Tesi sul tema "Algebraic kernel"
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1
Bhattacharjee, Papiya. "Minimal Prime Element Space of an Algebraic Frame". Bowling Green State University / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1243364652.
2
Peñaranda, Luis. "Géométrie algorithmique non linéaire et courbes algébriques planaires". Electronic Thesis or Diss., Nancy 2, 2010. http://www.theses.fr/2010NAN23002.
Abstract (sommario):
Nous abordons dans cette thèse le problème du calcul de la topologie de courbes algébriques planes. Nous présentons un algorithme qui, grâce à l’application d’outils algébriques comme les bases de Gröbner et les représentations rationnelles univariées, ne nécessite pas de traitement particulier de cas dégénérés. Nous avons implanté cet algorithme et démontré son efficacité par un ensemble de comparaisons avec les logiciels similaires. Nous présentons également une analyse de complexité sensible a la sortie de cet algorithme. Nous discutons ensuite des outils nécessaires pour l’implantation d’algorithmes de géométrie non-linéaire dans CGAL, la bibliothèque de référence de la communauté de géométrie algorithmique. Nous présentons un noyau univarié pour CGAL, un ensemble de fonctions nécessaires pour le traitement d’objets courbes définis par des polynômes univariés. Nous avons validé notre approche en la comparant avec les implantations similairesWe tackle in this thesis the problem of computing the topology of plane algebraic curves. We present an algorithm that avoids special treatment of degenerate cases, based on algebraic tools such as Gröbner bases and rational univariate representations. We implemented this algorithm and showed its performance by comparing to simi- lar existing programs. We also present an output-sensitive complexity analysis of this algorithm. We then discuss the tools that are necessary for the implementation of non- linear geometric algorithms in CGAL, the reference library in the computational geom- etry community. We present an univariate algebraic kernel for CGAL, a set of functions aimed to handle curved objects defined by univariate polynomials. We validated our approach by comparing it to other similar implementations
3
Laske, Michael. "Le K1 des courbes sur les corps globaux : conjecture de Bloch et noyaux sauvages". Thesis, Bordeaux 1, 2009. http://www.theses.fr/2009BOR13861/document.
Abstract (sommario):
Pour X une courbe sur un corps global k, lisse, projective et géométriquement connexe, nous déterminons la Q-structure du groupe de Quillen K1(X) : nous démontrons que dimQ K1(X) ? Q =2r, où r désigne le nombre de places archimédiennes de k (y compris le cas r = 0 pour un corps de fonctions). Cela con?rme une conjecture de Bloch annoncée dans les années 1980. Dans le langage de la K-théorie de Milnor, que nous dé?nissons pour les variétés algébriques via les groupes de Somekawa, le premier K-groupe spécial de Milnor SKM1 (X) est de torsion. Pour la preuve, nous développons une théorie des hauteurs applicable aux K-groupes de Milnor, et nous généralisons l’approche de base de facteurs de Bass-Tate. Une structure plus ?ne de SKM 1 (X) émerge en localisant le corps de base k, et une description explicite de la décomposition correspondante est donnée. En particulier, nous identi?ons un sous-groupe WKl(X):= ker (SKM 1 (X) ? Zl ? Lv SKM 1 (Xv) ? Zl) pour chaque entier rationnel l, nommé noyau sauvage, dont nous croyons qu’il est ?niFor a smooth projective geometrically connected curve X over a global ?eld k, we determine the Q-structure of its ?rst Quillen K-group K1(X) by showing that dimQ K1(X) ? Q =2r, where r denotes the number of archimedean places of k (including the case r = 0 for k a function ?eld). This con?rms a conjecture of Bloch. In the language of Milnor K-theory, which we de?ne for varieties via Somekawa groups, the ?rst special Milnor K-group SKM 1 (X) is torsion. For the proof, we develop a theory of heights applicable to Milnor K-groups, and generalize the factor basis approach of Bass-Tate. A ?ner structure of SKM 1 (X) emerges when localizing the ground ?eld k, and we give an explicit description of the resulting decomposition. In particular, we identify a potentially ?nite subgroup WKl(X):= ker (SKM 1 (X) ? Zl ? Lv SKM 1 (Xv) ? Zl) for each rational prime l, named wild kernel
4
Sondecker, Victoria L. "Kernel-trace approach to congruences on regular and inverse semigroups". Instructions for remote access. Click here to access this electronic resource. Access available to Kutztown University faculty, staff, and students only, 1994. http://www.kutztown.edu/library/services/remote_access.asp.
Abstract (sommario):
Thesis (M.A.)--Kutztown University of Pennsylvania, 1994.Source: Masters Abstracts International, Volume: 45-06, page: 3173. Abstract precedes thesis as [2] preliminary leaves. Typescript. Includes bibliographical references (leaves 52-53).
5
Speck, Robert [Verfasser]. "Generalized Algebraic Kernels and Multipole Expansions for massively parallel Vortex Particle Methods / Robert Speck". Wuppertal : Universitätsbibliothek Wuppertal, 2011. http://d-nb.info/1018299866/34.
6
Kumar, Suraj. "Scheduling of Dense Linear Algebra Kernels on Heterogeneous Resources". Thesis, Bordeaux, 2017. http://www.theses.fr/2017BORD0572/document.
