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2 years Postdoc project: Grammar-guided multigrid solver optimization for the Stokes equations

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Required Education : PhD
Mission duration : 2 ans

2 years Postdoc project: Grammar-guided multigrid solver optimization for the Stokes equations

The solution of large sparse linear systems is one of the core topics in numerical linear algebra and an active research area in academia and industry. For new and existing applications, the solver usually has to be developed with care, including for example preconditioning techniques. The goal nowadays is to reach large-scale computations. One candidate of solution algorithms to reach this goal are multigrid methods. However, an exponential number of parameter combinations (number of levels, relaxation steps, V or W cycle, …) are possible.  Trial and error to find a first idea for an adequate solver becomes thus tedious or even unfeasible. In general, deep domain knowledge is required to find a performant solver for a given problem. But how would it be if an automatic optimization of a solver was possible, that could be used as a first step and then further adapted by the user?

This project aims to develop concepts and a toolchain based on artificial intelligence to find the ‘best' preconditioner for a Krylov subspace iterative solver for the solution of the linear system coming from a discretized Stokes problem. We will focus in particular on a geometric multigrid preconditioner for the (1,1)-block, here, the Laplace operator, to ensure the scalability of the solver when passing to large-scale problems. We will extend the work in the references [1,2], in which a grammar-guided approach for the optimization of multigrid solvers for matrices from a finite difference discretization of the Laplacian operator is formulated. The particular difficulty for the extension to saddle point systems is the large number of possible combinations of solution techniques. These include, for example, preconditioned Krylov subspace solvers that make use of the block structure of the system, Schur complement approaches, deflation and augmentation techniques, or also monolithic multigrid solvers on the whole block system.

Required profile: PhD including one or several of the following domains: applied mathematics, computer science, iterative solvers, numerical linear algebra, artificial intelligence, high performance computing

Starting date: As soon as possible.

Contract length: 24 months


Carola Kruse (carola.kruse@cerfacs.fr)

Gabriel Staffelbach( gabriel.staffelbach@cerfacs.fr)

[1] J. Schmitt, S. Kuckuk, H. Koestler, Constructing Efficient Multigrid Solvers with Genetic Programming, In Proceedings of the 2020 Genetic and Evolutionary Computation Conference, GECCO '20, New York, NY, USA, 2020, Association for Computing Machinery, p.1012–1020.