Imperial College London

Dr Lawrence Mitchell

Faculty of EngineeringDepartment of Computing

Honorary Research Fellow
 
 
 
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Contact

 

+44 (0)20 7594 8399lawrence.mitchell Website

 
 
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Location

 

William Penney LaboratorySouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
to

51 results found

Ham DA, Kelly PHJ, Mitchell L, Cotter C, Kirby RC, Sagiyama K, Bouziani N, Vorderwuelbecke S, Gregory T, Betteridge J, Shapero DR, Nixon-Hill R, Ward C, Farrell PE, Brubeck PD, Marsden I, Gibson TH, Homolya M, Sun T, McRae ATT, Luporini F, Gregory A, Lange M, Funke SW, Rathgeber F, Bercea G-T, Markall GRet al., 2023, Firedrake user manual, Firedrake User Manual

Report

Farrell PE, Mitchell L, Scott LR, Wechsung Fet al., 2022, Robust multigrid methods for nearly incompressible elasticity using macro elements, IMA Journal of Numerical Analysis, Vol: 42, Pages: 3306-3329, ISSN: 0272-4979

We present a mesh-independent and parameter-robust multigrid solver for the Scott–Vogelius discretisation of the nearly incompressible linear elasticity equations on meshes with a macro element structure. The discretisation achieves exact representation of the limiting divergence constraint at moderate polynomial degree. Both the relaxation and multigrid transfer operators exploit the macro structure for robustness and efficiency. For the relaxation we use the existence of local Fortin operators on each macro cell to construct a local space decomposition with parameter-robust convergence. For the transfer we construct a robust prolongation operator by performing small local solves over each coarse macro cell. The necessity of both components of the algorithm is confirmed by numerical experiments.

Journal article

Crum J, Cheng C, Ham DA, Mitchell L, Kirby RC, Levine JA, Gillette Aet al., 2022, Bringing trimmed serendipity methods to computational practice in Firedrake, ACM Transactions on Mathematical Software, Vol: 48, ISSN: 0098-3500

We present an implementation of the trimmed serendipity finite element family, using the open-source finite element package Firedrake. The new elements can be used seamlessly within the software suite for problems requiring H1, H(curl), or H(div)-conforming elements on meshes of squares or cubes. To test how well trimmed serendipity elements perform in comparison to traditional tensor product elements, we perform a sequence of numerical experiments including the primal Poisson, mixed Poisson, and Maxwell cavity eigenvalue problems. Overall, we find that the trimmed serendipity elements converge, as expected, at the same rate as the respective tensor product elements, while being able to offer significant savings in the time or memory required to solve certain problems.

Journal article

Laakmann F, Farrell PE, Mitchell L, 2022, AN AUGMENTED LAGRANGIAN PRECONDITIONER FOR THE MAGNETOHYDRODYNAMICS EQUATIONS AT HIGH REYNOLDS AND COUPLING NUMBERS, SIAM JOURNAL ON SCIENTIFIC COMPUTING, Vol: 44, Pages: B1018-B1044, ISSN: 1064-8275

Journal article

Farrell PE, Knepley MG, Mitchell L, Wechsung Fet al., 2021, PCPATCH: Software for the Topological Construction of Multigrid Relaxation Methods, ACM Transactions on Mathematical Software, Vol: 47, ISSN: 0098-3500

Effective relaxation methods are necessary for good multigrid convergence. For many equations, standard Jacobi and Gauß-Seidel are inadequate, and more sophisticated space decompositions are required; examples include problems with semidefinite terms or saddle point structure. In this article, we present a unifying software abstraction, PCPATCH, for the topological construction of space decompositions for multigrid relaxation methods. Space decompositions are specified by collecting topological entities in a mesh (such as all vertices or faces) and applying a construction rule (such as taking all degrees of freedom in the cells around each entity). The software is implemented in PETSc and facilitates the elegant expression of a wide range of schemes merely by varying solver options at runtime. In turn, this allows for the very rapid development of fast solvers for difficult problems.

