Quasi-matrix-free Hybrid Multigrid on Dynamically Adaptive Cartesian Grids

ACM Trans. Math. Softw.

2019

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We discuss the design decisions, design alternatives and rationale behind the third generation of Peano, a framework for dynamically adaptive Cartesian meshes derived from spacetrees. Peano ties the mesh traversal to the mesh storage and supports only one element-wise traversal order resulting from space-filling curves. The user is not free to choose a traversal order herself. The traversal can exploit regular grid subregions and shared memory as well as distributed memory systems with almost no modifications to a serial application code. We formalize the software design by means of two interacting automata-one automaton for the multiscale grid traversal and one for the applicationspecific algorithmic steps. This yields a callback-based programming paradigm. We further sketch the supported application types and the two data storage schemes realized, before we detail high-performance computing aspects and lessons learned. Special emphasis is put on observations regarding the used programming idioms and algorithmic concepts. This transforms our report from a "one way to implement things" code description into a generic discussion and summary of some alternatives, rationale and design decisions to be made for any tree-based adaptive mesh refinement software.

Section: Observationmentioning

confidence: 99%

Section: Supported Application Typesmentioning

confidence: 99%

Section: A Software Basementioning

confidence: 99%

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The Peano Software—Parallel, Automaton-based, Dynamically Adaptive Grid Traversals

ACM Trans. Math. Softw.

2019

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“…F I G U R E 1 Multigrid implementation challenges: (1) An implementation has to be correct, that is, yield the result of the underlying mathematics and thus be consistent with it, (2) there has to be minimal data structures, while the cost per cycle (3), the assembly/setup cost (4) and the memory footprint (5) have to be small, too.…”

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confidence: 99%

“…(3),(4), (5) are the core areas where we make a contribution, while implications on (1) and (2) are studied Dirichlet conditions here. A Ritz-Galerkin finite element discretization over a mesh Ω h that geometrically discretizes Ω yields an equation system A h u h = f h .…”

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confidence: 99%

Delayed approximate matrix assembly in multigrid with dynamic precisions

Murray

Concurrency and Computation

2020

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The accurate assembly of the system matrix is an important step in any code that solves partial differential equations on a mesh. We either explicitly set up a matrix, or we work in a matrix-free environment where we have to be able to quickly return matrix entries upon demand. Either way, the construction can become costly due to nontrivial material parameters entering the equations, multigrid codes requiring cascades of matrices that depend upon each other, or dynamic adaptive mesh refinement that necessitates the recomputation of matrix entries or the whole equation system throughout the solve. We propose that these constructions can be performed concurrently with the multigrid cycles. Initial geometric matrices and low accuracy integrations kickstart the multigrid iterations, while improved assembly data is fed to the solver as and when it becomes available. The time to solution is improved as we eliminate an expensive preparation phase traditionally delaying the actual computation. We eliminate algorithmic latency. Furthermore, we desynchronize the assembly from the solution process. This anarchic increase in the concurrency level improves the scalability. Assembly routines are notoriously memory-and bandwidth-demanding. As we work with iteratively improving operator accuracies, we finally propose the use of a hierarchical, lossy compression scheme such that the memory footprint is brought down aggressively where the system matrix entries carry little information or are not yet available with high accuracy. K E Y W O R D S algebraic-geometric multigrid, asynchronous multigrid, delayed operator computation, dynamically adaptive Cartesian grids, finite element assembly, mixed precision computing 1 INTRODUCTION Multigrid algorithms are among the fastest solvers known for elliptic partial differential equations (PDEs) of the type −∇ (∇) u = f (1) on a d-dimensional, well-shaped domain Ω. An approximation of the function u : Ω  → R is what we are searching for with : Ω  → R + as a material parameter, and f : Ω  → R constituting the right-hand side. The system is closed by appropriate boundary conditions. We restrict ourselves to This paper is an extended version of C.D. Murray and T. Weinzierl: Lazy stencil integration in multigrid algorithms as introduced and published at the PPAM'19 Conference. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Stabilized asynchronous fast adaptive composite multigrid using additive damping

Murray

Numerical Linear Algebra App

2020

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Summary Multigrid solvers face multiple challenges on parallel computers. Two fundamental ones read as follows: Multiplicative solvers issue coarse grid solves which exhibit low concurrency and many multigrid implementations suffer from an expensive coarse grid identification phase plus adaptive mesh refinement overhead. We propose a new additive multigrid variant for spacetrees, that is, meshes as they are constructed from octrees and quadtrees: It is an additive scheme, that is, all multigrid resolution levels are updated concurrently. This ensures a high concurrency level, while the transfer operators between the mesh levels can still be constructed algebraically. The novel flavor of the additive scheme is an augmentation of the solver with an additive, auxiliary damping parameter per grid level per vertex that is in turn constructed through the next coarser level—an idea which utilizes smoothed aggregation principles or the motivation behind AFACx: Per level, we solve an additional equation whose purpose is to damp too aggressive solution updates per vertex which would otherwise, in combination with all the other levels, yield an overcorrection and, eventually, oscillations. This additional equation is constructed additively as well, that is, is once more solved concurrently to all other equations. This yields improved stability, closer to what is seen with multiplicative schemes, while pipelining techniques help us to write down the additive solver with single‐touch semantics for dynamically adaptive meshes.