Missions

Solving systems of linear equations is an integral part of many applications in high-performance computing (HPC). Multigrid methods are widely used for this purpose. They compute an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. This post-doc will study the effects of approximate coarsest-level solves on the convergence and performance of multigrid methods. In particular, she/he will focus on designing effective, computable stopping criteria for iterative coarsest-level solvers tailored to multigrid methods. The objective of this post-doc is to provide new theoretical results, implement the designed methods in state of the art HPC linear algebra libraries, and evaluate their performance on modern supercomputers.

Activités

The choice and configuration of the coarsest-level solver in multigrid methods can significantly affect the overall performance. The typical approach is to use a direct solver based on LU or Cholesky factorization. There are, however, settings where using approximate solvers, such as (preconditioned) Krylov subspace methods or block low-rank direct approximate solvers leads to better performance. Achieving good performance with these solvers requires finding the right balance: their accuracy must be sufficient not to slow down the overall convergence, while the computational cost remains low. This post-doc will aim to achieve this balance by designing effective computable stopping criteria for iterative coarsest-level solvers tailored to multigrid methods. In particular, the recruited person will focus on the preconditioned conjugate gradient method (PCG) as the coarsest-level solver stopped using a criterion based on the approximation of the energy norm of the error.

Planned actions
- Numerical analysis of multigrid methods with relative coarsest-level stopping criteria.
- Designed of coarsest-level stopping criteria for PCG based on the approximation of the energy of the error.
- Implementation of the stopping criteria in Ginkgo and Hyteg linear algebra libraries and evaluation of the performance of the resulting multigrid methods on supercomputers
- Comparison of multigrid methods with PCG and with direct approximate block low-rank solvers on the coarsest level.

Compétences

-Linear Algebra
- Numerical analysis
- Computer arithmetics
- Parallel algorithms
- Parallel programming (MPI, OpenMP, PGAS...)
- C/C++/Fortran

Contexte de travail

The agent will work within the APO team (Parallel Algorithms and Optimization) of the IRIT laboratory (Toulouse Institute for Research in Computer Science), on the premises of the ENSEEIHT school, in Toulouse. The contract will take place in the context of the PEPR NumPEx Exa-SofT project (https://numpex.org/), more specifically task 4.3 of work-package 4.

Le poste se situe dans un secteur relevant de la protection du potentiel scientifique et technique (PPST), et nécessite donc, conformément à la réglementation, que votre arrivée soit autorisée par l'autorité compétente du MESR.