Abstract (sommario):
Du fait des énormes capacités de calculs des accélérateurs tels que les GPUs et les Xeon Phi, l’utilisation de machines multicoques pourvues d’accélérateurs est devenue commune dans le domaine du calcul haute performance (HPC). La complexité induite par ces accélérateurs a suscité le développement de systèmes d’exécution à base de tâches, dans lesquels les dépendances entre les applications sont exprimées sous la forme de graphe de tâches et où les tâches sont ordonnancées dynamiquement sur les ressources de calcul. La difficulté est alors de concevoir des stratégies d’ordonnancement qui font une utilisation efficace des ressources de calculs et le développement de telles stratégies, même pour un unique noeud hybride, est un enjeu essentiel de la performance des systèmes HPC. Nous considérons dans cette thèse l’ordonnancement de noyaux d’algèbre linéaire dense sur des noeuds complètement hétérogènes et constitués de CPUs et de GPUs. Les performances relatives des accélérateurs par rapport aux coeurs classique dépend très fortement du noyau considéré. Par exemple, les accélérateurs sont beaucoup plus efficaces pour les produits de matrices, par exemple, que pour les factorisations. Dans cette thèse, nous analysons les performances de stratégies statiques et dynamiques d’ordonnancement et nous proposons un ensemble de stratégies intermédiaires, en ajoutant des composantes statiques (respectivement dynamiques) à des stratégies d’ordonnancements dynamique (respectivement statiques). Récemment, une stratégie appelée HeteroPrio a été proposée, qui s’appuie sur les affinités entre les tâches et les ressources pour un petit ensemble de tâches différentes s’exécutant sur deux types de ressources. Nous avons étendu cette stratégie d’ordonnancement pour des graphes de tâches généraux pour deux types de ressources puis pour plus de deux types. De manière complémentaire, nous avons également démontré des facteurs d’approximation et des pires cas pour HeteroPrio dans le cas d’un ensemble de tâches indépendantes sur différents types de plates-formesDue to massive computation power of accelerators such as GPU, Xeon phi, multicore machines equipped with accelerators are becoming popular in High Performance Computing (HPC). The added complexity led to the development of different task-based runtime systems, which allow computations to be expressed as graphs of tasks and rely on runtime systems to schedule those tasks among all resources of the platform. The real challenge is to design efficient schedulers for such runtimes to make effective utilization of all resources. Developing good schedulers, even for a single hybrid node, and analyzing them can thus have a strong impact on the performance of current HPC systems. We consider the problem of scheduling dense linear algebra applications on fully hybrid platforms made of CPUs and GPUs. The relative performance of CPU and GPU highly depends on the sub-routine. For instance, GPUs are much more efficient to process matrix-matrix multiplications than matrix factorizations. In this thesis, we analyze the performance of static and dynamic scheduling strategies and we propose a set of intermediate strategies, by adding static (resp. dynamic) features into dynamic (resp. static) strategies. A resource centric dynamic scheduler, HeteroPrio, which is based on affinity between tasks and resources, has been proposed recently for a set of small independent tasks on two types of resources. We extend and analyze this scheduler for general task graphs first on two types of resources and then on more than two types of resources. Additionally, we provide approximation ratios and worst case examples of HeteroPrio for a set of independent tasks on different platform sizes
7
Tachibana, Kanta, Takeshi Furuhashi, Tomohiro Yoshikawa, Eckhard Hitzer e MINH TUAN PHAM. "Clustering of Questionnaire Based on Feature Extracted by Geometric Algebra". 日本知能情報ファジィ学会, 2008. http://hdl.handle.net/2237/20676.
Abstract (sommario):
Session ID: FR-G2-2Joint 4th International Conference on Soft Computing and Intelligent Systems and 9th International Symposium on advanced Intelligent Systems, September 17-21, 2008, Nagoya University, Nagoya, Japan
8
Good, Jennifer Rose. "Weighted interpolation over W*-algebras". Diss., University of Iowa, 2015. https://ir.uiowa.edu/etd/1843.
Abstract (sommario):
An operator-theoretic formulation of the interpolation problem posed by Nevanlinna and Pick in the early twentieth century asks for conditions under which there exists a multiplier of a reproducing kernel Hilbert space that interpolates a specified set of data. Paul S. Muhly and Baruch Solel have shown that their theory for operator algebras built from W*-correspondences provides an appropriate context for generalizing this classic question. Their reproducing kernel W*-correspondences are spaces of functions that generalize the reproducing kernel Hilbert spaces. Their Nevanlinna-Pick interpolation theorem, which is proved using commutant lifting, implies that the algebra of multipliers of the reproducing kernel W*-correspondence associated with a certain W*-version of the classic Szegö kernel may be identified with their primary operator algebra of interest, the Hardy algebra.
To provide a context for generalizing another familiar topic in operator theory, the study of the weighted Hardy spaces, Muhly and Solel have recently expanded their theory to include operator-valued weights. This creates a new family of reproducing kernel W*-correspondences that includes certain, though not all, classic weighted Hardy spaces. It is the purpose of this thesis to generalize several of Muhly and Solel's results to the weighted setting and investigate the function-theoretic properties of the resulting spaces.
We give two principal results. The first is a weighted version of Muhly and Solel's commutant lifting theorem, which we obtain by making use of Parrott's lemma. The second main result, which in fact follows from the first, is a weighted Nevanlinna-Pick interpolation theorem. Other results, several of which follow from the two primary results, include the construction of an orthonormal basis for the nonzero tensor product of two W*-corrrespondences, a double commutant theorem, the identification of several function-theoretic properties of the elements in the reproducing kernel W*-correspondence associated with a weighted W*-Szegö kernel as well as the elements in its algebra of mutlipliers, and the presentation of a relationship between this algebra of multipliers and a weighted Hardy algebra. In addition, we consider a candidate for a W*-version of the complete Pick property and investigate the aforementioned weighted W*-Szegö kernel in its light.
9
Shinde, Sachin Dilip. "SuperTaco : Taco Tensor Algebra kernels on distributed systems using Legion". Thesis, Massachusetts Institute of Technology, 2019. https://hdl.handle.net/1721.1/121683.
Abstract (sommario):
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 89-91).
