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Timed State Tree Structures: Supervisory Control and Fault DiagnosisSaadatpoor, Ali 15 March 2010 (has links)
It is well known that the optimal nonblocking supervisory control problem of timed discrete event systems is NP-hard, subject in particular to state space explosion that is exponential in the number of system components. In this thesis, we propose to manage complexity by organizing the system as a Timed State Tree Structure (TSTS). TSTS are an adaptation of STS to timed Supervisory Control Theory (SCT). Based on TSTS we present an e±cient recursive symbolic algorithm that can perform nonblocking supervisory control design for systems of state size 10^12 and higher.
Failure diagnosis is the process of detecting and identifying deviations of a system from its normal behavior using the information available through sensors. A method for fault diagnosis of the TSTS model is proposed. A state based diagnoser is constructed for each timed holon of TSTS. Fault diagnosis is accomplished using the state estimates provided by the timed holon diagnosers. The diagnosers may communicate among each other in order to update their state estimates. At any given time, only a subset of the diagnosers are operational, and as a result, instead of the entire model of the system, only the models of the timed holons associated with the operational diagnosers are used.
It is shown that the computational complexity of constructing and storing the transition systems required for diagnosis in the proposed approach is polynomial in the number of system components, whereas in the original monolithic approach the computational complexity is exponential.
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Timed State Tree Structures: Supervisory Control and Fault DiagnosisSaadatpoor, Ali 15 March 2010 (has links)
It is well known that the optimal nonblocking supervisory control problem of timed discrete event systems is NP-hard, subject in particular to state space explosion that is exponential in the number of system components. In this thesis, we propose to manage complexity by organizing the system as a Timed State Tree Structure (TSTS). TSTS are an adaptation of STS to timed Supervisory Control Theory (SCT). Based on TSTS we present an e±cient recursive symbolic algorithm that can perform nonblocking supervisory control design for systems of state size 10^12 and higher.
Failure diagnosis is the process of detecting and identifying deviations of a system from its normal behavior using the information available through sensors. A method for fault diagnosis of the TSTS model is proposed. A state based diagnoser is constructed for each timed holon of TSTS. Fault diagnosis is accomplished using the state estimates provided by the timed holon diagnosers. The diagnosers may communicate among each other in order to update their state estimates. At any given time, only a subset of the diagnosers are operational, and as a result, instead of the entire model of the system, only the models of the timed holons associated with the operational diagnosers are used.
It is shown that the computational complexity of constructing and storing the transition systems required for diagnosis in the proposed approach is polynomial in the number of system components, whereas in the original monolithic approach the computational complexity is exponential.
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A Framework for Hierarchical Perception–Action Learning Utilizing Fuzzy ReasoningWindridge, David, Felsberg, Michael, Shaukat, Affan January 2013 (has links)
Perception-action (P-A) learning is an approach to cognitive system building that seeks to reduce the complexity associated with conventional environment-representation/action-planning approaches. Instead, actions are directly mapped onto the perceptual transitions that they bring about, eliminating the need for intermediate representation and significantly reducing training requirements. We here set out a very general learning framework for cognitive systems in which online learning of the P-A mapping may be conducted within a symbolic processing context, so that complex contextual reasoning can influence the P-A mapping. In utilizing a variational calculus approach to define a suitable objective function, the P-A mapping can be treated as an online learning problem via gradient descent using partial derivatives. Our central theoretical result is to demonstrate top-down modulation of low-level perceptual confidences via the Jacobian of the higher levels of a subsumptive P-A hierarchy. Thus, the separation of the Jacobian as a multiplying factor between levels within the objective function naturally enables the integration of abstract symbolic manipulation in the form of fuzzy deductive logic into the P-A mapping learning. We experimentally demonstrate that the resulting framework achieves significantly better accuracy than using P-A learning without top-down modulation. We also demonstrate that it permits novel forms of context-dependent multilevel P-A mapping, applying the mechanism in the context of an intelligent driver assistance system. / DIPLECS / GARNICS / CUAS
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[en] HIERARCHICAL NEURO-FUZZY MODELS / [pt] MODELOS NEURO-FUZZY HIERÁRQUICOSFLAVIO JOAQUIM DE SOUZA 13 December 2005 (has links)
[pt] Esta dissertação apresenta uma nova proposta de sistemas
(modelos) neuro-fuzzy que possuem, além do tradicional
aprendizado dos parâmetros, comuns às redes neurais e aos
sistemas nero-fuzzy, as seguintes características:
aprendizado de estrutura, a partir do uso de
particionamentos recursisvos; número maior de entradas que
o comumente encontrado nos sistemas neuro-fuzzy; e regras
com hierarquia. A definição da estrutura é uma necessidade
que surge quando da implementação de um determinado
modelo. Pode-se citar o caso das redes neurais, em que se
deve determinar (ou arbitrar) a priori sua estrutura
(número de camadas e quantidade de neurônios por camadas)
antes de qualquer teste. Um método automático de
aprendizado da estrutura é, portanto, uma característica
importante em qualquer modelo. Um sistema que também
permita o uso de um número maior de entradas é
interessante para se abranger um maior número de
aplicações. As regras com hierarquia são um subproduto do
método de aprendizado de estrutura desenvolvido nestes
novos modelos.
