Third generation full-waveform (FW) LIDAR systems image an entire scene by emitting laser pulses in particular directions and measuring the echoes. Each of these echoes provides range measurements about the objects intercepted by the laser pulse along a specified direction. By scanning through a specified region using a series of emitted pulses and observing their echoes, connected 1D profiles of 3D scenes can be readily obtained. This extra information has proven helpful in providing additional insight into the scene structure which can be used to construct effective characterizations and classifications. Unfortunately, massive amounts of data are typically collected which impose storage, processing and transmission limitations. To address these problems, a number of compression approaches have been developed in the literature. These, however, generally require the initial acquisition of large amounts of data only to later discard most of it by exploiting redundancies, thus sampling inefficiently. Based on this, our main goal is to apply efficient and effective LIDAR sampling schemes that achieve acceptable reconstruction quality of the 3D scenes. To achieve this goal, we propose on using compressive sampling by emitting pulses only into random locations within the scene and collecting only the corresponding returned FW signals. Under this framework, the number of emissions would typically be much smaller than what traditional LIDAR systems require. Application of this requires, however, that scenes contain many degrees of freedom. Fortunately, such a requirement is satisfied in most natural and man-made scenes. Here, we propose to use a measure of rank as the measure of degrees of freedom. To recover the connected 1D profiles of the 3D scene, matrix completion is applied to the tensor slices. In this paper, we test our approach by showing that recovery of compressively sampled 1D profiles of actual 3D scenes is possible using only a subset of measurements.

Acoustic and electromagnetic scattering from arbitrarily shaped structures can be numerically characterized by solving various surface integral equations (SIEs). One of the most effective techniques to solve SIEs is the Nyström method. Compared to other existing methods,the Nyström method is easier to implement especially when the geometrical discretization is non-conforming and higher-order representations of the geometry and unknowns are desired. However,singularities of the Green’s function are more difficult to”manage”since they are not ”smoothened” through the use of a testing function. This dissertation describes purely numerical schemes to account for different orders of singularities that appear in acoustic and electromagnetic SIEs when they are solved by a high-order Nyström method utilizing a mesh of curved discretization elements. These schemes make use of two sets of basis functions to smoothen singular integrals: the grid robust high-order Lagrange and the high-order Silvester-Lagrange interpolation basis functions. Numerical results comparing the convergence of two schemes are presented. Moreover, an extremely scalable implementation of fast multipole method (FMM) is developed to efficiently (and iteratively) solve the linear system resulting from the discretization of the acoustic SIEs by the Nyström method. The implementation results in O(N log N) complexity for high-frequency scattering problems. This FMM-accelerated solver can handle N =2 billion on a 200,000-core Cray XC40 with 85% strong scaling efficiency. Iterative solvers are often ineffective for ill-conditioned problems. Thus, a fast direct (LU)solver,which makes use of low-rank matrix approximations,is also developed. This solver relies on tile low rank (TLR) data compression format, as implemented in the hierarchical computations on many corearchitectures (HiCMA) library. This requires to taskify the underlying SIE kernels to expose fine-grained computations. The resulting asynchronous execution permit to weaken the artifactual synchronization points,while mitigating the overhead of data motion. We compare the obtained performance results of our TLRLU factorization against the state-of-the-art dense factorizations on shared memory systems. We achieve up to a fourfold performance speedup on a 3D acoustic problem with up to 150 K unknowns in double complex precision arithmetics.

