porting of FFT libraries to GPU systems. We show that, on CPUs, thanks to vectorization, the performance of the FFTc-generated FFT is comparable to performance of FFTW, a state-of-the-art FFT libraries. We also present the initial performance results for FFTc on Nvidia GPUs.

cs.MS↗

arXiv2023

Anderson Accelerated PMHSS for Complex-Symmetric Linear Systems

This paper presents the design and development of an Anderson Accelerated Preconditioned Modified Hermitian and Skew-Hermitian Splitting (AA-PMHSS) method for solving complex-symmetric linear systems with application to electromagnetics problems, such as wave scattering and eddy currents. While it has been shown that the Anderson Acceleration of real linear systems is essentially equivalent to GMRES, we show here that the formulation using Anderson acceleration leads to a more performant method. We show relatively good robustness compared to existing preconditioned GMRES methods and significantly better performance due to the faster evaluation of the preconditioner. In particular, AA-PMHSS can be applied to solve problems and equations arising from electromagnetics, such as time-harmonic eddy current simulations discretized with the Finite Element Method. We also evaluate three test systems present in previous literature. We show that the method is competitive with two types of preconditioned GMRES. One of the significant advantages of these methods is that the convergence rate is independent of the discretization size.

math.NA↗

arXiv2023

Boosting the Performance of Object Tracking with a Half-Precision Particle Filter on GPU

High-performance GPU-accelerated particle filter methods are critical for object detection applications, ranging from autonomous driving, robot localization, to time-series prediction. In this work, we investigate the design, development and optimization of particle-filter using half-precision on CUDA cores and compare their performance and accuracy with single- and double-precision baselines on Nvidia V100, A100, A40 and T4 GPUs. To mitigate numerical instability and precision losses, we introduce algorithmic changes in the particle filters. Using half-precision leads to a performance improvement of 1.5-2x and 2.5-4.6x with respect to single- and double-precision baselines respectively, at the cost of a relatively small loss of accuracy.

cs.DC↗

arXiv2023

Quantum Computer Simulations at Warp Speed: Assessing the Impact of GPU Acceleration

Quantum computer simulators are crucial for the development of quantum computing. In this work, we investigate the suitability and performance impact of GPU and multi-GPU systems on a widely used simulation tool - the state vector simulator Qiskit Aer. In particular, we evaluate the performance of both Qiskit's default Nvidia Thrust backend and the recent Nvidia cuQuantum backend on Nvidia A100 GPUs. We provide a benchmark suite of representative quantum applications for characterization. For simulations with a large number of qubits, the two GPU backends can provide up to 14x speedup over the CPU backend, with Nvidia cuQuantum providing further 1.5-3x speedup over the default Thrust backend. Our evaluation on a single GPU identifies the most important functions in Nvidia Thrust and cuQuantum for different quantum applications and their compute and memory bottlenecks. We also evaluate the gate fusion and cache-blocking optimizations on different quantum applications. Finally, we evaluate large-number qubit quantum applications on multi-GPU and identify data movement between host and GPU as the limiting factor for the performance.

cs.PF↗

arXiv2023

Leveraging HPC Profiling & Tracing Tools to Understand the Performance of Particle-in-Cell Monte Carlo Simulations

Large-scale plasma simulations are critical for designing and developing next-generation fusion energy devices and modeling industrial plasmas. BIT1 is a massively parallel Particle-in-Cell code designed for specifically studying plasma material interaction in fusion devices. Its most salient characteristic is the inclusion of collision Monte Carlo models for different plasma species. In this work, we characterize single node, multiple nodes, and I/O performances of the BIT1 code in two realistic cases by using several HPC profilers, such as perf, IPM, Extrae/Paraver, and Darshan tools. We find that the BIT1 sorting function on-node performance is the main performance bottleneck. Strong scaling tests show a parallel performance of 77% and 96% on 2,560 MPI ranks for the two test cases. We demonstrate that communication, load imbalance and self-synchronization are important factors impacting the performance of the BIT1 on large-scale runs.

cs.DC↗

arXiv2023

Physics-based adaptivity of a spectral method for the Vlasov-Poisson equations based on the asymmetrically-weighted Hermite expansion in velocity space

We propose a spectral method for the 1D-1V Vlasov-Poisson system where the discretization in velocity space is based on asymmetrically-weighted Hermite functions, dynamically adapted via a scaling $α$ and shifting $u$ of the velocity variable. Specifically, at each time instant an adaptivity criterion selects new values of $α$ and $u$ based on the numerical solution of the discrete Vlasov-Poisson system obtained at that time step. Once the new values of the Hermite parameters $α$ and $u$ are fixed, the Hermite expansion is updated and the discrete system is further evolved for the next time step. The procedure is applied iteratively over the desired temporal interval. The key aspects of the adaptive algorithm are: the map between approximation spaces associated with different values of the Hermite parameters that preserves total mass, momentum and energy; and the adaptivity criterion to update $α$ and $u$ based on physics considerations relating the Hermite parameters to the average velocity and temperature of each plasma species. For the discretization of the spatial coordinate, we rely on Fourier functions and use the implicit midpoint rule for time stepping. The resulting numerical method possesses intrinsically the property of fluid-kinetic coupling, where the low-order terms of the expansion are akin to the fluid moments of a macroscopic description of the plasma, while kinetic physics is retained by adding more spectral terms. Moreover, the scheme features conservation of total mass, momentum and energy associated in the discrete, for periodic boundary conditions. A set of numerical experiments confirms that the adaptive method outperforms the non-adaptive one in terms of accuracy and stability of the numerical solution.

