Solvers for coupled sparse/dense FEM/BEM linear systems
Table of Contents
1. Information htmlonly
This study is also available as an article and as an Inria research report.
2. Information htmlonly
See reproducing guidelines for this study here.
3. Introduction
This is an example experimental study relying on the test_FEMBEM solver
test suite. Here, we are especially interested in solving coupled sparse/dense
FEM/BEM linear systems arising in the domain of aeroacoustics. The idea is to
evaluate the solvers available in the open-source version of test_FEMBEM
for the solution of this kind of linear systems.
4. Experimental study
Unfortunately, the open-source version of test_FEMBEM testFEMBEM does
not implement couplings of sparse and dense direct solvers which is normally our
go-to method for solving sparse/dense FEM/BEM systems. Therefore, we rely here
only on dense direct solvers, namely HMAT-OSS and Chameleon.
HMAT-OSS hmat-oss is an open-source and sequential version of the compressed hierarchical \(\mathcal{H}\)-Matrix dense direct solver HMAT Lize14 developed at Airbus. Chameleon chameleon is a fully open-source dense direct solver without compression.
As of the test case, we consider a simplified short pipe which is still close enough to real-life models (see Figure 1).
Note that all the benchmarks were conducted on a single quad-core Intel(R) Xeon(R) CPU W3520 @ 2.67GHz machine with Hyper-Threading and 8 GiB of RAM .
Figure 1: A short pipe mesh counting 20,000 vertices.
4.1. Data compression
In the first part, we want to know to which extent can data compression improve the computation time. For this, we compare sequential executions of both HMAT-OSS, the compressed solver, and Chameleon, the non-compressed solver, on coupled FEM/BEM systems of different sizes (see Figure 2). The results clearly show the advantage of using data compression, especially with increasing size of the target linear system.
Figure 2: Computation times of sequential runs of HMAT-OSS and Chameleon on coupled sparse/dense FEM/BEM linear systems of varying size.
For these experiments, we have considered the precision parameter \(\epsilon\) for the HMAT-OSS solver to be 10-3. In Figure 3, the relative error curve for the runs presented in Figure 2 verifies that the threshold is respected and that the error of the solutions computed by HMAT-OSS is even smaller than \(\epsilon\).
Figure 3: Relative error of sequential runs of HMAT-OSS and Chameleon on coupled sparse/dense FEM/BEM linear systems of varying size.
4.2. Multi-threaded execution
To study the impact of parallel execution on the time to solution, we limit ourselves to the Chameleon solver as HMAT-OSS is sequential-only. In Figure 4, we comapre the computation times of Chameleon on coupled FEM/BEM systems of different sizes using either one or four threads. According to the results, we can observe a significant decrease in computation time in case of parallel executions. Moreover the parallel efficiency of the run on the largest linear system considered (8000 unknowns) is approximately 79%.
Figure 4: Computation times of sequential and parallel runs of Chameleon on coupled sparse/dense FEM/BEM linear systems of varying size.
5. Conclusion
We have evaluated the performance of the solvers branched to the
test_FEMBEM test suite on coupled sparse/dense FEM/BEM linear systems.
The solvers considered were HMAT-OSS, a sequential compressed dense direct
solver and Chameleon, a multi-threaded non-compressed dense direct solver.
The comparison of sequential runs of HMAT-OSS and Chameleon showed an important positive impact of data compression on the time to solution. In addition, the comparison of sequential and parallel runs of Chameleon as well as the computed parallel efficiency showed a considerable speed-up of the parallel execution.
6. Notes on reproducibility
With the aim of keeping the experimental environment of the study reproducible,
we manage the associated software framework with the GNU Guix transactional
package manager guix. Moreover, relying on the principles of literate
programming Knuth84, we provide a full documentation on the construction
process of the experimental environment, the execution of benchmarks, the
collection and the visualization of results as well as on producing the final
manuscripts in a dedicated technical report associated with this study
RT-EXAMPLE. A public companion contains all of the source code, guidelines
and other material required for reproducing the study:
https://gitlab.inria.fr/tuto-techno-guix-hpc/test_fembem/advanced-setup,
archived on https://archive.softwareheritage.org/ under the identifier
swh:1:snp:79f450e0f43828f56f261d81b3e86aaab18362eb.
Bibliography
- [testFEMBEM] @misctestFEMBEM, title = test\_FEMBEM, a simple application for testing dense and sparse solvers with pseudo-FEM or pseudo-BEM matrices , howpublished = \urlhttps://gitlab.inria.fr/solverstack/test_fembem
- [hmat-oss] @mischmat-oss, title = hmat-oss, howpublished = \urlhttps://github.com/jeromerobert/hmat-oss
- [Lize14] @phdthesisLize14, title = Résolution Directe Rapide pour les Éléments Finis de Frontière en Électromagnétisme et Acoustique : \(\mathcalH\)-Matrices. Parallélisme et Applications Industrielles. , author = Benoît Lizé, year = 2014, school = Université Paris 13
- [chameleon] @miscchameleon, title = Chameleon, a dense linear algebra software for heterogeneous architectures , howpublished = \urlhttps://gitlab.inria.fr/solverstack/chameleon
- [guix] @miscguix, title = GNU Guix software distribution and transactional package manager, howpublished = \urlhttps://guix.gnu.org
- [Knuth84] Knuth, Literate Programming, Comput. J., 27(2), 97–111 (1984). link. doi.
- [RT-EXAMPLE] Fel\v s\"oci, Solvers for coupled sparse/dense FEM/BEM linear systems: guidelines for reproducing the study , Inria Bordeaux Sud-Ouest, (3014).