Iterative strong coupling of dimensionally heterogeneous models

In this article we address decomposition strategies especially tailored to perform strong coupling of dimensionally heterogeneous models, under the hypothesis that one wants to solve each submodel separately and implement the interaction between subdomains by boundary conditions alone. The novel methodology takes full advantage of the small number of interface unknowns in this kind of problems. Existing algorithms can be viewed as variants of the `natural` staggered algorithm in which each domain transfers function values to the other, and receives fluxes (or forces), and vice versa. This natural algorithm is known as Dirichlet-to-Neumann in the Domain Decomposition literature. Essentially, we propose a framework in which this algorithm is equivalent to applying Gauss-Seidel iterations to a suitably defined (linear or nonlinear) system of equations. It is then immediate to switch to other iterative solvers such as GMRES or other Krylov-based method. which we assess through numerical experiments showing the significant gain that can be achieved. indeed. the benefit is that an extremely flexible, automatic coupling strategy can be developed, which in addition leads to iterative procedures that are parameter-free and rapidly converging. Further, in linear problems they have the finite termination property. Copyright (C) 2009 John Wiley & Sons, Ltd.

Paralelizações de métodos numéricos em clusters empregando as bibliotecas MPICH, DECK e Pthread

Este trabalho tem como objetivo desenvolver e empregar técnicas e estruturas de dados agrupadas visando paralelizar os métodos do subespaço de Krylov, fazendo-se uso de diversas ferramentas e abordagens. A partir dos resultados é feita uma análise comparativa de desemvpenho destas ferramentas e abordagens. As paralelizações aqui desenvolvidas foram projetadas para serem executadas em um arquitetura formada por um agregado de máquinas indepentes e multiprocessadas (Cluster), ou seja , são considerados o paralelismo e intra-nodos. Para auxiliar a programação paralela em clusters foram, e estão sendo, desenvolvidas diferentes ferramentas (bibliotecas) que visam a exploração dos dois níveis de paralelismo existentes neste tipo de arquitetura. Neste trabalho emprega-se diferentes bibliotecas de troca de mensagens e de criação de threads para a exploração do paralelismo inter-nodos e intra-nodos. As bibliotecas adotadas são o DECK e o MPICH e a Pthread. Um dos itens a serem analisados nestes trabalho é acomparação do desempenho obtido com essas bibliotecas.O outro item é a análise da influência no desemepnho quando quando tulizadas múltiplas threads no paralelismo em clusters multiprocessados. Os métodos paralelizados nesse trabalho são o Gradiente Conjugação (GC) e o Resíduo Mínmo Generalizado (GMRES), quepodem ser adotados, respectivamente, para solução de sistemas de equações lineares sintéticos positivos e definidos e não simétricas. Tais sistemas surgem da discretização, por exemplo, dos modelos da hidrodinâmica e do transporte de massa que estão sendo desenvolvidos no GMCPAD. A utilização desses métodos é justificada pelo fato de serem métodos iterativos, o que os torna adequados à solução de sistemas de equações esparsas e de grande porte. Na solução desses sistemas através desses métodos iterativos paralelizados faz-se necessário o particionamento do domínio do problema, o qual deve ser feito visando um bom balanceamento de carga e minimização das fronteiras entre os sub-domínios. A estrutura de dados desenvolvida para os métodos paralelizados nesse trabalho permite que eles sejam adotados para solução de sistemas de equações gerados a partir de qualquer tipo de particionamento, pois o formato de armazenamento de dados adotado supre qualquer tipo de dependência de dados. Além disso, nesse trabalho são adotadas duas estratégias de ordenação para as comunicações, estratégias essas que podem ser importantes quando se considera a portabilidade das paralelizações para máquinas interligadas por redes de interconexão com buffer de tamanho insuficiente para evitar a ocorrência de dealock. Os resultados obtidos nessa dissertação contribuem nos trabalhos do GMCPAD, pois as paralelizações são adotadas em aplicações que estão sendo desenvolvidas no grupo.

Algoritmos adaptativos para o método GMRES(m)

Nesse trabalho apresentamos algoritmos adaptativos do M´etodo do Res´ıduo M´ınimo Generalizado (GMRES) [Saad e Schultz, 1986], um m´etodo iterativo para resolver sistemas de equa¸c˜oes lineares com matrizes n˜ao sim´etricas e esparsas, o qual baseia-se nos m´etodos de proje¸c˜ao ortogonal sobre um subespa¸co de Krylov. O GMRES apresenta uma vers˜ao reinicializada, denotada por GMRES(m), tamb´em proposta por [Saad e Schultz, 1986], com o intuito de permitir a utiliza¸c˜ao do m´etodo para resolver grandes sistemas de n equa¸c˜oes, sendo n a dimens˜ao da matriz dos coeficientes do sistema, j´a que a vers˜ao n˜ao-reinicializada (“Full-GMRES”) apresenta um gasto de mem´oria proporcional a n2 e de n´umero de opera¸c˜oes de ponto-flutuante proporcional a n3, no pior caso. No entanto, escolher um valor apropriado para m ´e dif´ıcil, sendo m a dimens˜ao da base do subespa¸co de Krylov, visto que dependendo do valor do m podemos obter a estagna¸c˜ao ou uma r´apida convergˆencia. Dessa forma, nesse trabalho, acrescentamos ao GMRES(m) e algumas de suas variantes um crit´erio que tem por objetivo escolher, adequadamente, a dimens˜ao, m da base do subespa¸co de Krylov para o problema o qual deseja-se resolver, visando assim uma mais r´apida, e poss´ıvel, convergˆencia. Aproximadamente duas centenas de experimentos foram realizados utilizando as matrizes da Cole¸c˜ao Harwell-Boeing [MCSD/ITL/NIST, 2003], que foram utilizados para mostrar o comportamento dos algoritmos adaptativos. Foram obtidos resultados muito bons; isso poder´a ser constatado atrav´es da an´alise das tabelas e tamb´em da observa ¸c˜ao dos gr´aficos expostos ao longo desse trabalho.

