Linear Scaling Tests

For kernel matrices for which matrix technique is effective, H2Pack has asymptotically linear costs in both computation and storage costs. Furthermore, H2Pack is extremely efficient in practical calculations, e.g., it can compute a matrix-vector multiplication for a kernel matrix within 2 seconds while using 3 GB of memory in the example below.

Hardware and software configuration

• 2 * Intel Xeon Gold 6226 CPU @ 2.7GHz (2 * 12 cores, 2 * 12 * 2 threads, hyperthreading disabled)
• 6 * 32 GB DDR4 memory
• Red Hat Enterprise Linux 7.6 (kernel 3.10.0-957.12.1.el7)
• Intel Parallel Studio Cluster version 2019.5
• ICC optimization flags: -O3 -xHost
• OpenMP environment variables
  • OMP_NUM_THREADS=24
  • OMP_PLACES=cores
  • OMP_PROC_BIND=close

Test settings

• Point sets: uniformly and randomly distributed points in a 3D scaled cube with the fixed point density
• Relative error threshold: 1e-6
• Running mode: JIT
• -construction and -matvec timings in seconds, and matrix representation storage size in MB.

Numerical results

3D Coulomb Kernel

3D Gaussian Kernel with exponent parameter 0.1

3D Stokes Kernel

Note:

• Kernel matrices defined by the Stokes kernel is of dimension for points. Thus the corresponding computation and storage costs are more expensive than the other scalar kernel functions.