O trabalho envolveu três partes principais: um
levantamento sobre os sistemas neuro-fuzzy existentes e
sobre os métodos mais comuns de ajuste de parâmetros; a
definição e implementação de dois modelos neuro-fuzzy
hierárquicos; e o estudo de casos.
No estudo sobre os sistemas neuro-fuzzy(SNF) fez-se um
levantamento na bibliografia da área sobre as
características principais desses sistemas, incluindo suas
virtudes e deficiências. Este estudo gerou a proposta de
uma taxonomia para os SNF, em função das características
fuzzy neurais. Em virtude deste estudo constataram-se
limitações quanto à capacidade de criação de sua própria
estrutura e quanto ao número reduzido de entradas
possíveis.
No que se refere aos métodos de ajuste dos parâmetros
abordou-se os métodos mais comuns utilizados nos SNF, a
saber: o método dos mínimos quadrados com sua solução
através de métodos numéricos iterativos; e o método
gradient descent e seus derivados como o BackPropagation e
o RProp(Resilient BackPropagation).
A definição dos dois novos modelos neuro-fuzzy foi feita a
partir do estudo das características desejáveis e das
limitações dos SNF até então desenvolvidos. Observou-se
que a base de regras dos SNF juntamente com os seus
formatos de particionamento dos espaços de entrada e saída
têm grande influência sobre o desempenho e as limitações
destes modelos. Assim sendo, decidiu-se utilizar uma nova
forma de particionamento que eliminasse ou reduzisse as
limitações existentes- os particionamentos recursivos.
Optou-se pelo uso dos particionamentos Quadtree e BSP,
gerando os dois modelos NFHQ (Neuro-Fuzzy Hierárquico
Quadree) e NFHB (Neiro-Fuzzy Hierárquico BSP). Com o uso
de particionamentos obteve-se um nova classe de SNF que
permitiu além do aprendizado dos parâmetros, também o
aprendizado dos parâmetros. Isto representa um grande
diferencial em relação aos SNF tradicionais, além do fato
de se conseguir extender o limite do número de entradas
possíveis para estes sistemas.
No estudo de casos, os dois modelos neurofuzzy
hierárquicos foram testados 16 casos diferentes, entre as
aplicações benchmarks mais tradicionais da área e
problemas com maior número de entradas. Entre os casos
estudados estão: o conjunto de dados IRIS; o problema das
duas espirais; a previsão da série caótica de Mackey-
Glass; alguns sistemas de diagnóstico e classificação
gerados a partir de conjuntos de dados comumente
utilizados em artigos de machine learning e uma aplicação
de previsão de carga elétrica. A implementação dos dois
novos modelos neuro-fuzzy foi efetuada em linguagem pascal
e com o uso de um compilador de 32 bits para micros da
linha PC (Pentium) com sistema operacional DOS 32 bits,
Windows, ou Linux.
Os testes efetuados demostraram que: esses novos modelos
se ajustam bem a qualquer conjunto de dados; geram sua
própria estrutura; ajustam seus parâmetros com boa
generalização e extraem / [en] This dissertation presents a new proposal of neurofuzzy
systems (models), which present, in addition to the
learning capacity (which are common to the neural networks
and neurofuzzy systems) the following features: learning
of the structure; the use of recursive partitioning; a
greater number of inputs than usually allowed in
neurofuzzy systems; and hierarchical rules. The
structure´s definition is needed when implementing a
certain model. In the neural network case, for example,
one must, first of all, estabilish its structure (number
of layers and number of neurons per layers) before any
test is performed. So, an important feature for any model
is the existence of an automatic learning method for
creating its structure. A system that allows a larger
number of inputs is also important, in order to extend the
range of possible applications. The hierarchical rules
feature results from the structure learning method
developed for these two models.