NI-GMRES precondicionado

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Deconvolution of the EPR spectra of vanadium oxide nanotubes

In this work we report results of continuous wave (CW) electron paramagnetic resonance (EPR) spectroscopy of vanadium oxide nanotubes. The observed EPR spectra are composed of a weak well-resolved spectrum of isolated V4+ ions on top of an intense and broad structure-less line shape, attributed to spin-spin exchanged V4+ clusters. With the purpose to deconvolute the structured weak spectrum from the composed broad line, a new approach based on the Krylov basis diagonalization method (KBDM) is introduced. It is based on the discrimination between broad and sharp components with respect to a selectable threshold and can be executed with few adjustable parameters, without the need of a priori information on the shape and structure of the lines. This makes the method advantageous with respect to other procedures and suitable for fast and routine spectral analysis, which, in conjunction with simulation techniques based on the spin Hamiltonian parameters, can provide a full characterization of the EPR spectrum. Results demonstrate and characterize the coexistence of two V4+ species in the nanotubes and show good progress toward the goal of obtaining high fidelity deconvoluted spectra from complex signals with overlapping broader line shapes. (C) 2012 Elsevier Inc. All rights reserved.

Phylogenetic identification of marine bacteria isolated from deep-sea sediments of the eastern South Atlantic Ocean

The deep-sea environments of the South Atlantic Ocean are less studied in comparison to the North Atlantic and Pacific Oceans. With the aim of identifying the deep-sea bacteria in this less known ocean, 70 strains were isolated from eight sediment samples (depth range between 1905 to 5560 m) collected in the eastern part of the South Atlantic, from the equatorial region to the Cape Abyssal Plain, using three different culture media. The strains were classified into three phylogenetic groups, Gammaproteobacteria, Firmicutes and Actinobacteria, by the analysis of 16s rRNA gene sequences. Gammaproteobacteria and Firmicutes were the most frequently identified groups, with Halomonas the most frequent genus among the strains. Microorganisms belonging to Firmicutes were the only ones observed in all samples. Sixteen of the 41 identified operational taxonomic units probably represent new species. The presence of potentially new species reinforces the need for new studies in the deep-sea environments of the South Atlantic.

Calculation of the eigenfunctions of the two-group neutron diffusion equation and application to modal decomposition of BWR instabilities

In this thesis, numerical methods aiming at determining the eigenfunctions, their adjoint and the corresponding eigenvalues of the two-group neutron diffusion equations representing any heterogeneous system are investigated. First, the classical power iteration method is modified so that the calculation of modes higher than the fundamental mode is possible. Thereafter, the Explicitly-Restarted Arnoldi method, belonging to the class of Krylov subspace methods, is touched upon. Although the modified power iteration method is a computationally-expensive algorithm, its main advantage is its robustness, i.e. the method always converges to the desired eigenfunctions without any need from the user to set up any parameter in the algorithm. On the other hand, the Arnoldi method, which requires some parameters to be defined by the user, is a very efficient method for calculating eigenfunctions of large sparse system of equations with a minimum computational effort. These methods are thereafter used for off-line analysis of the stability of Boiling Water Reactors. Since several oscillation modes are usually excited (global and regional oscillations) when unstable conditions are encountered, the characterization of the stability of the reactor using for instance the Decay Ratio as a stability indicator might be difficult if the contribution from each of the modes are not separated from each other. Such a modal decomposition is applied to a stability test performed at the Swedish Ringhals-1 unit in September 2002, after the use of the Arnoldi method for pre-calculating the different eigenmodes of the neutron flux throughout the reactor. The modal decomposition clearly demonstrates the excitation of both the global and regional oscillations. Furthermore, such oscillations are found to be intermittent with a time-varying phase shift between the first and second azimuthal modes.

Nonlinear/electromagnetic co-simulation of microwaves and millimeter-waves links

The Ph.D. thesis describes the simulations of different microwave links from the transmitter to the receiver intermediate-frequency ports, by means of a rigorous circuit-level nonlinear analysis approach coupled with the electromagnetic characterization of the transmitter and receiver front ends. This includes a full electromagnetic computation of the radiated far field which is used to establish the connection between transmitter and receiver. Digitally modulated radio-frequency drive is treated by a modulation-oriented harmonic-balance method based on Krylov-subspace model-order reduction to allow the handling of large-size front ends. Different examples of links have been presented: an End-to-End link simulated by making use of an artificial neural network model; the latter allows a fast computation of the link itself when driven by long sequences of the order of millions of samples. In this way a meaningful evaluation of such link performance aspects as the bit error rate becomes possible at the circuit level. Subsequently, a work focused on the co-simulation an entire link including a realistic simulation of the radio channel has been presented. The channel has been characterized by means of a deterministic approach, such as Ray Tracing technique. Then, a 2x2 multiple-input multiple-output antenna link has been simulated; in this work near-field and far-field coupling between radiating elements, as well as the environment factors, has been rigorously taken into account. Finally, within the scope to simulate an entire ultra-wideband link, the transmitting side of an ultrawideband link has been designed, and an interesting Front-End co-design technique application has been setup.

Harnack inequalities in sub-Riemannian settings

In the present thesis, we discuss the main notions of an axiomatic approach for an invariant Harnack inequality. This procedure, originated from techniques for fully nonlinear elliptic operators, has been developed by Di Fazio, Gutiérrez, and Lanconelli in the general settings of doubling Hölder quasi-metric spaces. The main tools of the approach are the so-called double ball property and critical density property: the validity of these properties implies an invariant Harnack inequality. We are mainly interested in the horizontally elliptic operators, i.e. some second order linear degenerate-elliptic operators which are elliptic with respect to the horizontal directions of a Carnot group. An invariant Harnack inequality of Krylov-Safonov type is still an open problem in this context. In the thesis we show how the double ball property is related to the solvability of a kind of exterior Dirichlet problem for these operators. More precisely, it is a consequence of the existence of some suitable interior barrier functions of Bouligand-type. By following these ideas, we prove the double ball property for a generic step two Carnot group. Regarding the critical density, we generalize to the setting of H-type groups some arguments by Gutiérrez and Tournier for the Heisenberg group. We recognize that the critical density holds true in these peculiar contexts by assuming a Cordes-Landis type condition for the coefficient matrix of the operator. By the axiomatic approach, we thus prove an invariant Harnack inequality in H-type groups which is uniform in the class of the coefficient matrices with prescribed bounds for the eigenvalues and satisfying such a Cordes-Landis condition.

Preconditioning strategies for the numerical solution of convection-diffusion partial differential equations

Il trattamento numerico dell'equazione di convezione-diffusione con le relative condizioni al bordo, comporta la risoluzione di sistemi lineari algebrici di grandi dimensioni in cui la matrice dei coefficienti è non simmetrica. Risolutori iterativi basati sul sottospazio di Krylov sono ampiamente utilizzati per questi sistemi lineari la cui risoluzione risulta particolarmente impegnativa nel caso di convezione dominante. In questa tesi vengono analizzate alcune strategie di precondizionamento, atte ad accelerare la convergenza di questi metodi iterativi. Vengono confrontati sperimentalmente precondizionatori molto noti come ILU e iterazioni di tipo inner-outer flessibile. Nel caso in cui i coefficienti del termine di convezione siano a variabili separabili, proponiamo una nuova strategia di precondizionamento basata sull'approssimazione, mediante equazione matriciale, dell'operatore differenziale di convezione-diffusione. L'azione di questo nuovo precondizionatore sfrutta in modo opportuno recenti risolutori efficienti per equazioni matriciali lineari. Vengono riportati numerosi esperimenti numerici per studiare la dipendenza della performance dei diversi risolutori dalla scelta del termine di convezione, e dai parametri di discretizzazione.

Weighted geometric mean of large-scale matrices: numerical analysis and algorithms

Computing the weighted geometric mean of large sparse matrices is an operation that tends to become rapidly intractable, when the size of the matrices involved grows. However, if we are not interested in the computation of the matrix function itself, but just in that of its product times a vector, the problem turns simpler and there is a chance to solve it even when the matrix mean would actually be impossible to compute. Our interest is motivated by the fact that this calculation has some practical applications, related to the preconditioning of some operators arising in domain decomposition of elliptic problems. In this thesis, we explore how such a computation can be efficiently performed. First, we exploit the properties of the weighted geometric mean and find several equivalent ways to express it through real powers of a matrix. Hence, we focus our attention on matrix powers and examine how well-known techniques can be adapted to the solution of the problem at hand. In particular, we consider two broad families of approaches for the computation of f(A) v, namely quadrature formulae and Krylov subspace methods, and generalize them to the pencil case f(A\B) v. Finally, we provide an extensive experimental evaluation of the proposed algorithms and also try to assess how convergence speed and execution time are influenced by some characteristics of the input matrices. Our results suggest that a few elements have some bearing on the performance and that, although there is no best choice in general, knowing the conditioning and the sparsity of the arguments beforehand can considerably help in choosing the best strategy to tackle the problem.

Judenthum und Christenthum

von I. Lichtenstein