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Simulating the dynamics of discretized interacting structures whose relationship is dictated by a kernel function gives rise to a large dense matrix. We propose a multigrid solver for such a matrix that exploits not only its data-sparsity resulting from the decay of the kernel function but also the regularity of the geometry of the structures and the quantities of interest distributed on them. Like the well-known multigrid method for large sparse matrices arising from boundary-value problems, our method requires a smoother for removing high-frequency terms in solution errors, a strategy for coarsening a grid, and a pair of transfer operators for exchanging information between two grids. We develop new techniques for these processes that are tailored to a kernel function acting on discretized interacting structures. They are matrix-free in the sense that there is no need to construct the large dense matrix. Numerical experiments on a variety of bio-inspired microswimmers immersed in a Stokes flow demonstrate the effectiveness and efficiency of the proposed multigrid solver. In the case of free swimmers that must maintain force and torque balance, additional sparse rows and columns need to be appended to the dense matrix above. We develop a matrix-free fast solver for this bordered matrix as well, in which the multigrid method is a key component.
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Matrix-free SBP-SAT finite difference methods and the multigrid preconditioner on GPUs
Summation-by-parts (SBP) finite difference methods are widely used in scientific applications alongside a special treatment of boundary conditions through the simultaneous-approximate-term (SAT) technique which enables the valuable proof of numerical stability. Our work is motivated by multi-scale earthquake cycle simulations described by partial differential equations (PDEs) whose discretizations lead to huge systems of equations and often rely on iterative schemes and parallel implementations to make the nu- merical solutions tractable. In this study, we consider 2D, variable coefficient elliptic PDEs in complex geometries discretized with the SBP-SAT method. The multigrid method is a well-known, efficient solver or preconditioner for traditional numerical discretizations, but they have not been well-developed for SBP-SAT methods on HPC platforms. We propose a custom geometric-multigrid pre- conditioned conjugate-gradient (MGCG) method that applies SBP- preserving interpolations. We then present novel, matrix-free GPU kernels designed specifically for SBP operators whose differences from traditional methods make this task nontrivial but that perform 3× faster than SpMV while requiring only a fraction of memory. The matrix-free GPU implementation of our MGCG method per- forms 5× faster than the SpMV counterpart for the largest problems considered (67 million degrees of freedom). When compared to off- the-shelf solvers in the state-of-the-art libraries PETSc and AmgX, our implementation achieves superior performance in both itera- tions and overall runtime. The method presented in this work offers an attractive solver for simulations using the SBP-SAT method.
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- PAR ID:
- 10530614
- Publisher / Repository:
- ACM
- Date Published:
- ISBN:
- 9798400706103
- Page Range / eLocation ID:
- 400 to 412
- Format(s):
- Medium: X
- Location:
- Kyoto Japan
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3650200.3656614
