Adaptive GDSW Coarse Spaces of Reduced Dimension for Overlapping Schwarz Methods

Heinlein, Alexander; Klawonn, Axel; Knepper, Jascha; Rheinbach, Oliver; Widlund, Olof B.

doi:10.1137/20m1364540

Cited by 8 publications

(3 citation statements)

References 36 publications

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“…(4.4). Di↵erent scaling operators D i lead to di↵erent variants of RGDSW coarse spaces, e.g., Options 1, 2.1, and 2.2, introduced in [16] and another variant introduced in [25]. Here, we will only consider the algebraic variant, Option 1, where an inverse multiplicity scaling…”

Section: Preconditionersmentioning

confidence: 99%

FROSch Preconditioners for Land Ice Simulations of Greenland and Antarctica

Heinlein¹,

Perego²,

Rajamanickam³

2022

SIAM J. Sci. Comput.

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Numerical simulations of Greenland and Antarctic ice sheets involve the solution of large-scale highly nonlinear systems of equations on complex shallow geometries. This work is concerned with the construction of Schwarz preconditioners for the solution of the associated tangent problems, which are challenging for solvers mainly because of the strong anisotropy of the meshes and wildly changing boundary conditions that can lead to poorly constrained problems on large portions of the domain. Here, two-level GDSW (Generalized Dryja-Smith-Widlund) type Schwarz preconditioners are applied to di↵erent land ice problems, i.e., a velocity problem, a temperature problem, as well as the coupling of the former two problems. We employ the MPI-parallel implementation of multi-level Schwarz preconditioners provided by the package FROSch (Fast and Robust Schwarz) from the Trilinos library. The strength of the proposed preconditioner is that it yields out-of-the-box scalable and robust preconditioners for the single physics problems.To our knowledge, this is the first time two-level Schwarz preconditioners are applied to the ice sheet problem and a scalable preconditioner has been used for the coupled problem. The preconditioner for the coupled problem di↵ers from previous monolithic GDSW preconditioners in the sense that decoupled extension operators are used to compute the values in the interior of the subdomains. Several approaches for improving the performance, such as reuse strategies and shared memory OpenMP parallelization, are explored as well.In our numerical study we target both uniform meshes of varying resolution for the Antarctic ice sheet as well as non uniform meshes for the Greenland ice sheet are considered. We present several weak and strong scaling studies confirming the robustness of the approach and the parallel scalability of the FROSch implementation. Among the highlights of the numerical results are a weak scaling study for up to 32 K processor cores (8 K MPI-ranks and 4 OpenMP threads) and 566 M degrees of freedom for the velocity problem as well as a strong scaling study for up to 4 K processor cores (and MPI-ranks) and 68 M degrees of freedom for the coupled problem.

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Section: Preconditionersmentioning

confidence: 99%

FROSch Preconditioners for Land Ice Simulations of Greenland and Antarctica

Heinlein¹,

Perego²,

Rajamanickam³

2022

SIAM J. Sci. Comput.

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“…In this paper, we will always apply the preconditioners from FROSch in fully algebraic mode; this implies that no explicit information on the block structure is provided in the construction of the preconditioner. Note that recent RGDSW methods with adaptive coarse spaces are not fully algebraic; e.g., [40] and cannot be used here.…”

Section: Introductionmentioning

confidence: 99%

Monolithic parallel overlapping Schwarz methods in fully-coupled nonlinear chemo-mechanics problems

et al. 2023

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We consider the swelling of hydrogels as an example of a chemo-mechanical problem with strong coupling between the mechanical balance relations and the mass diffusion. The problem is cast into a minimization formulation using a time-explicit approach for the dependency of the dissipation potential on the deformation and the swelling volume fraction to obtain symmetric matrices, which are typically better suited for iterative solvers. The MPI-parallel implementation uses the software libraries deal.II, p4est and FROSch (Fast of Robust Overlapping Schwarz). FROSch is part of the Trilinos library and is used in fully algebraic mode, i.e., the preconditioner is constructed from the monolithic system matrix without making explicit use of the problem structure. Strong and weak parallel scalability is studied using up to 512 cores, considering the standard GDSW (Generalized Dryja-Smith-Widlund) coarse space and the newer coarse space with reduced dimension. The FROSch solver is applicable to the coupled problems within in the range of processor cores considered here, although numerical scalablity cannot be expected (and is not observed) for the fully algebraic mode. In our strong scalability study, the average number of Krylov iterations per Newton iteration is higher by a factor of up to six compared to a linear elasticity problem. However, making mild use of the problem structure in the preconditioner, this number can be reduced to a factor of two and, importantly, also numerical scalability can then be achieved experimentally. Nevertheless, the fully algebraic mode is still preferable since a faster time to solution is achieved.

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“…The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner [37] with an energy-minimizing coarse space and wellunderstood convergence for a wide range of model problems [6,7,9,20]. Moreover, there has been recent development regarding adaptive GDSW coarse spaces [14], which are related to the GenEO (Generalized Eigenvalues in the Overlaps) coarse spaces [35,36]. An important feature of the GDSW coarse space is that it can be constructed algebraically from the fully assembled stiffness matrix; in particular, it does neither require a coarse triangulation nor Neumann matrices for the subdomains.…”

Section: Introductionmentioning

confidence: 99%

A Multilevel Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions

Heinlein

Rheinbach

Röver

2023

Computational Methods in Applied Mathematics

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Multilevel extensions of overlapping Schwarz domain decomposition preconditioners of Generalized Dryja–Smith–Widlund (GDSW) type are considered in this paper. The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner, which can be constructed algebraically from the fully assembled stiffness matrix. The FROSch software, which belongs to the ShyLU package of the Trilinos software library, provides parallel implementations of different variants of GDSW preconditioners. The coarse problem can limit the parallel scalability of two-level GDSW preconditioners. As a remedy, in the past, three-level GDSW approaches have been proposed, which can significantly extend the range of scalability. Here, a multilevel extension of the GDSW preconditioner is introduced and analyzed. Finally, parallel results for the implementation in FROSch for up to 40 000 cores of the SuperMUC-NG supercomputer at Leibniz Supercomputing Centre (LRZ) and to 48 000 cores of the JUWELS supercomputer at Jülich Supercomputing Centre (JSC) are presented.

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Adaptive GDSW Coarse Spaces of Reduced Dimension for Overlapping Schwarz Methods

Cited by 8 publications

References 36 publications

FROSch Preconditioners for Land Ice Simulations of Greenland and Antarctica

FROSch Preconditioners for Land Ice Simulations of Greenland and Antarctica

Monolithic parallel overlapping Schwarz methods in fully-coupled nonlinear chemo-mechanics problems

A Multilevel Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions

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