Joseph Kump | Searcharxiv

arXiv2025

A Two-Level Direct Solver for the Hierarchical Poincar\'e-Steklov Method

We introduce a two-level direct solver for the Hierarchical Poincar\'e-Steklov (HPS) method for solving linear elliptic PDEs. HPS combines multidomain spectral collocation with a direct solver, enabling high-order discretizations for highly oscillatory solutions while preserving computational efficiency. Our method employs batched linear algebra routines with GPU acceleration to reduce the problem to subdomain interfaces, yielding a block-sparse linear system. This system is then factorized using a sparse direct solver that employs pivoting to achieve better numerical stability than the original HPS scheme. For a discretization of local order $p$ involving a total of $N$ degrees of freedom, the initial reduction step has asymptotic complexity $O(N p^6)$ in three dimensions. Nevertheless, the high efficiency of batched GPU routines makes the overall cost for practical purposes independent of polynomial order (for order $p=20$ or even higher). Additionally, the cost of the sparse direct solver is independent of the polynomial order. We present a description and justification of our method, along with numerical experiments on three-dimensional problems to evaluate its accuracy and performance.

math.NA↗

arXiv2020

Multichannel Analysis of Surface Waves Accelerated (MASWAccelerated): Software for Efficient Surface Wave Inversion Using MPI and GPUs

Multichannel Analysis of Surface Waves (MASW) is a technique frequently used in geotechnical engineering and engineering geophysics to infer layered models of seismic shear wave velocities in the top tens to hundreds of meters of the subsurface. We aim to accelerate MASW calculations by capitalizing on modern computer hardware available in the workstations of most engineers: multiple cores and graphics processing units (GPUs). We propose new parallel and GPU accelerated algorithms for evaluating MASW data, and provide software implementations in C using Message Passing Interface (MPI) and CUDA. These algorithms take advantage of sparsity that arises in the problem, and the work balance between processes considers typical data trends. We compare our methods to an existing open source Matlab MASW tool. Our serial C implementation achieves a 2x speedup over the Matlab software, and we continue to see improvements by parallelizing the problem with MPI. We see nearly perfect strong and weak scaling for uniform data, and improve strong scaling for realistic data by repartitioning the problem to process mapping. By utilizing GPUs available on most modern workstations, we observe an additional 1.3x speedup over the serial C implementation on the first use of the method. We typically repeatedly evaluate theoretical dispersion curves as part of an optimization procedure, and on the GPU the kernel can be cached for faster reuse on later runs. We observe a 3.2x speedup on the cached GPU runs compared to the serial C runs. This work is the first open-source parallel or GPU-accelerated software tool for MASW imaging, and should enable geotechnical engineers to fully utilize all computer hardware at their disposal.

cs.CE↗