The work has involved three main parts: study of the
existing neurofuzzy systems and of the most commom methods
to adjust its parameters; definition and implementation of
two hierarchical neurofuzzy models; and case studies.
The study of neurofuzzy systems (NFS) was accomplished by
creating a survey on this area, including advantages,
drawbacks and the main features of NFS. A taxonomy about
NFS was then proposed, taking into account the neural and
fuzzy features of the existing systems. This study pointed
out the limitations of neurofuzzy systems, mainly their
poor capability of creating its own structure and the
reduced number of allowed inputs.
The study of the methods for parameter adjustment has
focused on the following algorithms: Least Square
estimator (LSE) and its solutions by numerical iterative
methods; and the basic gradient descent method and its
offsprings such as Backpropagation and Rprop (Resilient
Backpropagation).
The definition of two new neurofuzzy models was
accomplished by considering desirable features and
limitations of the existing NFS. It was observed that the
partitioning formats and rule basis of the NFS have great
influence on its performance and limitations. Thus, the
decision to use a new partitioning method to remove or
reduce the existing limitations - the recursive
partitioning. The Quadtree and BSP partitioning were then
adopted, generating the so called Quadree Hierarchical
Neurofuzzy model (NFHQ) and the BSP hierarchical
Neurofuzzy model (NFHB). By using these kind os
partitioning a new class of NFS was obtained allowing the
learning of the structure in addition to parameter
learning. This Feature represents a great differential in
relation to the traditional NFS, besides overcoming the
limitation in the number of allowed inputs.
In the case studies, the two neurofuzzy models were tested
in 16 differents cases, such as traditional benchmarks and
problems with a greater number of inputs. Among the cases
studied are: the IRIS DATA set; the two spirals problem;
the forecasting of Mackey-Glass chaotic time series; some
diagnosis and classifications problems, found in papers
about machine learning; and a real application involving
load forecasting. The implementation of the two new
neurofuzzy models was carried out using a 32 bit Pascal
compiler for PC microcomputers using DOS or Linux
operating system.
The tests have shown that: these new models are able to
adjust well any data sets; they create its own struture;
they adjust its parameters, presenting a good
generalization performance; and automatically extract the
fuzzy rules. Beyond that, applications with a greater
number of inputs for these neurofuzzy models. In short two
neurofuzzy models were developed with the capability of
structure learning, in addition to parameter learning.
Moreover, these new models have good interpretability
through hierarchical fuzzy rules. They are not black coxes
as the neural networks.
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Modelagem e simulação de redes em chip sem fio. / Wireless network on chip modeling and simulation.Ferreira, Jefferson Chaves 17 March 2015 (has links)
O paradigma das redes em chip (NoCs) surgiu a fim de permitir alto grau de integração entre vários núcleos de sistemas em chip (SoCs), cuja comunicação é tradicionalmente baseada em barramentos. As NoCs são definidas como uma estrutura de switches e canais ponto a ponto que interconectam núcleos de propriedades intelectuais (IPs) de um SoC, provendo uma plataforma de comunicação entre os mesmos. As redes em chip sem fio (WiNoCs) são uma abordagem evolucionária do conceito de rede em chip (NoC), a qual possibilita a adoção dos mecanismos de roteamento das NoCs com o uso de tecnologias sem fio, propondo a otimização dos fluxos de tráfego, a redução de conectores e a atuação em conjunto com as NoCs tradicionais, reduzindo a carga nos barramentos. O uso do roteamento dinâmico dentro das redes em chip sem fio permite o desligamento seletivo de partes do hardware, o que reduz a energia consumida. Contudo, a escolha de onde empregar um link sem fio em uma NoC é uma tarefa complexa, dado que os nós são pontes de tráfego os quais não podem ser desligados sem potencialmente quebrar uma rota preestabelecida. Além de fornecer uma visão sobre as arquiteturas de NoCs e do estado da arte do paradigma emergente de WiNoC, este trabalho também propõe um método de avaliação baseado no já consolidado simulador ns-2, cujo objetivo é testar cenários híbridos de NoC e WiNoC. A partir desta abordagem é possível avaliar diferentes parâmetros das WiNoCs associados a aspectos de roteamento, aplicação e número de nós envolvidos em redes hierárquicas. Por meio da análise de tais simulações também é possível investigar qual estratégia de roteamento é mais recomendada para um determinado cenário de utilização, o que é relevante ao se escolher a disposição espacial dos nós em uma NoC. Os experimentos realizados são o estudo da dinâmica de funcionamento dos protocolos ad hoc de roteamento sem fio em uma topologia hierárquica de WiNoC, seguido da análise de tamanho da rede e dos padrões de tráfego na WiNoC. / The network on chip (NoC) paradigm was conceived in order to allow a high-level integration among several system-on-chip (SoC) cores whose communication