Nous considérons la résolution de très grands systèmes linéaires creux à l'aide d'une méthode de factorisation directe appelée méthode multifrontale. Bien que numériquement robustes et faciles à utiliser (elles ne nécessitent que des informations algébriques : la matrice d'entrée A et le second membre b, même si elles peuvent exploiter des stratégies de prétraitement basées sur des informations géométriques), les méthodes directes sont très coûteuses en termes de mémoire et d'opérations, ce qui limite leur applicabilité à des problèmes de taille raisonnable (quelques millions d'équations). Cette étude se concentre sur l'exploitation des approximations de rang-faible dans la méthode multifrontale, pour réduire sa consommation mémoire et son volume d'opérations, dans des environnements séquentiel et à mémoire distribuée, sur une large classe de problèmes. D'abord, nous examinons les formats rang-faible qui ont déjà été développé pour représenter efficacement les matrices denses et qui ont été utilisées pour concevoir des solveurs rapides pour les équations aux dérivées partielles, les équations intégrales et les problèmes aux valeurs propres. Ces formats sont hiérarchiques (les formats H et HSS sont les plus répandus) et il a été prouvé, en théorie et en pratique, qu'ils permettent de réduire substantiellement les besoins en mémoire et opération des calculs d'algèbre linéaire. Cependant, de nombreuses contraintes structurelles sont imposées sur les problèmes visés, ce qui peut limiter leur efficacité et leur applicabilité aux solveurs multifrontaux généraux. Nous proposons un format plat appelé Block Rang-Faible (BRF) basé sur un découpage naturel de la matrice en blocs et expliquons pourquoi il fournit toute la flexibilité nécéssaire à son utilisation dans un solveur multifrontal général, en terme de pivotage numérique et de parallélisme. Nous comparons le format BRF avec les autres et montrons que le format BRF ne compromet que peu les améliorations en mémoire et opération obtenues grâce aux approximations rang-faible. Une étude de stabilité montre que les approximations sont bien contrôlées par un paramètre numérique explicite appelé le seuil rang-faible, ce qui est critique dans l'optique de résoudre des systèmes linéaires creux avec précision. Ensuite, nous expliquons comment les factorisations exploitant le format BRF peuvent être efficacement implémentées dans les solveurs multifrontaux. Nous proposons plusieurs algorithmes de factorisation BRF, ce qui permet d'atteindre différents objectifs. Les algorithmes proposés ont été implémentés dans le solveur multifrontal MUMPS. Nous présentons tout d'abord des expériences effectuées avec des équations aux dérivées partielles standardes pour analyser les principales propriétés des algorithmes BRF et montrer le potentiel et la flexibilité de l'approche ; une comparaison avec un code basé sur le format HSS est également fournie. Ensuite, nous expérimentons le format BRF sur des problèmes variés et de grande taille (jusqu'à une centaine de millions d'inconnues), provenant de nombreuses applications industrielles. Pour finir, nous illustrons l'utilisation de notre approche en tant que préconditionneur pour la méthode du Gradient Conjugué. / We consider the solution of large sparse linear systems by means of direct factorization based on a multifrontal approach. Although numerically robust and easy to use (it only needs algebraic information: the input matrix A and a right-hand side b, even if it can also digest preprocessing strategies based on geometric information), direct factorization methods are computationally intensive both in terms of memory and operations, which limits their scope on very large problems (matrices with up to few hundred millions of equations). This work focuses on exploiting low-rank approximations on multifrontal based direct methods to reduce both the memory footprints and the operation count, in sequential and distributed-memory environments, on a wide class of problems. We first survey the low-rank formats which have been previously developed to efficiently represent dense matrices and have been widely used to design fast solutions of partial differential equations, integral equations and eigenvalue problems. These formats are hierarchical (H and Hierarchically Semiseparable matrices are the most common ones) and have been (both theoretically and practically) shown to substantially decrease the memory and operation requirements for linear algebra computations. However, they impose many structural constraints which can limit their scope and efficiency, especially in the context of general purpose multifrontal solvers. We propose a flat format called Block Low-Rank (BLR) based on a natural blocking of the matrices and explain why it provides all the flexibility needed by a general purpose multifrontal solver in terms of numerical pivoting for stability and parallelism. We compare BLR format with other formats and show that BLR does not compromise much the memory and operation improvements achieved through low-rank approximations. A stability study shows that the approximations are well controlled by an explicit numerical parameter called low-rank threshold, which is critical in order to solve the sparse linear system accurately. Details on how Block Low-Rank factorizations can be efficiently implemented within multifrontal solvers are then given. We propose several Block Low-Rank factorization algorithms which allow for different types of gains. The proposed algorithms have been implemented within the MUMPS (MUltifrontal Massively Parallel Solver) solver. We first report experiments on standard partial differential equations based problems to analyse the main features of our BLR algorithms and to show the potential and flexibility of the approach; a comparison with a Hierarchically SemiSeparable code is also given. Then, Block Low-Rank formats are experimented on large (up to a hundred millions of unknowns) and various problems coming from several industrial applications. We finally illustrate the use of our approach as a preconditioning method for the Conjugate Gradient.

Nous nous intéressons à l'utilisation d'approximations de rang faible pour réduire le coût des solveurs creux directs multifrontaux. Parmi les différents formats matriciels qui ont été proposés pour exploiter la propriété de rang faible dans les solveurs multifrontaux, nous nous concentrons sur le format Block Low-Rank (BLR) dont la simplicité et la flexibilité permettent de l'utiliser facilement dans un solveur multifrontal algébrique et généraliste. Nous présentons différentes variantes de la factorisation BLR, selon comment les mises à jour de rang faible sont effectuées, et comment le pivotage numérique est géré. D'abord, nous étudions la complexité théorique du format BLR qui, contrairement à d'autres formats comme les formats hiérarchiques, était inconnue jusqu'à présent. Nous prouvons que la complexité théorique de la factorisation multifrontale BLR est asymptotiquement inférieure à celle du solveur de rang plein. Nous montrons ensuite comment les variantes BLR peuvent encore réduire cette complexité. Nous étayons nos bornes de complexité par une étude expérimentale. Après avoir montré que les solveurs multifrontaux BLR peuvent atteindre une faible complexité, nous nous intéressons au problème de la convertir en gains de performance réels sur les architectures modernes. Nous présentons d'abord une factorisation BLR multithreadée, et analysons sa performance dans des environnements multicœurs à mémoire partagée. Nous montrons que les variantes BLR sont cruciales pour exploiter efficacement les machines multicœurs en améliorant l'intensité arithmétique et la scalabilité de la factorisation. Nous considérons ensuite à la factorisation BLR sur des architectures à mémoire distribuée. Les algorithmes présentés dans cette thèse ont été implémentés dans le solveur MUMPS. Nous illustrons l'utilisation de notre approche dans trois applications industrielles provenant des géosciences et de la mécanique des structures. Nous comparons également notre solveur avec STRUMPACK, basé sur des approximations Hierarchically Semi-Separable. Nous concluons cette thèse en rapportant un résultat sur un problème de très grande taille (130 millions d'inconnues) qui illustre les futurs défis posés par le passage à l'échelle des solveurs multifrontaux BLR. / We investigate the use of low-rank approximations to reduce the cost of sparse direct multifrontal solvers. Among the different matrix representations that have been proposed to exploit the low-rank property within multifrontal solvers, we focus on the Block Low-Rank (BLR) format whose simplicity and flexibility make it easy to use in a general purpose, algebraic multifrontal solver. We present different variants of the BLR factorization, depending on how the low-rank updates are performed and on the constraints to handle numerical pivoting. We first investigate the theoretical complexity of the BLR format which, unlike other formats such as hierarchical ones, was previously unknown. We prove that the theoretical complexity of the BLR multifrontal factorization is asymptotically lower than that of the full-rank solver. We then show how the BLR variants can further reduce that complexity. We provide an experimental study with numerical results to support our complexity bounds. After proving that BLR multifrontal solvers can achieve a low complexity, we turn to the problem of translating that low complexity in actual performance gains on modern architectures. We first present a multithreaded BLR factorization, and analyze its performance in shared-memory multicore environments on a large set of real-life problems. We put forward several algorithmic properties of the BLR variants necessary to efficiently exploit multicore systems by improving the arithmetic intensity and the scalability of the BLR factorization. We then move on to the distributed-memory BLR factorization, for which additional challenges are identified and addressed. The algorithms presented throughout this thesis have been implemented within the MUMPS solver. We illustrate the use of our approach in three industrial applications coming from geosciences and structural mechanics. We also compare our solver with the STRUMPACK package, based on Hierarchically Semi-Separable approximations. We conclude this thesis by reporting results on a very large problem (130 millions of unknowns) which illustrates future challenges posed by BLR multifrontal solvers at scale.