Journal article

Farrell PE, Mitchell L, Scott R, Wechsung Fet al., 2021, A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations, SMAI Journal of Computational Mathematics, Vol: 7, Pages: 75-96

Augmented Lagrangian preconditioners have successfully yielded Reynolds-robust preconditioners for the stationary incompressible Navier-Stokes equations, but only for specific discretizations. The discretizations for which these preconditioners have been designed possess error estimates which depend on the Reynolds number, with the discretization error deteriorating as the Reynolds number is increased. In this paper we present an augmented Lagrangian preconditioner for the Scott-Vogelius discretization on barycentrically-refined meshes. This achieves both Reynolds-robust performance and Reynolds-robust error estimates. A key consideration is the design of a suitable space decomposition that captures the kernel of the grad-div term added to control the Schur complement; the same barycentric refinement that guarantees inf-sup stability also provides a local decomposition of the kernel of the divergence. The robustness of the scheme is confirmed by numerical experiments in two and three dimensions.

Journal article

Sun T, Mitchell L, Kulkarni K, Klockner A, Ham DA, Kelly PHJet al., 2020, A study of vectorization for matrix-free finite element methods, International Journal of High Performance Computing Applications, Vol: 34, Pages: 629-644, ISSN: 1094-3420

Vectorization is increasingly important to achieve high performance on modern hardware with SIMD instructions. Assembly of matrices and vectors in the finite element method, which is characterized by iterating a local assembly kernel over unstructured meshes, poses difficulties to effective vectorization. Maintaining a user-friendly high-level interface with a suitable degree of abstraction while generating efficient, vectorized code for the finite element method is a challenge for numerical software systems and libraries. In this work, we study cross-element vectorization in the finite element framework Firedrake via code transformation and demonstrate the efficacy of such an approach by evaluating a wide range of matrix-free operators spanning different polynomial degrees and discretizations on two recent CPUs using three mainstream compilers. Our experiments show that our approaches for cross-element vectorization achieve 30% of theoretical peak performance for many examples of practical significance, and exceed 50% for cases with high arithmetic intensities, with consistent speed-up over (intra-element) vectorization restricted to the local assembly kernels.

Journal article

Shipton J, Cotter C, Bendall T, Gibson T, Mitchell L, Ham D, Wingate Bet al., 2020, Compatible finite element methods and parallel-in-time schemes for numerical weather prediction.

<jats:p> &amp;lt;div&amp;gt;&amp;lt;div&amp;gt;&amp;lt;div&amp;gt;&amp;lt;p&amp;gt;I will describe Gusto, a dynamical core toolkit built on top of the Fire- drake finite element library; present recent results from a range of test cases and outline our plans for future code development.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;Gusto uses compatible finite element methods, a form of mixed finite element methods (meaning that different finite element spaces are used for different fields) that allow the exact representation of the standard vector calculus identities div-curl=0 and curl-grad=0. The popularity of these methods for numerical weather prediction is due to the flexibility to run on non-orthogonal grid, thus avoiding the communication bottleneck at the poles, while retaining the necessary convergence and wave propagation prop- erties required for accuracy.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;Although the flexibility of the compatible finite element spatial discreti- sation improves the parallel scalability of the model it does not solve the parallel scalability problem inherent in spatial domain decomposition: we need to find a way to perform parallel calculations in the time domain. Ex- ponential integrators, approximated by a near optimal rational expansion, offer a way to take large timesteps and form the basis for parallel timestep- ping schemes based on wave averaging. I will describe the progress we have made towards implementing these schemes in Gusto.&amp;lt;/p&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt; </jats:p>

Journal article

Gibson T, Mitchell L, Ham D, Cotter Cet al., 2020, Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond, Geoscientific Model Development, Vol: 13, Pages: 735-761, ISSN: 1991-959X

Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have becomeincreasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle-point andelliptic systems arising from these discretizations continue to be an on-going challenge. One possible approach for addressingthis issue is to employ a method known as hybridization, where the discrete equations are transformed such that classic staticcondensation and local post-processing methods can be employed. However, it is challenging to implement hybridization as performant parallel code within complex models, whilst maintaining separation of concerns between applications scientistsand software experts. In this paper, we introduce a domain-specific abstraction within the Firedrake finite element library thatpermits the rapid execution of these hybridization techniques within a code-generating framework. The resulting frameworkcomposes naturally with Firedrake’s solver environment, allowing for the implementation of hybridization and static condensa-tion as runtime-configurable preconditioners via the Python interface to PETSc, petsc4py. We provide examples derived from second order elliptic problems and geophysical fluid dynamics. In addition, we demonstrate that hybridization shows greatpromise for improving the performance of solvers for mixed finite element discretizations of equations related to large-scalegeophysical flows.

Journal article

Kirby RC, Mitchell L, 2019, Code generation for generally mapped finite elements, ACM Transactions on Mathematical Software, Vol: 45, ISSN: 0098-3500

Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using C1 discretizations.

Journal article

Logg A, Barton SN, Wells GN, Mitchell L, Homolya M, Ham D, Richardson C, Ring J, Blechta J, Rognes ME, Daversin-Catty C, Bergersen A, Wechsung F, Hale JS, Habera M, Li L, Gibson TH, Leoni M, Barton N, Baratta I, Mardal K-A, Yashchuk I, Rathgeber F, Fauske VT, Markall G, Kirby R, Cotter CJ, Gregory Aet al., 2019, firedrakeproject/ufl: The Unified Form Language

This release is specifically created to document the version of ufl used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html

Software

Homolya M, Kirby R, Blechta J, Ham D, Rognes ME, Logg A, Wells GN, Gibson TH, Ring J, Yashchuk I, Mitchell L, Richardson C, Schlömer N, Li L, Bergersen A, Cotter CJ, Hale JS, Knepley M, Rathgeber F, Terrel ARet al., 2019, firedrakeproject/fiat: The Finite Element Automated Tabulator

This release is specifically created to document the version of fiat used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html.

Software

Ham DA, Mitchell L, Paganini A, Wechsung Fet al., 2019, Automated shape differentiation in the Unified Form Language, STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION, Vol: 60, Pages: 1813-1820, ISSN: 1615-147X

Journal article

Gibson TH, Mitchell L, Ham DA, Cotter CJet al., 2019, Slate: extending Firedrake's domain-specific abstraction tohybridized solvers for geoscience and beyond

<jats:p>Abstract. Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have become increasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle-point and elliptic systems arising from these discretizations continue to be an on-going challenge. One possible approach for addressing this issue is to employ a method known as hybridization, where the discrete equations are transformed such that classic static condensation and local post-processing methods can be employed. However, it is challenging to implement hybridization as performant parallel code within complex models, whilst maintaining separation of concerns between applications scientists and software experts. In this paper, we introduce a domain-specific abstraction within the Firedrake finite element library that permits the rapid execution of these hybridization techniques within a code-generating framework. The resulting framework composes naturally with Firedrake's solver environment, allowing for the implementation of hybridization and static condensation as runtime-configurable preconditioners via the Python interface to PETSc, petsc4py. We provide examples derived from second order elliptic problems and geophysical fluid dynamics. In addition, we demonstrate that hybridization shows great promise for improving the performance of solvers for mixed finite element discretizations of equations related to large-scale geophysical flows. </jats:p>

Working paper

Ham D, Homolya M, Lange M, Kirby R, Mitchell Let al., 2019, FInAT/FInAT: a smarter library of finite elements

This release is specifically created to document the version of FInAT used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html

Software

Rathgeber F, Mitchell L, Luporini F, Ham D, Markall G, Homolya M, tj-sun, Barton N, Dearman H, Gibson T, gbts, Wechsung F, Kramer S, Funke SW, Shapero D, Kirby R, Lange M, Sato K, Russell F, Fangyi Zet al., 2019, OP2/PyOP2: Framework for performance-portable parallel computations on unstructured meshes

