Parallelisation of Multiscale-Based Grid Adaptation using Space-Filling Curves

Brix, Kolja; Melian, Silvia Sorana; Müller, Siegfried; Schieffer, Gero

doi:10.1051/proc/2009058

Cited by 41 publications

(22 citation statements)

References 33 publications

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“…Here the position of a cell along a space-filling curve is determined from the path of the cell using a finite state machine [18,24]. This has been applied for logically Cartesian grids in [36] and [10]. One of our targets is to generalize this in the future to triangular, tetrahedral, and hybrid grids.…”

Section: Id Bitstringsmentioning

confidence: 99%

Refinement and Connectivity Algorithms for Adaptive Discontinuous Galerkin Methods

Brix¹,

Massjung²,

Voß³

2011

SIAM J. Sci. Comput.

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Adaptive multiscale methods are among the many effective techniques for the numerical solution of partial differential equations. Efficient grid management is an important task in these solvers. In this paper we focus on this problem for discontinuous Galerkin discretization methods in 2 and 3 spatial dimensions and present a data structure for handling adaptive grids of different cell types in a unified approach. Instead of tree-based techniques where connectivity is stored via pointers, we associate each cell that arises in the refinement hierarchy with a cell identifier and construct algorithms that establish hierarchical and spatial connectivity. By means of bitwise operations, the complexity of the connectivity algorithms can be bounded independent of the level. The grid is represented by a hash table which results in a low-memory data structure and ensures fast access to cell data. The spatial connectivity algorithm also supports the application of quadrature rules for face integrals that occur in discontinuous Galerkin discretizations. The concept allows us to implement discontinuous Galerkin methods largely independent of spatial dimension and cell type. We demonstrate this by outlining how typical algorithmic tasks that arise in these implementations can be performed with our data structure. In computational tests we compare our approach with that of a classical implementation using pointers.

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Section: Id Bitstringsmentioning

confidence: 99%

Refinement and Connectivity Algorithms for Adaptive Discontinuous Galerkin Methods

Brix¹,

Massjung²,

Voß³

2011

SIAM J. Sci. Comput.

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“…We also denote by Y ∆t (U 0 ) the numerical solution of the reaction part, 4) with initial data U R (0) = U 0 . The two Lie approximation formulae of the solution of system (2.2) are then defined by 5) whereas the two Strang approximation formulae [32,33] are given by…”

Section: Time Operator Splittingmentioning

confidence: 99%

“…Therefore, an important amount of work is still in progress concerning programming features such as data structures, optimized routines and parallelization strategies for the time integration technique as well as for the multiresolution environment, even though the global CPU time is largely dominated by the time resolution for these stiff problems. For instance, some dedicated and efficient implementations have been recently developed for multiresolution applications [3,4]. Finally, when dealing with more complex systems such as complex or more detailed chemistry or stroke modeling in the brain, the source term involves many species (typically 50) and many reactions (typically several hundreds) or complex mechanisms.…”

Section: Concluding Remarks and Outlookmentioning

confidence: 99%

“…In all these illustrations, the finest grid of computation is previously settled and 4 It is important to notice that ROCK4 needs to save only 8 arrays of the size of the number of unknowns regardless the number of stages. One of these arrays contains the approximate solution used to estimate the local error in order to adapt the time step of integration within the prescribed tolerance.…”

Section: D Bz Equationmentioning

confidence: 99%

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New Resolution Strategy for Multiscale Reaction Waves using Time Operator Splitting, Space Adaptive Multiresolution, and Dedicated High Order Implicit/Explicit Time Integrators

Duarte¹,

Massot²,

Descombes³

et al. 2012

SIAM J. Sci. Comput.

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Abstract. We tackle the numerical simulation of reaction-diffusion equations modeling multiscale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. It considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is not restricted neither by the fastest scales in the source term nor by stability constraints of the diffusive steps, but only by the physics of the phenomenon. We aim thus at solving complete models including all time and space scales within a prescribed accuracy, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far, out of reach of standard methods in the field of nonlinear chemical dynamics for 2D spiral waves and 3D scroll waves as an illustration. Future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are finally discussed. Key words.Reaction-diffusion equations, multi-scale reaction waves, operator splitting, adaptive multiresolution AMS subject classifications. 33K57, 35A18, 65M50, 65M081. Introduction. Numerical simulations of multi-scale phenomena are commonly used for modeling purposes in many applications such as combustion, chemical vapor deposition, or air pollution modeling. In general, all these models raise several difficulties created by the high number of unknowns, the wide range of temporal scales due to large and detailed chemical kinetic mechanisms, as well as steep spatial gradients associated with very localized fronts of high chemical activity. Furthermore, a natural stumbling block to perform 3D simulations with all scales resolution is either the unreasonably small time step due to stability requirements or the unreasonable memory requirements for implicit methods. In this context, one can consider various numerical strategies in order to treat the induced stiffness for time dependent * This research was supported by a fundamental project grant from ANR (French National Research Agency -ANR Blancs) Séchelles (project leader S. Descombes -2009Descombes - -2013, by a CNRS PEPS Maths-ST2I project MIPAC (project leader V. Louvet -2009Louvet - -2010, and by a DIGITEO RTRA project MUSE (project leader M. Massot -2010Massot - -2014 problems. The most natural id...

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“…[28] extends the use of a multiscale analysis for grid adaptation to incorporate locally varying time stepping. Space filling curves are also used to transform multidimensional data for better parallelization of multiscale adaptation methods [29]. The differences between our approach and these works is the manipulation of the simulation space since they modify the shape of the simulation space at runtime.…”

Section: Comparison Against Loop Perforationmentioning

confidence: 99%

Adaptive Code Refinement: A Compiler Technique and Extensions to Generate Self-Tuning Applications

Schmitt

Helluy

Bastoul

2017

2017 IEEE 24th International Conference on High Performance Computing (HiPC)

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Abstract-Compiler high-level automatic optimization and parallelization techniques are well suited for some classes of simulation or signal processing applications, however they usually don't take into account domain-specific knowledge nor the possibility to change or to remove some computations to achieve "good enough" results. Differently, production simulation and signal processing codes have adaptive capabilities: they are designed to compute precise results only where it matters if the complete problem is not tractable or if computation time must be short. In this paper, we present a new way to provide adaptive capabilities to compute-intensive codes automatically. It relies on domain-specific knowledge provided through special pragmas by the programmer in the input code and on polyhedral compilation techniques to continuously regenerate at runtime a code that performs heavy computations only where it matters. We present experimental results on several applications where our strategy enables significant computation savings and speedup while maintaining a good precision, with a minimal effort from the programmer.

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Parallelisation of Multiscale-Based Grid Adaptation using Space-Filling Curves

Cited by 41 publications

References 33 publications

Refinement and Connectivity Algorithms for Adaptive Discontinuous Galerkin Methods

Refinement and Connectivity Algorithms for Adaptive Discontinuous Galerkin Methods

New Resolution Strategy for Multiscale Reaction Waves using Time Operator Splitting, Space Adaptive Multiresolution, and Dedicated High Order Implicit/Explicit Time Integrators

Adaptive Code Refinement: A Compiler Technique and Extensions to Generate Self-Tuning Applications

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