is traditionally based on buses. NoCs are defined as a switch structure with communication channels, which interconnect SoC Intellectual Property cores allowing data transfer among them. Wireless networks on chip (Wi-NoC) are an evolutionary approach from the network on chip (NoC) concept, proposing the traffic flow optimization among different modules by providing wireless shortcuts over a traditional NoC, reducing the bus load. Using dynamic routing within the WiNoC enables selective hardware power management, reducing power consumption. However, choosing where to deploy a wireless link over a NoC is a complex task given that those nodes are gateways that cannot be turned off without potentially breaking an established route. Besides providing an overview of NoC architectures and about the emerging WiNoC paradigm, this work proposes a method to use well known ns-2 network simulator to test mixed NoC-WiNoC scenarios. With this approach it is possible to evaluate different WiNoC parameters associated to routing, application and total number of nodes in hierarchical topologies. Simulation study can also point-out which routing strategy is more suitable for a given scenario, what is considered important when choosing wireless node placement over a NoC.We performed experiments to understand the dynamics of wireless ad hoc routing protocol functioning in a WiNoC hierarchical topology, followed by an analysis of network size and traffic patterns over WiNoC.
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Modelagem e simulação de redes em chip sem fio. / Wireless network on chip modeling and simulation.Jefferson Chaves Ferreira 17 March 2015 (has links)
O paradigma das redes em chip (NoCs) surgiu a fim de permitir alto grau de integração entre vários núcleos de sistemas em chip (SoCs), cuja comunicação é tradicionalmente baseada em barramentos. As NoCs são definidas como uma estrutura de switches e canais ponto a ponto que interconectam núcleos de propriedades intelectuais (IPs) de um SoC, provendo uma plataforma de comunicação entre os mesmos. As redes em chip sem fio (WiNoCs) são uma abordagem evolucionária do conceito de rede em chip (NoC), a qual possibilita a adoção dos mecanismos de roteamento das NoCs com o uso de tecnologias sem fio, propondo a otimização dos fluxos de tráfego, a redução de conectores e a atuação em conjunto com as NoCs tradicionais, reduzindo a carga nos barramentos. O uso do roteamento dinâmico dentro das redes em chip sem fio permite o desligamento seletivo de partes do hardware, o que reduz a energia consumida. Contudo, a escolha de onde empregar um link sem fio em uma NoC é uma tarefa complexa, dado que os nós são pontes de tráfego os quais não podem ser desligados sem potencialmente quebrar uma rota preestabelecida. Além de fornecer uma visão sobre as arquiteturas de NoCs e do estado da arte do paradigma emergente de WiNoC, este trabalho também propõe um método de avaliação baseado no já consolidado simulador ns-2, cujo objetivo é testar cenários híbridos de NoC e WiNoC. A partir desta abordagem é possível avaliar diferentes parâmetros das WiNoCs associados a aspectos de roteamento, aplicação e número de nós envolvidos em redes hierárquicas. Por meio da análise de tais simulações também é possível investigar qual estratégia de roteamento é mais recomendada para um determinado cenário de utilização, o que é relevante ao se escolher a disposição espacial dos nós em uma NoC. Os experimentos realizados são o estudo da dinâmica de funcionamento dos protocolos ad hoc de roteamento sem fio em uma topologia hierárquica de WiNoC, seguido da análise de tamanho da rede e dos padrões de tráfego na WiNoC. / The network on chip (NoC) paradigm was conceived in order to allow a high-level integration among several system-on-chip (SoC) cores whose communication is traditionally based on buses. NoCs are defined as a switch structure with communication channels, which interconnect SoC Intellectual Property cores allowing data transfer among them. Wireless networks on chip (Wi-NoC) are an evolutionary approach from the network on chip (NoC) concept, proposing the traffic flow optimization among different modules by providing wireless shortcuts over a traditional NoC, reducing the bus load. Using dynamic routing within the WiNoC enables selective hardware power management, reducing power consumption. However, choosing where to deploy a wireless link over a NoC is a complex task given that those nodes are gateways that cannot be turned off without potentially breaking an established route. Besides providing an overview of NoC architectures and about the emerging WiNoC paradigm, this work proposes a method to use well known ns-2 network simulator to test mixed NoC-WiNoC scenarios. With this approach it is possible to evaluate different WiNoC parameters associated to routing, application and total number of nodes in hierarchical topologies. Simulation study can also point-out which routing strategy is more suitable for a given scenario, what is considered important when choosing wireless node placement over a NoC.We performed experiments to understand the dynamics of wireless ad hoc routing protocol functioning in a WiNoC hierarchical topology, followed by an analysis of network size and traffic patterns over WiNoC.