This release is specifically created to document the version of PyOP2 used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html

Software

Farrell PE, Mitchell L, Wechsung F, 2019, An augmented lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at high Reynolds number, SIAM Journal on Scientific Computing, Vol: 41, Pages: A3073-A3096, ISSN: 1064-8275

In [M. Benzi and M. A. Olshanskii, SIAM J. Sci. Comput., 28 (2006), pp. 2095-2113] a preconditioner of augmented Lagrangian type was presented for the two-dimensional stationary incompressible Navier-Stokes equations that exhibits convergence almost independent of Reynolds number. The algorithm relies on a highly specialized multigrid method involving a custom prolongation operator and for robustness requires the use of piecewise constant finite elements for the pressure. However, the prolongation operator and velocity element used do not directly extend to three dimensions: the local solves necessary in the prolongation operator do not satisfy the inf-sup condition. In this work we generalize the preconditioner to three dimensions, proposing alternative finite elements for the velocity and prolongation operators for which the preconditioner works robustly. The solver is effective at high Reynolds number: on a three-dimensional lid-driven cavity problem with approximately one billion degrees of freedom, the average number of Krylov iterations per Newton step varies from 4.5 at Re = 10 to 3 at Re = 1000 and 5 at Re = 5000.

Journal article

Gibson T, McRae ATT, Cotter C, Mitchell L, Ham Det al., 2019, Compatible finite element methods for geophysical flows: Automation and implementation using Firedrake, Publisher: Springer International Publishing, ISBN: 9783030239565

This book introduces recently developed mixed finite element methods for large-scale geophysical flows that preserve essential numerical properties for accurate simulations. The methods are presented using standard models of atmospheric flows and are implemented using the Firedrake finite element library. Examples guide the reader through problem formulation, discretisation, and automated implementation.The so-called “compatible” finite element methods possess key numerical properties which are crucial for real-world operational weather and climate prediction. The authors summarise the theory and practical implications of these methods for model problems, introducing the reader to the Firedrake package and providing open-source implementations for all the examples covered.Students and researchers with engineering, physics, mathematics, or computer science backgrounds will benefit from this book. Those readers who are less familiar with the topic are provided with an overview of geophysical fluid dynamics.

Book

Ham D, Luporini F, Mitchell L, Homolya M, Gibson TH, Barton N, Rathgeber F, Lange M, Markall G, tj-sun, Blechta J, Russell F, Engwer Cet al., 2018, coneoproject/COFFEE: A Compiler for Fast Expression Evaluation

This release is specifically created to document the version of COFFEE used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html

Software

Mitchell L, Ham D, Gibson T, Rathgeber F, Homolya M, McRae ATT, Fangyi Z, Cotter CJ, Lange M, Andrej J, Luporini F, Kirby R, Wechsung F, Paganini A, Poulin FJ, Roy T, Hawkes C, Salwa T, Kalogirou A, Karna T, Greaves T, tj-sun, Gregory A, Kramer S, Funke SW, Busing H, McBain G, Mueller EHet al., 2018, firedrakeproject/firedrake: an automated finite element system

This release is specifically created to document the version of firedrake used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html

Software

Homolya M, Mitchell L, Barton NS, Ham D, tj-sun, Gisbson TH, Kirby R, Blechta J, Karna Tet al., 2018, firedrakeproject/tsfc: The Two Stage Form Compiler

This release is specifically created to document the version of tsfc used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html

Software

Homolya M, Kirby R, Blechta J, Rognes ME, Ham D, Logg A, Wells GN, Gibson TH, Ring J, Yaschuck I, Schlomer N, Mitchell L, Li L, Bergerson A, Cotter CJ, Richardson C, Knepley M, Hale JS, Rathberger F, Liertzer M, Terrel ARet al., 2018, firedrakeproject/fiat: The Finite Element Automated Tabulator

This release is specifically created to document the version of fiat used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html

