Peano—A Traversal and Storage Scheme for Octree-Like Adaptive Cartesian Multiscale Grids

Weinzierl, Tobias; Mehl, Miriam

doi:10.1137/100799071

Cited by 81 publications

(117 citation statements)

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“…However, there are a few major differences to other approaches: [55] and [56] both implement the lattice-Boltzmann method on top of a highly optimised octree grid implementation. I.e., the solver itself had to be developed more or less from scratch in a way that is adapted to the grid structure and its specific memory-optimal traversal in the Peano framework [57] or waLBerla [58]. Also, [56] provides a solver on statically adaptive meshes, which allows for much more sophisticated solutions in terms of run-time optimisation per core and on massively parallel hardware.…”

Section: Discussionmentioning

confidence: 99%

“…Even in worst-case tests where we assumed full dynamical adaptivity in every fine grid time step, we observe good performance and scalability. The flexibility and minimal invasiveness of the p4est mesh comes with an acceptable overhead when compared to specialised versions such as [58,57].…”

Section: Discussionmentioning

confidence: 99%

“…Figure 5 shows a twodimensional adaptive quadtree mesh ordered according to the Z-curve together with the cell tree and the linearised representation. In scientific simulation software, the combination of tree-structured grids and space-filling curves has been used in several ways, for example augmented by hashing [41], or for partial differential equation solvers with cache-optimised data administration [42,5]. Octor [43] and Dendro [6] are two examples of parallel octree libraries that have been scaled to 62,000 [44] and 32,000 [45] cores, operating on parent-child pointers and a linearised octant storage, respectively.…”

Section: Parallel Tree-structured Gridsmentioning

confidence: 99%

“…They are constructed by recursively refining mesh elements into a fixed number of equally sized and equally shaped child elements. They are very efficient, in particular in terms of memory requirements and memory access patterns, if represented in a linearised form according to a depth-first traversal of the underlying tree and an ordering of the children prescribed by a space-filling curve [5,6,7]. Depending on the aggressiveness of the encoding, however, this may impede traversal of the mesh or parts thereof in an arbitrary order.…”

Section: Introductionmentioning

confidence: 99%

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Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

Lahnert¹,

Burstedde²,

Holm³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. We present the minimally-invasive exchange of the regular Cartesian grid in the lattice-Boltzmann solver of ESPResSo by a dynamically-adaptive octree grid. Octree grids are favoured by computer scientists over other grid types as they are very memory-efficient. In addition, they represent a natural generalisation of regular Cartesian grids, such that most discretisation details of a regular grid solver can be maintained. Optimised codes, however, require a special tree-oriented grid traversal, which typically conflicts with existing simulation codes using various iterators, some for only parts of the grid, e.g., boundaries. ESPResSo is a large software package developed for soft-matter simulations involving fluid flow, electrostatic, and electrokinetic effects, and molecular dynamics. The currently used regular Cartesian grid hinders the simulation of realistic domain sizes and significant time periods, a problem that can be solved using grid adaptivity. In a first step, we focus on the lattice-Boltzmann flow solver in ESPResSo.p4est is a grid framework, that already provides dynamically adaptive quadtree and octree grids together with high-level interfaces for flexible grid traversals with direct neighbour access in all grid components. In this paper, we first describe extensions of p4est that were necessary to fulfill certain application requirements. The second part of our work consists of the minimally-invasive changes in ESPResSo preserving the expertise accumulated in the software's implementation over years. Our numerical results demonstrate physical correctness of the implementation, good parallel scalability and low overhead of the dynamical grid adaptivity. These are prerequisites to actually profit from grid adaptivity in terms of being able to simulate larger domains over longer time periods with limited computational resources. Thus, the current status forms a solid basis for further steps such as the development of refinement criteria, the setup of more realistic application scenarios, and a GPU implementation.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Parallel Tree-structured Gridsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

Lahnert¹,

Burstedde²,

Holm³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

Self Cite

View full text Add to dashboard Cite

show abstract

“…Within one code base, we may choose a stable solver depending on the equations' characteristics. It combines the idea of additive multigrid [Bastian et al 1998] with full approximation storage (FAS) based upon a hierarchical generating system [Griebel 1990] which resembles the MLAT idea [Brandt 1973;1977], and it realises all data structures within a p-dimensional spacetree traversed by a depth-first search automaton [Weinzierl 2009;Weinzierl and Mehl 2011]. Our Helmholtz solver supports dynamically adaptive meshes resolving localised wave characteristics.…”

Section: Introductionmentioning

confidence: 99%

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Reps

Weinzierl

2017

ACM Trans. Math. Softw.

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We introduce a family of implementations of low order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based upon the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free.Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realise full approximation storage (FAS) within the additive environment where, amortised, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realise a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as enabler for full multigrid (FMG) cycling-the grid unfolds throughout the computation-allows us to reduce the cost per unknown per order of accuracy.The present paper primary contributes towards software realisation and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR and vectorisation-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.

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Delayed approximate matrix assembly in multigrid with dynamic precisions

Murray

Weinzierl

2020

Concurrency and Computation

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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.

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Peano—A Traversal and Storage Scheme for Octree-Like Adaptive Cartesian Multiscale Grids

Cited by 81 publications

References 4 publications

Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Delayed approximate matrix assembly in multigrid with dynamic precisions

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