In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that is proven to stay proportional to the number of unknowns. Key issues are the second compression, that reduces the near field complexity significantly, and an additional a-posteriori compression. The latter one is based on a general result concerning an optimal work balance, that applies, in particular, to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time. The theoretical results are illustrated by a 3D example on a nontrivial domain.

In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that is proven to stay proportional to the number of unknowns. Key issues are the second compression, that reduces the near field complexity significantly, and an additional a-posteriori compression. The latter one is based on a general result concerning an optimal work balance, that applies, in particular, to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time. The theoretical results are illustrated by a 3D example on a nontrivial domain.

The goal of this thesis is to design algorithms that run with better complexity when compiling or parallelizing loop programs. The framework within which our algorithms operate is the polyhedral model of compilation which has been successful in the design and implementation of complex loop nest optimizers and parallelizing compilers. The algorithmic complexity and scalability limitations of the above framework remain one important weakness. We address it by introducing sub-polyhedral compilation by using (Unit-)Two-Variable-Per-Inequality or (U)TVPI Polyhedra, namely polyhedrawith restricted constraints of the type ax_{i}+bx_{j}\le c (\pm x_{i}\pm x_{j}\le c). A major focus of our sub-polyhedral compilation is the introduction of sub-polyhedral scheduling, where we propose a technique for scheduling using (U)TVPI polyhedra. As part of this, we introduce algorithms that can be used to construct under-aproximations of the systems of constraints resulting from affine scheduling problems. This technique relies on simple polynomial time algorithms to under approximate a general polyhedron into (U)TVPI polyhedra. The above under-approximation algorithms are generic enough that they can be used for many kinds of loop parallelization scheduling problems, reducing each of their complexities to asymptotically polynomial time. We also introduce sub-polyhedral code-generation where we propose algorithms to use the improved complexities of (U)TVPI sub-polyhedra in polyhedral code generation. In this problem, we show that the exponentialities associated with the widely used polyhedral code generators could be reduced to polynomial time using the improved complexities of (U)TVPI sub-polyhedra. The above presented sub-polyhedral scheduling techniques are evaluated in an experimental framework. For this, we modify the state-of-the-art PLuTo compiler which can parallelize for multi-core architectures using permutation and tiling transformations. We show that using our scheduling technique, the above under-approximations yield polyhedra that are non-empty for 10 out of 16 benchmarks from the Polybench (2.0) kernels. Solving the under-approximated system leads to asymptotic gains in complexity, and shows practically significant improvements when compared to a traditional LP solver. We also verify that code generated by our sub-polyhedral parallelization prototype matches the performance of PLuTo-optimized code when the under-approximation preserves feasibility.

Notre étude de la compilation sous-polyédrique est dominée par l’introduction de la notion l’ordonnancement affine sous-polyédrique, pour laquelle nous proposons une technique utilisant des sous-polyèdres (U)TVPI. Dans ce cadre, nous introduisons des algorithmes capables de construire des sous-approximations de systèmes de contraintes résultant de problèmes d’ordonnancement affine. Cette technique repose sur des algorithmes polynomiaux simples pour approcher un polyèdre quelconque par un polyèdre (U)TVPI. Nos algorithmes sont suffisamment génériques pour s’appliquer à de nombreux problèmes d’ordonnancement, de parallélisation, et d’optimisation de boucles, réduisant leur complexité temporelle à des fonctions polynomiales. Nous introduisons également une méthode pour la génération de code utilisant des algorithmes sous-polyédriques, tirant parti de la faible complexité des sous-polyèdres (U)TVPI. Dans ce cadre, nous montrons comment réduire la complexité associée aux générateurs de code les plus populaires, ramenant la complexité de plusieurs facteurs exponentiels à des fonctions polynomiales. Nombre de ces techniques sont évaluées expérimentalement. Pour cela, nous avons réalisé une version modifiée du compilateur PLuTo, capable de paralléliser et d’optimiser des nids de boucles pour des architectures multi-cœurs à l’aide de transformations affines, et notamment de partitionnement (tiling). Nous montrons qu’une majorité des noyaux de calcul de la suite Polybench (2.0) peut être manipulée à l’aide de notre technique d’ordonnancement, en préservant la faisabilité des polyèdres lors des sous-approximations. L’utilisation des systèmes approchés par des sous-polyèdres conduit à des gains asymptotiques en complexité, qui se traduit par des réductions significatives en temps de compilation, par rapport à un solveur de programmation linéaire de référence. Nous vérifions également que le code généré par notre prototype de parallélisation sous-polyédrique est compétitif par rapport à la performance du code généré par Pluto. / The goal of this thesis is to design algorithms that run with better complexity when compiling or parallelizing loop programs. The framework within which our algorithms operate is the polyhedral model of compilation which has been successful in the design and implementation of complex loop nest optimizers and parallelizing compilers. The algorithmic complexity and scalability limitations of the above framework remain one important weakness. We address it by introducing sub-polyhedral compilation by using (Unit-)Two-Variable-Per-Inequality or (U)TVPI Polyhedra, namely polyhedrawith restricted constraints of the type ax_{i}+bx_{j}\le c (\pm x_{i}\pm x_{j}\le c). A major focus of our sub-polyhedral compilation is the introduction of sub-polyhedral scheduling, where we propose a technique for scheduling using (U)TVPI polyhedra. As part of this, we introduce algorithms that can be used to construct under-aproximations of the systems of constraints resulting from affine scheduling problems. This technique relies on simple polynomial time algorithms to under approximate a general polyhedron into (U)TVPI polyhedra. The above under-approximation algorithms are generic enough that they can be used for many kinds of loop parallelization scheduling problems, reducing each of their complexities to asymptotically polynomial time. We also introduce sub-polyhedral code-generation where we propose algorithms to use the improved complexities of (U)TVPI sub-polyhedra in polyhedral code generation. In this problem, we show that the exponentialities associated with the widely used polyhedral code generators could be reduced to polynomial time using the improved complexities of (U)TVPI sub-polyhedra. The above presented sub-polyhedral scheduling techniques are evaluated in an experimental framework. For this, we modify the state-of-the-art PLuTo compiler which can parallelize for multi-core architectures using permutation and tiling transformations. We show that using our scheduling technique, the above under-approximations yield polyhedra that are non-empty for 10 out of 16 benchmarks from the Polybench (2.0) kernels. Solving the under-approximated system leads to asymptotic gains in complexity, and shows practically significant improvements when compared to a traditional LP solver. We also verify that code generated by our sub-polyhedral parallelization prototype matches the performance of PLuTo-optimized code when the under-approximation preserves feasibility.