Automated Translation and Accelerated Solving of Differential Equations on Multiple GPU Platforms

We demonstrate a high-performance vendor-agnostic method for massively parallel solving of ensembles of ordinary differential equations (ODEs) and stochastic differential equations (SDEs) on GPUs. The method is integrated with a widely used differential equation solver library in a high-level language (Julia's DifferentialEquations.jl) and enables GPU acceleration without requiring code changes by the user. Our approach achieves state-of-the-art performance compared to hand-optimized CUDA-C++ kernels, while performing $20-100\times$ faster than the vectorized-map (\texttt{vmap}) approach implemented in JAX and PyTorch. Performance evaluation on NVIDIA, AMD, Intel, and Apple GPUs demonstrates performance portability and vendor-agnosticism. We show composability with MPI to enable distributed multi-GPU workflows. The implemented solvers are fully featured, supporting event handling, automatic differentiation, and incorporating of datasets via the GPU's texture memory, allowing scientists to take advantage of GPU acceleration on all major current architectures without changing their model code and without loss of performance.Comment: 11 figure

Recent progress and challenges in exploiting graphics processors in computational fluid dynamics

The progress made in accelerating simulations of fluid flow using GPUs, and the challenges that remain, are surveyed. The review first provides an introduction to GPU computing and programming, and discusses various considerations for improved performance. Case studies comparing the performance of CPU- and GPUbased solvers for the Laplace and incompressible Navier–Stokes equations are performed in order to demonstrate the potential improvement even with simple codes. Recent efforts to accelerate CFD simulations using GPUs are reviewed for laminar, turbulent, and reactive flow solvers. Also, GPU implementations of the lattice Boltzmann method are reviewed. Finally, recommendations for implementing CFD codes on GPUs are given and remaining challenges are discussed, such as the need to develop new strategies and redesign algorithms to enable GPU acceleration.Keywords: Graphics processing unit (GPU), Reactive flow, Computational fluid dynamics (CFD), Laminar flows, Turbulent flow, CUD

General Purpose Flow Visualization at the Exascale

Exascale computing, i.e., supercomputers that can perform 1018 math operations per second, provide significant opportunity for improving the computational sciences. That said, these machines can be difficult to use efficiently, due to their massive parallelism, due to the use of accelerators, and due to the diversity of accelerators used. All areas of the computational science stack need to be reconsidered to address these problems. With this dissertation, we consider flow visualization, which is critical for analyzing vector field data from simulations. We specifically consider flow visualization techniques that use particle advection, i.e., tracing particle trajectories, which presents performance and implementation challenges. The dissertation makes four primary contributions. First, it synthesizes previous work on particle advection performance and introduces a high-level analytical cost model. Second, it proposes an approach for performance portability across accelerators. Third, it studies expected speedups based on using accelerators, including the importance of factors such as duration, particle count, data set, and others. Finally, it proposes an exascale-capable particle advection system that addresses diversity in many dimensions, including accelerator type, parallelism approach, analysis use case, underlying vector field, and more

Fast Dynamic 1D Simulation of Divertor Plasmas with Neural PDE Surrogates

Managing divertor plasmas is crucial for operating reactor scale tokamak devices due to heat and particle flux constraints on the divertor target. Simulation is an important tool to understand and control these plasmas, however, for real-time applications or exhaustive parameter scans only simple approximations are currently fast enough. We address this lack of fast simulators using neural PDE surrogates, data-driven neural network-based surrogate models trained using solutions generated with a classical numerical method. The surrogate approximates a time-stepping operator that evolves the full spatial solution of a reference physics-based model over time. We use DIV1D, a 1D dynamic model of the divertor plasma, as reference model to generate data. DIV1D's domain covers a 1D heat flux tube from the X-point (upstream) to the target. We simulate a realistic TCV divertor plasma with dynamics induced by upstream density ramps and provide an exploratory outlook towards fast transients. State-of-the-art neural PDE surrogates are evaluated in a common framework and extended for properties of the DIV1D data. We evaluate (1) the speed-accuracy trade-off; (2) recreating non-linear behavior; (3) data efficiency; and (4) parameter inter- and extrapolation. Once trained, neural PDE surrogates can faithfully approximate DIV1D's divertor plasma dynamics at sub real-time computation speeds: In the proposed configuration, 2ms of plasma dynamics can be computed in $\approx$0.63ms of wall-clock time, several orders of magnitude faster than DIV1D.Comment: Published in Nuclear Fusio

Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling. Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms