Optimizations for quadrature representations of finite element tensors through automated code generation

Ølgaard, Kristian Breum; Wells, Garth N.

doi:10.1145/1644001.1644009

Cited by 102 publications

(86 citation statements)

References 15 publications

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“…It includes several components such as DOLFIN (Logg and Wells, 2010;Logg et al, 2012c), FFC (Kirby and Logg, 2006;Logg et al, 2012b;Ølgaard and Wells, 2010) and FIAT (Kirby, 2004(Kirby, , 2012, which enable automatic solution of linear and non-linear problems once the variational forms of the PDEs are expressed in the Unified Form Language (UFL; Alnaes et al, 2014;Alnaes, 2012).…”

Section: Fenics Softwarementioning

confidence: 99%

Modelling debris transport within glaciers by advection in a full-Stokes ice flow model

Wirbel¹,

Jarosch²,

Nicholson³

2018

The Cryosphere

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Abstract. Glaciers with extensive surface debris cover respond differently to climate forcing than those without supraglacial debris. In order to include debris-covered glaciers in projections of glaciogenic runoff and sea level rise and to understand the paleoclimate proxy recorded by such glaciers, it is necessary to understand the manner and timescales over which a supraglacial debris cover develops. Because debris is delivered to the glacier by processes that are heterogeneous in space and time, and these debris inclusions are altered during englacial transport through the glacier system, correctly determining where, when and how much debris is delivered to the glacier surface requires knowledge of englacial transport pathways and deformation. To achieve this, we present a model of englacial debris transport in which we couple an advection scheme to a full-Stokes ice flow model. The model performs well in numerical benchmark tests, and we present both 2-D and 3-D glacier test cases that, for a set of prescribed debris inputs, reproduce the englacial features, deformation thereof and patterns of surface emergence predicted by theory and observations of structural glaciology. In a future step, coupling this model to (i) a debris-aware surface mass balance scheme and (ii) a supraglacial debris transport scheme will enable the co-evolution of debris cover and glacier geometry to be modelled.

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Section: Fenics Softwarementioning

confidence: 99%

Modelling debris transport within glaciers by advection in a full-Stokes ice flow model

Wirbel¹,

Jarosch²,

Nicholson³

2018

The Cryosphere

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“…As the mathematical structure of the PDE is available for analysis, important equation-specific optimisations may be performed that low level compilers cannot automate, or that would be too laborious to implement by hand [7,8]. UFL is remarkably compact: a finite element discretisation that might take thousands or tens of thousands of lines of FORTRAN or C++ code to implement can be cleanly expressed in just a handful of lines of UFL.…”

Section: Automated Code Generation In Computational Sciencementioning

confidence: 99%

Rapid development and adjoining of transient finite element models

Maddison

Farrell

2014

Computer Methods in Applied Mechanics and Engineering

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Recent advances in high level finite element systems have allowed for the symbolic representation of discretisations and their efficient automated implementation as model source code. This allows for the extremely compact implementation of complex non-linear models in a handful of lines of high level code. In this work we extend the high level finite element FEniCS system to introduce an abstract representation of the temporal discretisation: this enables the similarly rapid development of transient finite element models. Efficiency is achieved via aggressive optimisations that exploit the temporal structure, such as automated pre-assembly and caching of forms, and the robust re-use of matrix factorisations and preconditioner data. The resulting models are as fast or faster than hand-optimised finite element codes. The high level representation of the system remains extremely compact and easily manipulated. This structure is exploited to derive the associated discrete adjoint model automatically, with the adjoint model inheriting the performance advantages of the forward model. Combined, this provides a system for the rapid development of efficient transient models, together with their discrete adjoints.

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“…The FEniCS form compiler also performs other optimizations for for quadrature based code generation [Ølgaard and Wells 2010]. Firstly, if a basis function or its derivatives evaluate to zero at all quadrature points, operations on those values can be eliminated.…”

Section: Quadraturementioning

confidence: 99%

“…Work by Ølgaard and Wells [Ølgaard and Wells 2010] has shown that for some variational forms, the tensor contraction representation of assembly outperforms quadrature by a factor of 300 in terms of operation count. However, for other forms, quadrature representation can outperform tensor contraction by a factor of 100.…”

Section: Introductionmentioning

confidence: 99%

Optimized code generation for finite element local assembly using symbolic manipulation

Russell

2013

ACM Trans. Math. Softw.

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Automated code generators for finite element local assembly have facilitated exploration of alternative implementation strategies within generated code. However, even for a theoretical performance indicator such as operation count, an optimal strategy for local assembly is unknown. We explore a code generation strategy based on symbolic integration and polynomial common sub-expression elimination (CSE). We present our implementation of a local assembly code generator using these techniques. We systematically evaluate the approach, measuring operation count, execution time and numerical error using a benchmark suite of synthetic variational forms, comparing against the FEniCS Form Compiler (FFC). Our benchmark forms span complexities chosen to expose the performance characteristics of different code generation approaches. We show that it is possible with additional computational cost, to consistently achieve much of, and sometimes substantially exceed, the performance of alternative approaches without compromising precision. Although the approach of using symbolic integration and CSE for optimizing local assembly is not new, we distinguish our work through our strategies for maintaining numerical precision and detecting common sub-expressions. We discuss the benefits of the symbolic approach for inferring numerical relationships, and analyze the relationship to other proposed techniques which also have greater computational complexity than those of FFC.

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Optimizations for quadrature representations of finite element tensors through automated code generation

Cited by 102 publications

References 15 publications

Modelling debris transport within glaciers by advection in a full-Stokes ice flow model

Modelling debris transport within glaciers by advection in a full-Stokes ice flow model

Rapid development and adjoining of transient finite element models

Optimized code generation for finite element local assembly using symbolic manipulation

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