9 research outputs found
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Optimisations for quadrature representations of finite element tensors through automated code generation
We examine aspects of the computation of finite element matrices and vectors
which are made possible by automated code generation. Given a variational form
in a syntax which resembles standard mathematical notation, the low-level
computer code for building finite element tensors, typically matrices, vectors
and scalars, can be generated automatically via a form compiler. In particular,
the generation of code for computing finite element matrices using a quadrature
approach is addressed. For quadrature representations, a number of optimisation
strategies which are made possible by automated code generation are presented.
The relative performance of two different automatically generated
representations of finite element matrices is examined, with a particular
emphasis on complicated variational forms. It is shown that approaches which
perform best for simple forms are not tractable for more complicated problems
in terms of run time performance, the time required to generate the code or the
size of the generated code. The approach and optimisations elaborated here are
effective for a range of variational forms
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Supporting material for the paper 'Optimisations for quadrature representations of finite element tensors through automated code generation'
Supporting computer code for the paper 'Optimisations for quadrature representations of finite element tensors through automated code generation' in ACM Transactions on Mathematical Software
Formoptimierung für Fluid-Struktur Interaktionsprobleme
In this thesis, shape optimization for unsteady fluid-structure interaction problems via the method of mappings is investigated theoretically and numerically. New existence and regularity results are proven. A framework for deriving differentiability for the solution of nonlinear, unsteady, parameter-dependent partial differential equations is developed and applied to show differentiability of the states with respect to domain variations.In dieser Arbeit wird Formoptimierung für instationäre Fluid-Struktur Interaktionsprobleme mittels der Method of Mappings theoretisch und numerisch untersucht. Es werden neue Existenz- und Regularitätsaussagen bewiesen. Des Weiteren wird ein theoretischer Rahmen entwickelt, womit Differenzierbarkeit für die Lösung von nichtlinearen, instationären und parameterabhängigen Differentialgleichungen gezeigt werden kann, und angewandt, um Differenzierbarkeit des Zustandes bezüglich Gebietsvariationen zu zeigen
An Active-Library Based Investigation into the Performance Optimisation of Linear Algebra and the Finite Element Method
In this thesis, I explore an approach called "active libraries". These are libraries that take
part in their own optimisation, enabling both high-performance code and the presentation of
intuitive abstractions.
I investigate the use of active libraries in two domains. Firstly, dense and sparse linear algebra,
particularly, the solution of linear systems of equations. Secondly, the specification and solution
of finite element problems.
Extending my earlier (MEng) thesis work, I describe the modifications to my linear algebra
library "Desola" required to perform sparse-matrix code generation. I show that optimisations
easily applied in the dense case using code-transformation must be applied at a higher level of
abstraction in the sparse case. I present performance results for sparse linear system solvers
generated using Desola and compare against an implementation using the Intel Math Kernel
Library. I also present improved dense linear-algebra performance results.
Next, I explore the active-library approach by developing a finite element library that captures
runtime representations of basis functions, variational forms and sequences of operations between
discretised operators and fields. Using captured representations of variational forms and
basis functions, I demonstrate optimisations to cell-local integral assembly that this approach
enables, and compare against the state of the art.
As part of my work on optimising local assembly, I extend the work of Hosangadi et al. on
common sub-expression elimination and factorisation of polynomials. I improve the weight
function presented by Hosangadi et al., increasing the number of factorisations found. I present
an implementation of an optimised branch-and-bound algorithm inspired by reformulating the
original matrix-covering problem as a maximal graph biclique search problem. I evaluate the
algorithm's effectiveness on the expressions generated by our finite element solver