math.NA↗

arXiv2023

QHDL: a Low-Level Circuit Description Language for Quantum Computing

This paper proposes a descriptive language called QHDL, akin to VHDL, to program gate-based quantum computing systems. Unlike other popular quantum programming languages, QHDL targets low-level quantum computing programming and aims to provide a common framework for programming FPGAs and gate-based quantum computing systems. The paper presents an initial implementation and design principles of the QHDL framework, including a compiler and quantum computer simulator. We discuss the challenges of low-level integration of streaming models and quantum computing for programming FPGAs and gate-based quantum computing systems.

cs.ET↗

arXiv2023

Parallel Cholesky Factorization for Banded Matrices using OpenMP Tasks

Cholesky factorization is a widely used method for solving linear systems involving symmetric, positive-definite matrices, and can be an attractive choice in applications where a high degree of numerical stability is needed. One such application is numerical optimization, where direct methods for solving linear systems are widely used and often a significant performance bottleneck. An example where this is the case, and the specific type of optimization problem motivating this work, is radiation therapy treatment planning, where numerical optimization is used to create individual treatment plans for patients. To address this bottleneck, we propose a task-based multi-threaded method for Cholesky factorization of banded matrices with medium-sized bands. We implement our algorithm using OpenMP tasks and compare our performance with state-of-the-art libraries such as Intel MKL. Our performance measurements show a performance that is on par or better than Intel MKL (up to ~26%) for a wide range of matrix bandwidths on two different Intel CPU systems.

math.NA↗

arXiv2023

Making Applications Faster by Asynchronous Execution: Slowing Down Processes or Relaxing MPI Collectives

Comprehending the performance bottlenecks at the core of the intricate hardware-software interactions exhibited by highly parallel programs on HPC clusters is crucial. This paper sheds light on the issue of automatically asynchronous MPI communication in memory-bound parallel programs on multicore clusters and how it can be facilitated. For instance, slowing down MPI processes by deliberate injection of delays can improve performance if certain conditions are met. This leads to the counter-intuitive conclusion that noise, independent of its source, is not always detrimental but can be leveraged for performance improvements. We employ phase-space graphs as a new tool to visualize parallel program dynamics. They are useful in spotting certain patterns in parallel execution that will easily go unnoticed with traditional tracing tools. We investigate five different microbenchmarks and applications on different supercomputer platforms: an MPI-augmented STREAM Triad, two implementations of Lattice-Boltzmann fluid solvers, and the LULESH and HPCG proxy applications.

cs.DC↗

arXiv2022

On Physics-Informed Neural Networks for Quantum Computers

Physics-Informed Neural Networks (PINN) emerged as a powerful tool for solving scientific computing problems, ranging from the solution of Partial Differential Equations to data assimilation tasks. One of the advantages of using PINN is to leverage the usage of Machine Learning computational frameworks relying on the combined usage of CPUs and co-processors, such as accelerators, to achieve maximum performance. This work investigates the design, implementation, and performance of PINNs, using the Quantum Processing Unit (QPU) co-processor. We design a simple Quantum PINN to solve the one-dimensional Poisson problem using a Continuous Variable (CV) quantum computing framework. We discuss the impact of different optimizers, PINN residual formulation, and quantum neural network depth on the quantum PINN accuracy. We show that the optimizer exploration of the training landscape in the case of quantum PINN is not as effective as in classical PINN, and basic Stochastic Gradient Descent (SGD) optimizers outperform adaptive and high-order optimizers. Finally, we highlight the difference in methods and algorithms between quantum and classical PINNs and outline future research challenges for quantum PINN development.

quant-ph↗

arXiv2022

Breaking Down the Parallel Performance of GROMACS, a High-Performance Molecular Dynamics Software

GROMACS is one of the most widely used HPC software packages using the Molecular Dynamics (MD) simulation technique. In this work, we quantify GROMACS parallel performance using different configurations, HPC systems, and FFT libraries (FFTW, Intel MKL FFT, and FFT PACK). We break down the cost of each GROMACS computational phase and identify non-scalable stages, such as MPI communication during the 3D FFT computation when using a large number of processes. We show that the Particle-Mesh Ewald phase and the 3D FFT calculation significantly impact the GROMACS performance. Finally, we discuss performance opportunities with a particular interest in developing GROMACS for the FFT calculations.

physics.comp-ph↗

arXiv2022

Distributed Objective Function Evaluation for Optimization of Radiation Therapy Treatment Plans

The modern workflow for radiation therapy treatment planning involves mathematical optimization to determine optimal treatment machine parameters for each patient case. The optimization problems can be computationally expensive, requiring iterative optimization algorithms to solve. In this work, we investigate a method for distributing the calculation of objective functions and gradients for radiation therapy optimization problems across computational nodes. We test our approach on the TROTS dataset -- which consists of optimization problems from real clinical patient cases -- using the IPOPT optimization solver in a leader/follower type approach for parallelization. We show that our approach can utilize multiple computational nodes efficiently, with a speedup of approximately 2-3.5 times compared to the serial version.

cs.MS↗

arXiv2022

FFTc: An MLIR Dialect for Developing HPC Fast Fourier Transform Libraries

Discrete Fourier Transform (DFT) libraries are one of the most critical software components for scientific computing. Inspired by FFTW, a widely used library for DFT HPC calculations, we apply compiler technologies for the development of HPC Fourier transform libraries. In this work, we introduce FFTc, a domain-specific language, based on Multi-Level Intermediate Representation (MLIR), for expressing Fourier Transform algorithms. We present the initial design, implementation, and preliminary results of FFTc.

cs.MS↗