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Celulární polymerní nanokompozity / Cellular polymer nanocompositesZárybnická, Klára January 2022 (has links)
Tato dizertační práce se zabývá přípravou a charakterizací nanokompozitních polymerních pěn se zaměřením na strukturu materiálu a aplikaci v 3D tisku. Cílem práce je studium materiálu s vysoce organizovanou hierarchickou strukturou – od nanoměřítka, přes mikroskopickou strukturu po makroskopická tělesa. V první části práce byly řešeny strukturní vlastnosti nanokompozitů připravených z polymerních skel roztokovou metodou. Byl hledán obecně platný trend, pomocí kterého by bylo možné předpovídat disperzi nanočástic v kompozitu. Ukázalo se, že řídícím faktorem může být závislost na rozdílu parametrů rozpustnosti polymeru a rozpouštědla. Tento poznatek byl ověřen na systémech obsahujících různé nanočástice, polymery a rozpouštědla. Se znalostí principů pro řízení struktury nanokompozitů byly připraveny nanokompozity impaktního polystyrenu plněného nanosilikou. Tyto nanokompozity posloužily jako základ pro přípravu polymerních nakompozitních pěn. Porézní struktury bylo dosaženo pomocí termálního chemického nadouvadla azodikarbonamidu. Z těchto materiálů byly extrudovány filamenty, které byly následně zpracovány pomocí 3D tisku do požadovaných tvarů a vypěněny. Výsledkem byla hierarchická struktura s organizací struktury od nano (organizace nanočástic), přes mikro (struktura dvoukomponentní polymerní směsi a struktura pěny) po makroměřítko (struktura pěny a design 3D tisku). Byl pozorován vliv nanočástic na strukturu a termální a mechanické vlastnosti polymerních pěn. Nanočástice fungují při tvorbě pěny jako nukleační činidlo, na jejich povrchu snadno dochází k tvorbě pórů, takže s obsahem nanočástic v materiálu bylo vytvořeno více menších pórů, což napomohlo k homogenitě pěnové struktury. Přítomnost nanočástic změnila povrchovou energii zrn nadouvadla, díky čemuž docházelo k jeho rozkladu za nižích teplot a pěnění bylo i rychlejší. Nanočástice mají zároveň potenciál vyztužit stěny pěny a zlepšit tak mechanické vlastnosti. 3D tisk je oblíbená a hojně rozšířená technika, díky své jednoduchosti je v mnoha laboratořích a zkušebnách, proto roste poptávka po filamentech se speciálními vlastnostmi. Materiál vyvinutý v této dizertační práci je v podstatě hotovým a charakterizovaným produktem, který by mohl přispět k uspokojení této pohledávky.
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Resource Management in Large-scale SystemsPaya, Ashkan 01 January 2015 (has links)
The focus of this thesis is resource management in large-scale systems. Our primary concerns are energy management and practical principles for self-organization and self-management. The main contributions of our work are: 1. Models. We proposed several models for different aspects of resource management, e.g., energy-aware load balancing and application scaling for the cloud ecosystem, hierarchical architecture model for self-organizing and self-manageable systems and a new cloud delivery model based on auction-driven self-organization approach. 2. Algorithms. We also proposed several different algorithms for the models described above. Algorithms such as coalition formation, combinatorial auctions and clustering algorithm for scale-free organizations of scale-free networks. 3. Evaluation. Eventually we conducted different evaluations for the proposed models and algorithms in order to verify them. All the simulations reported in this thesis had been carried out on different instances and services of Amazon Web Services (AWS). All of these modules will be discussed in detail in the following chapters respectively.
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Dense matrix computations : communication cost and numerical stability / Calculs pour les matrices denses : coût de communication et stabilité numériqueKhabou, Amal 11 February 2013 (has links)
Cette thèse traite d’une routine d’algèbre linéaire largement utilisée pour la résolution des systèmes li- néaires, il s’agit de la factorisation LU. Habituellement, pour calculer une telle décomposition, on utilise l’élimination de Gauss avec pivotage partiel (GEPP). La stabilité numérique de l’élimination de Gauss avec pivotage partiel est caractérisée par un facteur de croissance qui est reste assez petit en pratique. Toutefois, la version parallèle de cet algorithme ne permet pas d’atteindre les bornes inférieures qui ca- ractérisent le coût de communication pour un algorithme donné. En effet, la factorisation d’un bloc de colonnes constitue un goulot d’étranglement en termes de communication. Pour remédier à ce problème, Grigori et al [60] ont développé une factorisation LU qui minimise la communication(CALU) au prix de quelques calculs redondants. En théorie la borne supérieure du facteur de croissance de CALU est plus grande que celle de l’élimination de Gauss avec pivotage partiel, cependant CALU est stable en pratique. Pour améliorer la borne supérieure du facteur de croissance, nous étudions une nouvelle stra- tégie de pivotage utilisant la factorisation QR avec forte révélation de rang. Ainsi nous développons un nouvel algorithme pour la factorisation LU par blocs. La borne supérieure du facteur de croissance de cet algorithme est plus petite que celle de l’élimination de Gauss avec pivotage partiel. Cette stratégie de pivotage est ensuite combinée avec le pivotage basé sur un tournoi pour produire une factorisation LU qui minimise la communication et qui est plus stable que CALU. Pour les systèmes hiérarchiques, plusieurs niveaux de parallélisme sont disponibles. Cependant, aucune des méthodes précédemment ci- tées n’exploite pleinement ces ressources. Nous proposons et étudions alors deux algorithmes récursifs qui utilisent les mêmes principes que CALU mais qui sont plus appropriés pour des architectures à plu- sieurs niveaux de parallélisme. Pour analyser d’une façon précise et réaliste / This dissertation focuses on a widely used linear algebra kernel to solve linear systems, that is the LU decomposition. Usually, to perform such a computation one uses the Gaussian elimination with partial pivoting (GEPP). The backward stability of GEPP depends on a quantity which is referred to as the growth factor, it is known that in general GEPP leads to modest element growth in practice. However its parallel version does not attain the communication lower bounds. Indeed the panel factorization rep- resents a bottleneck in terms of communication. To overcome this communication bottleneck, Grigori et al [60] have developed a communication avoiding LU factorization (CALU), which is asymptotically optimal in terms of communication cost at the cost of some redundant computation. In theory, the upper bound of the growth factor is larger than that of Gaussian elimination with partial pivoting, however CALU is stable in practice. To improve the upper bound of the growth factor, we study a new pivoting strategy based on strong rank revealing QR factorization. Thus we develop a new block algorithm for the LU factorization. This algorithm has a smaller growth factor upper bound compared to Gaussian elimination with partial pivoting. The strong rank revealing pivoting is then combined with tournament pivoting strategy to produce a communication avoiding LU factorization that is more stable than CALU. For hierarchical systems, multiple levels of parallelism are available. However, none of the previously cited methods fully exploit these hierarchical systems. We propose and study two recursive algorithms based on the communication avoiding LU algorithm, which are more suitable for architectures with multiple levels of parallelism. For an accurate and realistic cost analysis of these hierarchical algo- rithms, we introduce a hierarchical parallel performance model that takes into account processor and network hierarchies. This analysis enables us to accurately predict the performance of the hierarchical LU factorization on an exascale platform.
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