Tensor algebra is a powerful language for expressing computation on multidimensional data. While many tensor datasets are sparse, most tensor algebra libraries have limited support for handling sparsity. The Tensor Algebra Compiler (Taco) has introduced a taxonomy for sparse tensor formats that has allowed them to compile sparse tensor algebra expressions to performant C code, but they have not taken advantage of distributed systems. This work provides a code generation technique for creating Legion programs that distribute the computation of Taco tensor algebra kernels across distributed systems, and a scheduling language for controlling how this distributed computation is structured. This technique is implemented in the form of a command-line tool called SuperTaco. We perform a strong scaling analysis for the SpMV and TTM kernels under a row blocking distribution schedule, and find speedups of 9-10x when using 20 cores on a single node. For multi-node systems using 20 cores per node, SpMV achieves a 33.3x speedup at 160 cores and TTM achieves a 42.0x speedup at 140 cores.
by Sachin Dilip Shinde.
M. Eng.
M.Eng. Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science
10
Wilding, David. "Linear algebra over semirings". Thesis, University of Manchester, 2015. https://www.research.manchester.ac.uk/portal/en/theses/linear-algebra-over-semirings(1dfe7143-9341-4dd1-a0d1-ab976628442d).html.
Abstract (sommario):
Motivated by results of linear algebra over fields, rings and tropical semirings, we present a systematic way to understand the behaviour of matrices with entries in an arbitrary semiring. We focus on three closely related problems concerning the row and column spaces of matrices. This allows us to isolate and extract common properties that hold for different reasons over different semirings, yet also lets us identify which features of linear algebra are specific to particular types of semiring. For instance, the row and column spaces of a matrix over a field are isomorphic to each others' duals, as well as to each other, but over a tropical semiring only the first of these properties holds in general (this in itself is a surprising fact). Instead of being isomorphic, the row space and column space of a tropical matrix are anti-isomorphic in a certain order-theoretic and algebraic sense. The first problem is to describe the kernels of the row and column spaces of a given matrix. These equivalence relations generalise the orthogonal complement of a set of vectors, and the nature of their equivalence classes is entirely dependent upon the kind of semiring in question. The second, Hahn-Banach type, problem is to decide which linear functionals on row and column spaces of matrices have a linear extension. If they all do, the underlying semiring is called exact, and in this case the row and column spaces of any matrix are isomorphic to each others' duals. The final problem is to explain the connection between the row space and column space of each matrix. Our notion of a conjugation on a semiring accounts for the different possibilities in a unified manner, as it guarantees the existence of bijections between row and column spaces and lets us focus on the peculiarities of those bijections. Our main original contribution is the systematic approach described above, but along the way we establish several new results about exactness of semirings. We give sufficient conditions for a subsemiring of an exact semiring to inherit exactness, and we apply these conditions to show that exactness transfers to finite group semirings. We also show that every Boolean ring is exact. This result is interesting because it allows us to construct a ring which is exact (also known as FP-injective) but not self-injective. Finally, we consider exactness for residuated lattices, showing that every involutive residuated lattice is exact. We end by showing that the residuated lattice of subsets of a finite monoid is exact if and only if the monoid is a group.
11
Wu, Wenhao. "High-performance matrix multiplication hierarchical data structures, optimized kernel routines, and qualitative performance modeling /". Master's thesis, Mississippi State : Mississippi State University, 2003. http://library.msstate.edu/etd/show.asp?etd=etd-07092003-003633.
12
DiPaolo, Conner. "Randomized Algorithms for Preconditioner Selection with Applications to Kernel Regression". Scholarship @ Claremont, 2019. https://scholarship.claremont.edu/hmc_theses/230.
Abstract (sommario):
The task of choosing a preconditioner M to use when solving a linear system Ax=b with iterative methods is often tedious and most methods remain ad-hoc. This thesis presents a randomized algorithm to make this chore less painful through use of randomized algorithms for estimating traces. In particular, we show that the preconditioner stability || I - M-1A ||F, known to forecast preconditioner quality, can be computed in the time it takes to run a constant number of iterations of conjugate gradients through use of sketching methods. This is in spite of folklore which suggests the quantity is impractical to compute, and a proof we give that ensures the quantity could not possibly be approximated in a useful amount of time by a deterministic algorithm. Using our estimator, we provide a method which can provably select a quality preconditioner among n candidates using floating operations commensurate with running about n log(n) steps of the conjugate gradients algorithm. In the absence of such a preconditioner among the candidates, our method can advise the practitioner to use no preconditioner at all. The algorithm is extremely easy to implement and trivially parallelizable, and along the way we provide theoretical improvements to the literature on trace estimation. In empirical experiments, we show the selection method can be quite helpful. For example, it allows us to create to the best of our knowledge the first preconditioning method for kernel regression which never uses more iterations over the non-preconditioned analog in standard settings.
13
Kardell, Marcus. "Total positivity and oscillatory kernels : An overview, and applications to the spectral theory of the cubic string". Thesis, Linköpings universitet, Tillämpad matematik, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-58005.
Abstract (sommario):
In the study of the Degasperis-Procesi dierential equation, an eigenvalue problem called the cubic string occurs. This is a third order generalization of the second order problem describing the eigenmodes of a vibrating string. In this thesis we study the eigenfunctions of the cubic string for discrete and continuous mass distributions, using the theory of total positivity, via a combinatorial approach with planar networks.
14
Issa, Hassan A. [Verfasser], Wolfram [Akademischer Betreuer] Bauer e Ingo [Akademischer Betreuer] Witt. "The analysis of Toeplitz operators, commutative Toeplitz algebras and applications to heat kernel constructions / Hassan A. Issa. Gutachter: Wolfram Bauer ; Ingo Witt. Betreuer: Wolfram Bauer". Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2012. http://d-nb.info/1042970947/34.
15
Mascarenhas, Helena. "Convolution type operators on cones and asymptotic spectral theory". Doctoral thesis, [S.l. : s.n.], 2004. http://deposit.ddb.de/cgi-bin/dokserv?idn=970638809.
16
Kernert, David [Verfasser], Wolfgang [Akademischer Betreuer] [Gutachter] Lehner e Peter [Gutachter] Fischer. "Density-Aware Linear Algebra in a Column-Oriented In-Memory Database System / David Kernert ; Gutachter: Wolfgang Lehner, Peter Fischer ; Betreuer: Wolfgang Lehner". Dresden : Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2016. http://d-nb.info/1119362385/34.
17
Wood, Nicholas Linder. "Extension of Similarity Functions and their Application toChemical Informatics Problems". The Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1542299336598615.
18
Herrero, Zaragoza Jose Ramón. "A framework for efficient execution of matrix computations". Doctoral thesis, Universitat Politècnica de Catalunya, 2006. http://hdl.handle.net/10803/5991.
Abstract (sommario):
Matrix computations lie at the heart of most scientific computational tasks. The solution of linear systems of equations is a very frequent operation in many fields in science, engineering, surveying, physics and others. Other matrix operations occur frequently in many other fields such as pattern recognition and classification, or multimedia applications. Therefore, it is important to perform matrix operations efficiently. The work in this thesis focuses on the efficient execution on commodity processors of matrix operations which arise frequently in different fields.We study some important operations which appear in the solution of real world problems: some sparse and dense linear algebra codes and a classification algorithm. In particular, we focus our attention on the efficient execution of the following operations: sparse Cholesky factorization; dense matrix multiplication; dense Cholesky factorization; and Nearest Neighbor Classification.
A lot of research has been conducted on the efficient parallelization of numerical algorithms. However, the efficiency of a parallel algorithm depends ultimately on the performance obtained from the computations performed on each node. The work presented in this thesis focuses on the sequential execution on a single processor.
There exists a number of data structures for sparse computations which can be used in order to avoid the storage of and computation on zero elements. We work with a hierarchical data structure known as hypermatrix. A matrix is subdivided recursively an arbitrary number of times. Several pointer matrices are used to store the location of
submatrices at each level. The last level consists of data submatrices which are dealt with as dense submatrices. When the block size of this dense submatrices is small, the number of zeros can be greatly reduced. However, the performance obtained from BLAS3 routines drops heavily. Consequently, there is a trade-off in the size of data submatrices used for a sparse Cholesky factorization with the hypermatrix scheme. Our goal is that of reducing the overhead introduced by the unnecessary operation on zeros when a hypermatrix data structure is used to produce a sparse Cholesky factorization. In this work we study several techniques for reducing such overhead in order to obtain high performance.
One of our goals is the creation of codes which work efficiently on different platforms when operating on dense matrices. To obtain high performance, the resources offered by the CPU must be properly utilized. At the same time, the memory hierarchy must be exploited to tolerate increasing memory latencies. To achieve the former, we produce inner kernels which use the CPU very efficiently. To achieve the latter, we investigate nonlinear data layouts. Such data formats can contribute to the effective use of the memory system.
The use of highly optimized inner kernels is of paramount importance for obtaining efficient numerical algorithms. Often, such kernels are created by hand. However, we want to create efficient inner kernels for a variety of processors using a general approach and avoiding hand-made codification in assembly language. In this work, we present an alternative way to produce efficient kernels automatically, based on a set of simple codes written in a high level language, which can be parameterized at compilation time. The advantage of our method lies in the ability to generate very efficient inner kernels by means of a good compiler. Working on regular codes for small matrices most of the compilers we used in different platforms were creating very efficient inner kernels for matrix multiplication. Using the resulting kernels we have been able to produce high performance sparse and dense linear algebra codes on a variety of platforms.
In this work we also show that techniques used in linear algebra codes can be useful in other fields. We present the work we have done in the optimization of the Nearest Neighbor classification focusing on the speed of the classification process.
Tuning several codes for different problems and machines can become a heavy and unbearable task. For this reason we have developed an environment for development and automatic benchmarking of codes which is presented in this thesis.
As a practical result of this work, we have been able to create efficient codes for several matrix operations on a variety of platforms. Our codes are highly competitive with other state-of-art codes for some problems.
19
Fan, Yang, Hidehiko Masuhara, Tomoyuki Aotani, Flemming Nielson e Hanne Riis Nielson. "AspectKE*: Security aspects with program analysis for distributed systems". Universität Potsdam, 2010. http://opus.kobv.de/ubp/volltexte/2010/4136/.
Abstract (sommario):
Enforcing security policies to distributed systems is difficult, in particular, when a system contains untrusted components. We designed AspectKE*, a distributed AOP language based on a tuple space, to tackle this issue. In AspectKE*, aspects can enforce access control policies that depend on future behavior of running processes. One of the key language features is the predicates and functions that extract results of static program analysis, which are useful for defining security aspects that have to know about future behavior of a program. AspectKE* also provides a novel variable binding mechanism for pointcuts, so that pointcuts can uniformly specify join points based on both static and dynamic information about the program. Our implementation strategy performs fundamental static analysis at load-time, so as to retain runtime overheads minimal. We implemented a compiler for AspectKE*, and demonstrate usefulness of AspectKE* through a security aspect for a distributed chat system.
20
Hong, Guixiang. "Quelques problèmes en analyse harmonique non commutative". Phd thesis, Université de Franche-Comté, 2012. http://tel.archives-ouvertes.fr/tel-00979472.
Abstract (sommario):
Cette thèse présente quelques résultats de la théorie des probabilités quantiques et de l'analyse harmonique non commutative. Elle est constituée de trois parties. La première partie démontre l'analogue non commutatif de l'inégalité de John-Nirenberg et la décomposition atomique pour les martingales non commutatives. Ces résultats étendent et améliorent ceux qui existent déjà, et correspondent exactement à ceux que l'on connaît dans le cas classique. La deuxième partie est consacrée à l'étude des espaces de Hardy à valeurs opérateurs via la méthode d'ondelettes. Il est montré que les espaces de Hardy définis par ondelettes coïncident avec ceux définis par les fonctions carrées de Littlewood-Paley et Lusin. Cette approche est similaire à celle du cas des martingales non commutatives, mais l'utilisation des outils de martingales en analyse harmonique permet une démonstration plus rapide. Dans la troisième partie, nous nous tournons vers des applications de la théorie bien établie des espaces de Hardy, c'est-à-dire des opérateurs de Calderón-Zygmund (OCZ pour abréviation) associés à des noyaux à valeurs matricielles. On obtient des estimations de type faible (1, 1) pour des OCZ dyadiques parfaites et des shifts de Haar annulateurs associés à des noyaux non commutatifs, ainsi que des estimations de type H1 → L1 pour des OCZ arbitaires d'après une décomposition d'une fonction en ligne/colonne. En conjonction avec L∞ → BMO, nous établissons certaines estimations de type Lp. Cette approche s'applique aussi à des paraproduits et des transformées de martingales avec des symboles et coefficients non commutatifs respectivement.
21
Vasilescu, M. Alex O. "A Multilinear (Tensor) Algebraic Framework for Computer Graphics, Computer Vision and Machine Learning". Thesis, 2012. http://hdl.handle.net/1807/65327.
Abstract (sommario):
This thesis introduces a multilinear algebraic framework for computer graphics, computer vision, and machine learning, particularly for the fundamental purposes of image synthesis, analysis, and recognition. Natural images result from the multifactor interaction between the imaging process, the scene illumination, and the scene geometry. We assert that a principled mathematical approach to disentangling and explicitly representing these causal factors, which are essential to image formation, is through numerical multilinear algebra, the algebra of higher-order tensors.
Our new image modeling framework is based on(i) a multilinear generalization of principal components analysis (PCA), (ii) a novel multilinear generalization of independent components analysis (ICA), and (iii) a multilinear projection for use in recognition that maps images to the multiple causal factor spaces associated with their formation. Multilinear PCA employs a tensor extension of the conventional matrix singular value decomposition (SVD), known as the M-mode SVD, while our multilinear ICA method involves an analogous M-mode ICA algorithm.
As applications of our tensor framework, we tackle important problems in computer graphics, computer vision, and pattern recognition; in particular, (i) image-based rendering, specifically introducing the multilinear synthesis of images of textured surfaces under varying view and illumination conditions, a new technique that we call
``TensorTextures'', as well as (ii) the multilinear analysis and recognition of facial images under variable face shape, view, and illumination conditions, a new technique that we call ``TensorFaces''. In developing these applications, we introduce a multilinear image-based rendering algorithm and a multilinear appearance-based recognition algorithm. As a final, non-image-based application of our framework, we consider the analysis, synthesis and recognition of human motion data using multilinear methods, introducing a new technique that we call ``Human Motion Signatures''.
22
Kerber, Michael [Verfasser]. "Geometric algorithms for algebraic curves and surfaces / vorgelegt von Michael Kerber". 2009. http://d-nb.info/1002267331/34.
23
Issa, Hassan. "The analysis of Toeplitz operators, commutative Toeplitz algebras and applications to heat kernel constructions". Doctoral thesis, 2012. http://hdl.handle.net/11858/00-1735-0000-000D-F066-5.
24
"Accelerating Linear Algebra and Machine Learning Kernels on a Massively Parallel Reconfigurable Architecture". Master's thesis, 2019. http://hdl.handle.net/2286/R.I.55557.
Abstract (sommario):
abstract: This thesis presents efficient implementations of several linear algebra kernels, machine learning kernels and a neural network based recommender systems engine onto a massively parallel reconfigurable architecture, Transformer. The linear algebra kernels include Triangular Matrix Solver (TRSM), LU Decomposition (LUD), QR Decomposition (QRD), and Matrix Inversion. The machine learning kernels include an LSTM (Long Short Term Memory) cell, and a GRU (gated Recurrent Unit) cell used in recurrent neural networks. The neural network based recommender systems engine consists of multiple kernels including fully connected layers, embedding layer, 1-D batchnorm, Adam optimizer, etc.
Transformer is a massively parallel reconfigurable multicore architecture designed at the University of Michigan. The Transformer configuration considered here is 4 tiles and 16 General Processing Elements (GPEs) per tile. It supports a two level cache hierarchy where the L1 and L2 caches can operate in shared (S) or private (P) modes. The architecture was modeled using Gem5 and cycle accurate simulations were done to evaluate the performance in terms of execution times, giga-operations per second per Watt (GOPS/W), and giga-floating-point-operations per second per Watt (GFLOPS/W).
This thesis shows that for linear algebra kernels, each kernel achieves high performance for a certain cache mode and that this cache mode can change when the matrix size changes. For instance, for smaller matrix sizes, L1P, L2P cache mode is best for TRSM, while L1S, L2S is the best cache mode for LUD, and L1P, L2S is the best for QRD. For each kernel, the optimal cache mode changes when the matrix size is increased. For instance, for TRSM, the L1P, L2P cache mode is best for smaller matrix sizes ($N=64, 128, 256, 512$) and it changes to L1S, L2P for larger matrix sizes ($N=1024$). For machine learning kernels, L1P, L2P is the best cache mode for all network parameter sizes.
Gem5 simulations show that the peak performance for TRSM, LUD, QRD and Matrix Inverse in the 14nm node is 97.5, 59.4, 133.0 and 83.05 GFLOPS/W, respectively. For LSTM and GRU, the peak performance is 44.06 and 69.3 GFLOPS/W.
The neural network based recommender system was implemented in L1S, L2S cache mode. It includes a forward pass and a backward pass and is significantly more complex in terms of both computational complexity and data movement. The most computationally intensive block is the fully connected layer followed by Adam optimizer. The overall performance of the recommender systems engine is 54.55 GFLOPS/W and 169.12 GOPS/W.Dissertation/Thesis
Masters Thesis Electrical Engineering 2019
25
Abdelfattah, Ahmad. "Accelerating Scientific Applications using High Performance Dense and Sparse Linear Algebra Kernels on GPUs". Diss., 2015. http://hdl.handle.net/10754/346955.
Abstract (sommario):
High performance computing (HPC) platforms are evolving to more heterogeneous configurations to support the workloads of various applications. The current hardware landscape is composed of traditional multicore CPUs equipped with hardware accelerators that can handle high levels of parallelism. Graphical Processing Units (GPUs) are popular high performance hardware accelerators in modern supercomputers. GPU programming has a different model than that for CPUs, which means that many numerical kernels have to be redesigned and optimized specifically for this architecture. GPUs usually outperform multicore CPUs in some compute intensive and massively parallel applications that have regular processing patterns. However, most scientific applications rely on crucial memory-bound kernels and may witness bottlenecks due to the overhead of the memory bus latency. They can still take advantage of the GPU compute power capabilities, provided that an efficient architecture-aware design is achieved.
This dissertation presents a uniform design strategy for optimizing critical memory-bound kernels on GPUs. Based on hierarchical register blocking, double buffering and latency hiding techniques, this strategy leverages the performance of a wide range of standard numerical kernels found in dense and sparse linear algebra libraries. The work presented here focuses on matrix-vector multiplication kernels (MVM) as repre-
sentative and most important memory-bound operations in this context. Each kernel inherits the benefits of the proposed strategies. By exposing a proper set of tuning parameters, the strategy is flexible enough to suit different types of matrices, ranging from large dense matrices, to sparse matrices with dense block structures, while high performance is maintained. Furthermore, the tuning parameters are used to maintain the relative performance across different GPU architectures. Multi-GPU acceleration is proposed to scale the performance on several devices. The performance experiments show improvements ranging from 10% and up to more than fourfold speedup against competitive GPU MVM approaches. Performance impacts on high-level numerical libraries and a computational astronomy application are highlighted, since such memory-bound kernels are often located in innermost levels of the software chain. The excellent performance obtained in this work has led to the adoption of code in NVIDIAs widely distributed cuBLAS library.
26
Zhou, Wei. "Fast Order Basis and Kernel Basis Computation and Related Problems". Thesis, 2012. http://hdl.handle.net/10012/7326.
Abstract (sommario):
In this thesis, we present efficient deterministic algorithms
for polynomial matrix computation problems, including the computation
of order basis, minimal kernel basis, matrix inverse, column basis,
unimodular completion, determinant, Hermite normal form, rank and
rank profile for matrices of univariate polynomials over a field.
The algorithm for kernel basis computation also immediately provides
an efficient deterministic algorithm for solving linear systems. The
algorithm for column basis also gives efficient deterministic algorithms
for computing matrix GCDs, column reduced forms, and Popov normal
forms for matrices of any dimension and any rank.
We reduce all these problems to polynomial matrix multiplications.
The computational costs of our algorithms are then similar to the
costs of multiplying matrices, whose dimensions match the input matrix
dimensions in the original problems, and whose degrees equal the average
column degrees of the original input matrices in most cases. The use
of the average column degrees instead of the commonly used matrix
degrees, or equivalently the maximum column degrees, makes our computational
costs more precise and tighter. In addition, the shifted minimal bases
computed by our algorithms are more general than the standard minimal
bases.
27
Lee, Jia-Wei, e 李家瑋. "Application of the Clifford algebra valued boundary integral equations with Cauchy-type kernels to some engineering problems". Thesis, 2016. http://ndltd.ncl.edu.tw/handle/02351203973629353925.
Abstract (sommario):
博士國立臺灣海洋大學
河海工程學系
104
The conventional complex variable boundary integral equation (CVBIE) based on the conventional Cauchy integral formula is powerful and suitable to solve two-dimensional problems. In particular, the unknown function is a complex-valued holomorphic function. In other words, the unknown function satisfies the Cauchy-Riemann equations. However, the most part of practical engineering problems are three-dimensional problems and do not necessarily satisfies Cauchy-Riemann equations. Therefore, there are two targets in this dissertation. One is to extend the conventional CVBIE to solve two-dimensional problems for which the unknown function is not a complex-valued holomorphic function. The other is to extend to three-dimensions and derive an extended BIE still preserving some properties of complex variables in the three-dimensional state. For the extension of the conventional CVBIE, we employ the Borel-Pompeiu formula to derive the generalized CVBIE. In this way, the torsion problems can be solved in the state of two shear stress fields directly. In addition, the torsional rigidity can also be determined simultaneously. Since the theory of complex variables has a limitation that is only suitable for 2-dimensional problems, we introduce Clifford algebra and Clifford analysis to replace complex variables to deal with 3-dimensional problems. Clifford algebra can be seen as an extension of complex or quaternionic algebras. Clifford analysis is also known as hypercomplex analysis. We apply the Clifford algebra valued Stokes' theorem to derive Clifford algebra valued BIEs with Cauchy-type kernels. In this way, some three-dimensional problem with multiple unknown fields may be solved straightforward. Finally, several electromagnetic scattering problems are considered to check the validity of the derived Clifford algebra valued BIEs.
28
Biswas, Prasenjit. "Hardware Consolidation Of Systolic Algorithms On A Coarse Grained Runtime Reconfigurable Architecture". Thesis, 2011. https://etd.iisc.ac.in/handle/2005/2108.
Abstract (sommario):
Application domains such as Bio-informatics, DSP, Structural Biology, Fluid Dynamics, high resolution direction finding, state estimation, adaptive noise cancellation etc. demand high performance computing solutions for their simulation environments. The core computations of these applications are in Numerical Linear Algebra (NLA) kernels. Direct solvers are predominantly required in the domains like DSP, estimation algorithms like Kalman Filter etc, where the matrices on which operations need to be performed are either small or medium sized, but dense. Faddeev's Algorithm is often used for solving
dense linear system of equations. Modified Faddeev's algorithm (MFA) is a general algorithm on which LU decomposition, QR factorization or SVD of matrices can be realized. MFA has the good property of realizing a host of matrix operations by computing the Schur complements on four blocked matrices, thereby reducing the overall computation requirements. We will use MFA as a representative Direct Solver in this work. We further discuss Given's rotation based QR algorithm for Decomposition of any matrix, often used to solve the linear least square problem. Systolic Array Architectures are widely accepted ASIC solutions for NLA algorithms. But the \can of worms" associated with
this traditional solution spawns the need for alternative solutions. While popular custom hardware solution in form of systolic arrays can deliver high performance, but because of their rigid structure they are not scalable and reconfigurable, and hence not commercially viable. We show how a Reconfigurable computing platform can serve to contain the \can of worms". REDEFINE, a coarse grained runtime reconfigurable architecture has been used for systolic actualization of NLA kernels. We elaborate upon streaming NLA-specific enhancements to REDEFINE in order to meet expected performance goals. We explore the need for an algorithm aware custom compilation framework. We bring about a proposition to realize Faddeev's Algorithm on REDEFINE. We show that REDEFINE performs several times faster than traditional GPPs. Further we direct our interest to QR Decomposition to be the next NLA kernel as it ensures better stability than LU and other decompositions. We use QR Decomposition as a case study to explore the design space of the proposed solution on REDEFINE. We also investigate the architectural details of the Custom Functional Units (CFU) for these NLA kernels. We determine the right size
of the sub-array in accordance with the optimal pipeline depth of the core execution units and the number of such units to be used per sub-array. The framework used to realize QR Decomposition can be generalized for the realization of other algorithms dealing with decompositions like LU, Faddeev's Algorithm, Gauss-Jordon etc with different CFU definitions .
29
Biswas, Prasenjit. "Hardware Consolidation Of Systolic Algorithms On A Coarse Grained Runtime Reconfigurable Architecture". Thesis, 2011. http://etd.iisc.ernet.in/handle/2005/2108.
Abstract (sommario):
Application domains such as Bio-informatics, DSP, Structural Biology, Fluid Dynamics, high resolution direction finding, state estimation, adaptive noise cancellation etc. demand high performance computing solutions for their simulation environments. The core computations of these applications are in Numerical Linear Algebra (NLA) kernels. Direct solvers are predominantly required in the domains like DSP, estimation algorithms like Kalman Filter etc, where the matrices on which operations need to be performed are either small or medium sized, but dense. Faddeev's Algorithm is often used for solving
dense linear system of equations. Modified Faddeev's algorithm (MFA) is a general algorithm on which LU decomposition, QR factorization or SVD of matrices can be realized. MFA has the good property of realizing a host of matrix operations by computing the Schur complements on four blocked matrices, thereby reducing the overall computation requirements. We will use MFA as a representative Direct Solver in this work. We further discuss Given's rotation based QR algorithm for Decomposition of any matrix, often used to solve the linear least square problem. Systolic Array Architectures are widely accepted ASIC solutions for NLA algorithms. But the \can of worms" associated with
this traditional solution spawns the need for alternative solutions. While popular custom hardware solution in form of systolic arrays can deliver high performance, but because of their rigid structure they are not scalable and reconfigurable, and hence not commercially viable. We show how a Reconfigurable computing platform can serve to contain the \can of worms". REDEFINE, a coarse grained runtime reconfigurable architecture has been used for systolic actualization of NLA kernels. We elaborate upon streaming NLA-specific enhancements to REDEFINE in order to meet expected performance goals. We explore the need for an algorithm aware custom compilation framework. We bring about a proposition to realize Faddeev's Algorithm on REDEFINE. We show that REDEFINE performs several times faster than traditional GPPs. Further we direct our interest to QR Decomposition to be the next NLA kernel as it ensures better stability than LU and other decompositions. We use QR Decomposition as a case study to explore the design space of the proposed solution on REDEFINE. We also investigate the architectural details of the Custom Functional Units (CFU) for these NLA kernels. We determine the right size
of the sub-array in accordance with the optimal pipeline depth of the core execution units and the number of such units to be used per sub-array. The framework used to realize QR Decomposition can be generalized for the realization of other algorithms dealing with decompositions like LU, Faddeev's Algorithm, Gauss-Jordon etc with different CFU definitions .
30
Sen, Samrat. "Geometric invariants for a class of submodules of analytic Hilbert modules". Thesis, 2019. https://etd.iisc.ac.in/handle/2005/4455.
Abstract (sommario):
Let $\Omega \subseteq \mathbb C^m$ be a bounded connected open set and $\mathcal H \subseteq \mathcal O(\Omega)$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal H$ possesses a reproducing kernel $K$, the polynomial ring $\mathbb C[\underline{z}]\subseteq \mathcal H$ is dense and the point-wise multiplication induced by $p\in \mathbb C[\underline{z}]$ is bounded on $\mathcal H$. We fix an ideal $\mathcal I \subseteq \mathbb C[\underline{z}]$ generated by $p_1,\ldots,p_t$ and let $[\mathcal I]$ denote the completion of $\mathcal I$ in $\mathcal H$. Let $X:[\mathcal I] \to \mathcal H$ be the inclusion map.
Thus we have a short exact sequence of Hilbert modules
\begin{tikzcd}
0 \arrow{r} &\mbox{[}
\mathcal I \mbox{]}
\arrow{r}{X}
& {\mathcal H} \arrow{r}{\pi} & \mathcal Q \arrow{r}& 0 ,
\end{tikzcd}
where the module multiplication in the quotient $\mathcal Q:=[\mathcal I]^\perp$ is given by the formula $m_p f = P_{[\mathcal I]^\perp} (p f),$ $p\in \mathbb C[\underline{z}],\,f\in \mathcal Q$. The analytic Hilbert module $\mathcal H$ defines a subsheaf
$\mathcal S^\mathcal H$ of the sheaf $\mathcal O(\Omega)$ of holomorphic functions defined on $\Omega$. For any open $U \subset \Omega$, it is obtained by setting
$$\mathcal S^\mathcal H(U) := \Big \{\, \sum_{i=1}^n ({f_i|}_U) h_i : f_i \in \mathcal H, h_i \in \mathcal O(U), n\in\mathbb N\,\Big \}.$$
This is locally free and naturally gives rise to a holomorphic line bundle on $\Omega$. However, in general, the sheaf corresponding to the sub-module $[\mathcal I]$ is not locally free but only coherent.
Building on the earlier work of S. Biswas, a decomposition theorem is obtained for the kernel $K_{[\mathcal I]}$ along the zero set $V_{[\mathcal I]}:=\big\{z\in \mathbb C^m: f(z) = 0, f\in [\mathcal I]\big\}$ which is assumed to be a submanifold of codimension $t$: There exists anti-holomorphic maps $F_1, \ldots, F_t: V_{[\mathcal I]}\to [\mathcal I]$ such that
$$ K_{[\mathcal I]}(\cdot, u) = \overline{p_1(u)} F^1_w(u) + \cdots \overline{p_t(u)} F_w^t(u),\, u\in \Omega_w,$$
in some neighbourhood $\Omega_w$ of each fixed but arbitrary $w\in V_{[\mathcal I]}$ for some anti-holomorphic maps $F_w^1, \ldots, F^t_w: \Omega_w \to [\mathcal I]$ extending $F_1, \ldots,F_t$. The anti-holomorphic maps $F_1, \ldots,F_t$ are linearly independent on $V_{[\mathcal I]}$, defining a rank $t$ anti-holomorphic Hermitian vector bundle on it. This gives rise to complex geometric invariants for the pair $([\mathcal I], \mathcal H)$.
Next, using a decomposition formula obtained from an earlier work of Douglas, Misra and Varughese, the maps $F_1, \ldots, F_t: V_{[\mathcal I]}\to [\mathcal I]$ are explicitly determined with the additional assumption that $p_{i},p_{j}$ are relatively prime for $i\neq j$. Using this, a line bundle on $V_{[\mathcal I]}\times\mathbb{P}^{t-1}$ is constructed via the monoidal transformation around $V_{[\mathcal I]}$ which provides useful invariants for $([\mathcal I], \mathcal H)$.
Localising the modules $[\mathcal I]$ and $\mathcal H$ at $w\in \Omega$, we obtain the localization $X(w)$ of the module map $X$. The localizations are nothing but the quotient modules $[\mathcal I]/{[\mathcal I]_w}$ and $\mathcal H/{\mathcal H_w}$, where $[\mathcal I]_w$ and $\mathcal H_w$ are the maximal sub-modules of functions vanishing at $w$. These clearly define anti-holomorphic line bundles $E_{[\mathcal I]}$ and $E_\mathcal H$, respectively, on $\Omega\setminus V_{[\mathcal I]}$. However, there is a third line bundle, namely, ${\rm Hom}(E_\mathcal H, E_{[\mathcal I]})$ defined by the anti-holomorphic map $X(w)^*$. The curvature of a holomorphic line bundle $\mathcal L$ on $\Omega$, computed with respect to a holomorphic frame $\gamma$ is given by the formula
$$\mathcal K_\mathcal L(z) = \sum_{i,j=1}^{m}\tfrac{\partial^2}{\partial z_i \partial \bar{z}_j}\log\|\gamma(z)\|^2 dz_i \wedge d\bar{z}_j.$$
It is a complete invariant for the line bundle $\mathcal L$. The alternating sum
$$
\mathcal A_{[\mathcal I], \mathcal H}(w):=\mathcal K_X(w) - \mathcal K_{[\mathcal{I}]}(w) + \mathcal K_{\mathcal{H}}(w) = 0,\,\, w\in \Omega \setminus V_{[\mathcal I]},
$$
where $\mathcal K_X$, $\mathcal K_{[\mathcal{I}]}$ and $\mathcal K_{\mathcal{H}}$ denote the curvature $(1,1)$ form of the line bundles $E_X$, $E_{[\mathcal{I}]}$ and $E_{\mathcal{H}}$, respectively. Thus it is an invariant for the pair $([\mathcal I], \mathcal H)$. However, when $\mathcal I$ is principal, by taking distributional derivatives, $\mathcal A_{[\mathcal I], \mathcal H}(w)$ extends to all of $\Omega$ as a $(1,1)$ current. Consider the following diagram of short exact sequences of Hilbert modules:
$$(1)\,\,\,\,\,\,\,\,\,\,\, \begin{tikzcd}
0\arrow{r} &\mbox{[}\mathcal I\mbox{]} \arrow{d} \arrow{r} {X}
& {\mathcal H}\arrow{d}{L} \arrow{r}{\pi} & \mathcal Q\arrow{d} \arrow{r}& 0\\
0\arrow{r} &\mbox{[}\widetilde{\mathcal I}\mbox{]} \arrow{r}{\widetilde{X}} &\widetilde{\mathcal H} \arrow{r}{\tilde{\pi}}& \widetilde{\mathcal Q} \arrow{r}& 0,
\end{tikzcd} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (2)\,\,\,\,\,\,\,\,\,\,\, \begin{tikzcd}
\mbox{[}\mathcal I\mbox{]} \arrow{d} \arrow{r} {X}
& {\mathcal H}\arrow{d}{L}\\
\mbox{[}\widetilde{\mathcal I}\mbox{]} \arrow{r}{\widetilde{X}} &\widetilde{\mathcal H} \end{tikzcd}$$
It is shown that if $\mathcal A_{[\mathcal I], \mathcal H}(w)=\mathcal A_{[\widetilde{\mathcal I}], \widetilde{\mathcal H}}(w)$, then $L|_{[\mathcal I]}$ makes the second diagram commute. Hence, if $L$ is bijective, then $[\mathcal I]$ and $[\widetilde{\mathcal I]}$ are equivalent as Hilbert modules. It follows that the alternating sum is an invariant for the ``rigidity'' phenomenon.
31
Mascarenhas, Helena. "Convolution type operators on cones and asymptotic spectral theory". Doctoral thesis, 2003. https://monarch.qucosa.de/id/qucosa%3A18097.
Abstract (sommario):
Die Arbeit beschäftigt sich mit Faltungsoperatoren auf Kegeln, die in Lebesgueräumen L^p(R^2) (1