Software

Kärnä T, Kramer SC, Mitchell L, Ham DA, Piggott MD, Baptista AMet al., 2018, Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations, Geoscientific Model Development, Vol: 11, Pages: 4359-4382

<jats:p>Abstract. Unstructured grid ocean models are advantageous for simulating the coastal ocean and river–estuary–plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive, which limits their applicability to real-life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability-preserving time integration method and slope limiters. Compared to previous DG models, advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models. </jats:p>

Journal article

Kärnä T, Kramer SC, Mitchell L, Ham DA, Piggott MD, Baptista AMet al., 2018, Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations, Geoscientific Model Development, Vol: 11, Pages: 4359-4382, ISSN: 1991-959X

Unstructured grid ocean models are advantageous for simulating the coastal ocean and river-estuary-plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive which limits their applicability to real life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability preserving time integration method and slope limiters. Compared to previous DG models advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models.

Journal article

Homolya M, Mitchell L, Luporini F, Ham DAet al., 2018, TSFC: a structure-preserving form compiler, SIAM Journal on Scientific Computing, ISSN: 1064-8275

A form compiler takes a high-level description of the weak form of partialdifferential equations and produces low-level code that carries out the finiteelement assembly. In this paper we present the Two-Stage Form Compiler (TSFC),a new form compiler with the main motivation to maintain the structure of theinput expression as long as possible. This facilitates the application ofoptimizations at the highest possible level of abstraction. TSFC features anovel, structure-preserving method for separating the contributions of a formto the subblocks of the local tensor in discontinuous Galerkin problems. Thisenables us to preserve the tensor structure of expressions longer through thecompilation process than other form compilers. This is also achieved in part bya two-stage approach that cleanly separates the lowering of finite elementconstructs to tensor algebra in the first stage, from the scheduling of thosetensor operations in the second stage. TSFC also efficiently traversescomplicated expressions, and experimental evaluation demonstrates goodcompile-time performance even for highly complex forms.

Journal article

Kirby RC, Mitchell L, 2017, Solver composition across the PDE/linear algebra barrier, SIAM Journal on Scientific Computing, Vol: 40, Pages: C76-C98

The efficient solution of discretisations of coupled systems of partialdifferential equations (PDEs) is at the core of much of numerical simulation.Significant effort has been expended on scalable algorithms to preconditionKrylov iterations for the linear systems that arise. With few exceptions, thereported numerical implementation of such solution strategies is specific to aparticular model setup, and intimately ties the solver strategy to thediscretisation and PDE, especially when the preconditioner requires auxiliaryoperators. In this paper, we present recent improvements in the Firedrakefinite element library that allow for straightforward development of thebuilding blocks of extensible, composable preconditioners that decouple thesolver from the model formulation. Our implementation extends the algebraiccomposability of linear solvers offered by the PETSc library by augmentingoperators, and hence preconditioners, with the ability to provide any necessaryauxiliary operators. Rather than specifying up front the full solverconfiguration, tied to the model, solvers can be developed independently ofmodel formulation and configured at runtime. We illustrate with examples fromincompressible fluids and temperature-driven convection.

Journal article

Yamazaki H, Shipton J, Cullen MJP, Mitchell L, Cotter CJet al., 2017, Vertical slice modelling of nonlinear Eady waves using a compatible finite element method, Journal of Computational Physics, Vol: 343, Pages: 130-149, ISSN: 1090-2716

A vertical slice model is developed for the Euler–Boussinesq equations with a constant temperature gradient in the direction normal to the slice (the Eady–Boussinesq model). The model is a solution of the full three-dimensional equations with no variation normal to the slice, which is an idealised problem used to study the formation and subsequent evolution of weather fronts. A compatible finite element method is used to discretise the governing equations. To extend the Charney–Phillips grid staggering in the compatible finite element framework, we use the same node locations for buoyancy as the vertical part of velocity and apply a transport scheme for a partially continuous finite element space. For the time discretisation, we solve the semi-implicit equations together with an explicit strong-stability-preserving Runge–Kutta scheme to all of the advection terms. The model reproduces several quasi-periodic lifecycles of fronts despite the presence of strong discontinuities. An asymptotic limit analysis based on the semi-geostrophic theory shows that the model solutions are converging to a solution in cross-front geostrophic balance. The results are consistent with the previous results using finite difference methods, indicating that the compatible finite element method is performing as well as finite difference methods for this test problem. We observe dissipation of kinetic energy of the cross-front velocity in the model due to the lack of resolution at the fronts, even though the energy loss is not likely to account for the large gap on the strength of the fronts between the model result and the semi-geostrophic limit solution.

Journal article

Mitchell L, Ham DA, McRae ATT, Rathgeber F, Lange M, Luporini F, Bercea G-T, Markall G, Kelly PHJet al., 2017, Firedrake: automating the finite element method by composing abstractions, ACM Transactions on Mathematical Software, Vol: 43, Pages: 1-27, ISSN: 1557-7295

Firedrake is a new tool for automating the numerical solution of partial differential equations. Firedrakeadopts the domain-specific language for the finite element method of the FEniCS project, but with a purePython runtime-only implementation centred on the composition of several existing and new abstractions forparticular aspects of scientific computing. The result is a more complete separation of concerns which easesthe incorporation of separate contributions from computer scientists, numerical analysts and applicationspecialists. These contributions may add functionality, or improve performance.Firedrake benefits from automatically applying new optimisations. This includes factorising mixed functionspaces, transforming and vectorising inner loops, and intrinsically supporting block matrix operations.Importantly, Firedrake presents a simple public API for escaping the UFL abstraction. This allows users toimplement common operations that fall outside pure variational formulations, such as flux-limiters.

Journal article

Bercea G, McRae ATT, Ham DA, Mitchell L, Rathgeber F, Nardi L, Luporini F, Kelly PHJet al., 2016, A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in Firedrake, Geoscientific Model Development, Vol: 9, Pages: 3803-3815, ISSN: 1991-9603

We present a generic algorithm for numbering and then efficiently iterating over the data values attached to an extruded mesh. An extruded mesh is formed by replicating an existing mesh, assumed to be unstructured, to form layers of prismatic cells. Applications of extruded meshes include, but are not limited to, the representation of 3D high aspect ratio domains employed by geophysical finite element simulations. These meshes are structured in the extruded direction. The algorithm presented here exploits this structure to avoid the performance penalty traditionally associated with unstructured meshes. We evaluate the implementation of this algorithm in the Firedrake finite element system on a range of low compute intensity operations which constitute worst cases for data layout performance exploration. The experiments show that having structure along the extruded direction enables the cost of the indirect data accesses to be amortized after 10-20 layers as long as the underlying mesh is well-ordered. We characterise the resulting spatial and temporal reuse in a representative set of both continuous-Galerkin and discontinuous-Galerkin discretisations. On meshes with realistic numbers of layers the performance achieved is between 70% and 90% of a theoretical hardware-specific limit.

Journal article

Lange M, Mitchell L, Knepley M, Gorman Get al., 2016, Efficient Mesh Management in Firedrake Using PETSc DMPlex, SIAM Journal on Scientific Computing, Vol: 38, Pages: S143-S155, ISSN: 1095-7197

The use of composable abstractions allows the application of new and established algorithms to a wide range of problems, while automatically inheriting the benefits of well-known performance optimizations. This work highlights the composition of the PETSc DMPlex domain topology abstraction with the Firedrake automated finite element system to create a PDE solving environment that combines expressiveness, flexibility, and high performance. We describe how Firedrake utilizes DMPlex to provide the indirection maps required for finite element assembly, while supporting various mesh input formats and runtime domain decomposition. In particular, we describe how DMPlex and its accompanying data structures allow the generic creation of user-defined discretizations, while utilizing data layout optimizations that improve cache coherency and ensure overlapped communication during assembly computation.

Journal article

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