Στη παρούσα διπλωματική εργασία ασχολούμαστε με την αποδοτική επίλυση ταινιακών και γενικών, αραιών γραμμικών συστημάτων σε παράλληλες αρχιτεκτονικές μέσω της οικογένειας αλγορίθμων Spike. Ζητούμενο είναι η βελτίωση (μείωση) του χρόνου επίλυσης μέσω τεχνικών επίλυσης γραμμικών συστημάτων με πολλά δεξιά μέλη. Πιο συγκεκριμένα, επικεντρωνόμαστε στην επίλυση της εξίσωσης μητρώου $AX=F$ (1) όπου $A\in \mathbb{R}^{n\times n}$ είναι το μητρώο συντελεστών και το οποίο είναι αραιό ή/και ταινιακό, $F\in \mathbb{R}^{n\times s}$ είναι ένα μητρώο με $s$ στήλες το οποίο ονομάζεται μητρώο δεξιών μελών και $X\in \mathbb{R}^{n\times s}$ είναι η λύση του συστήματος. Μια σημαντική μέθοδος για την παράλληλη επίλυση της παραπάνω εξίσωσης, είναι η μέθοδος Spike και οι παραλλαγές της. Η μέθοδος Spike βασίζεται στη τεχνική διαίρει και βασίλευε και αποτελείται από δυο φάσεις: α) επίλυση ανεξάρτητων υπο-προβλημάτων τοπικά σε κάθε επεξεργαστή, και β) επίλυση ενός πολύ μικρότερου προβλήματος το οποίο απαιτεί επικοινωνία μεταξύ των επεξεργαστών. Οι δύο φάσεις συνδυάζονται ώστε να παραχθεί η τελική λύση $X$. Η συνεισφορά της διπλωματικής εργασίας έγκειται στην επιτάχυνση της οικογένειας αλγορίθμων Spike για την επίλυση της εξίσωσης (1) μέσω της μελέτης, το σχεδιασμό και την υλοποίηση νέων, περισσότερο αποδοτικών αλγοριθμικών σχημάτων τα οποία βασίζονται σε τεχνικές επίλυσης γραμμικών συστημάτων με πολλά δεξιά μέλη. Αυτά τα νέα αλγοριθμικά σχήματα έχουν ως στόχο τη βελτίωση του χρόνου επίλυσης των γραμμικών συστημάτων καθώς και άλλα οφέλη όπως η αποδοτικότερη χρήση μνήμης. / In this thesis we focus on the efficient solution of general banded and general sparse linear systems on parallel architectures by exploiting the Spike family of algorithms. The equation of interest can be written in matrix form as $ AX = F $ (1) where $ A \ in \ mathbb {R} ^ {n \ times n} $ is the coefficient matrix, which is also sparse and / or banded, $ F \ in \ mathbb {R} ^ {n \ times s} $ is a matrix with $ s $ columns called matrix of the right hand sides and $ X \ in \ mathbb {R} ^ {n \ times s} $ is the solution of the system. An important method for the parallel solution of the above equation, is the Spike method and its variants. The Spike method is based on the divide and conquer technique and consists of two phases: a) solution of local, independent sub-problems in each processor, and b) solution of a much smaller problem which requires communication among the processors. The two phases are combined to produce the final solution $ X $. The contribution of this thesis is the acceleration of the Spike method for the solution of the matrix equation in (1) by studying, designing and implementing new, more efficient algorithmic schemes which are based on techniques used for the effective solution of linear systems with multiple right hand sides. These new algorithmic schemes were designed to improve the solving time of the linear systems as well as to provide other benefits such as more efficient use of memory.

One purpose of breeding programs is the selection of the better test lines. The accuracy of selection can be improved by using optimal design and using models that fit the data well. Finding this is not easy, especially in large experiments which assess more than one hundred lines without the possibility of replication due to the limited material, area and high costs. Thus, the large number of parameters in the complex variance structure that needs to be fitted relies on the limited number of replicated check varieties. The main objectives of this thesis were to model 21 trials of sugarcane provided by \"Centro de Tecnologia Canavieira\" (CTC - Brazilian company of sugarcane) and to evaluate the design employed, which uses a large number of unreplicated test lines (new varieties) and systematic replicated check (commercial) lines. The mixed linear model was used to identify the three major components of spatial variation in the plot errors and the competition effects at the genetic and residual levels. The test lines were assumed as random effects and check lines as fixed, because they came from different processes. The single and joint analyses were developed because the trials could be grouped into two types: (i) one longitudinal data set (two cuts) and (ii) five regional groups of experiment (each group was a region which had three sites). In a study of alternative designs, a fixed size trial was assumed to evaluate the efficiency of the type of unreplicated design employed in these 21 trials comparing to spatially optimized unreplicated and p-rep designs with checks and a spatially optimized p-rep design. To investigate models and design there were four simulation studies to assess mainly the i) fitted model, under conditions of competition effects at the genetic level, ii) accuracy of estimation in the separate versus joint analysis; iii) relation between the sugarcane lodging and the negative residual correlation, and iv) design efficiency. To conclude, the main information obtained from the simulation studies was: the convergence number of the algorithm model analyzed; the variance parameter estimates; the correlations between the direct genetic EBLUPs and the true direct genetic effects; the assertiveness of selection or the average similarity, where similarity was measured as the percentage of the 30 test lines with the highest direct genetic EBLUPs that are in the true 30 best test lines (generated); and the heritability estimates or the genetic gain. / Um dos propósitos dos programas de melhoramento genético é a seleção de novos clones melhores (novos materiais). A acurácia de seleção pode ser melhorada usando delineamentos ótimos e modelos bem ajustados. Porém, descobrir isso não é fácil, especialmente, em experimentos grandes que possuem mais de cem clones sem a possibilidade de repetição devido à limitação de material, área e custos elevados, dadas as poucas repetições de parcelas com variedades comerciais (testemunhas) e o número de parâmetros de complexa variância estrutural que necessitam ser assumidos. Os principais objetivos desta tese foram modelar 21 experimentos de cana de açúcar fornecidos pelo Centro de Tecnologia Canavieira (CTC - empresa brasileira de cana de açúcar) e avaliar o delineamento empregado, o qual usa um número grande de clones não repetidos e testemunhas sistematicamente repetidas. O modelo linear misto foi usado, identificando três principais componentes de variação espacial nos erros de parcelas e efeitos de competição, em nível genético e residual. Os clones foram assumidos de efeitos aleatórios e as testemunhas de efeitos fixos, pois vieram de processos diferentes. As análises individuais e conjuntas foram desenvolvidas neste material pois os experimentos puderam ser agrupados em dois tipos: (i) um delineamento longitudinal (duas colheitas) e (ii) cinco grupos de experimentos (cada grupo uma região com três locais). Para os estudos de delineamentos, um tamanho fixo de experimento foi assumido para se avaliar a eficiência do delineamento não replicado (empregado nesses 21 experimentos) com os não replicados otimizado espacialmente, os parcialmente replicados com testemunhas e os parcialmente replicados otimizado espacialmente. Quatro estudos de simulação foram feitos para avaliar i) os modelos ajustados, sob condições de efeito de competição em nível genético, ii) a acurácia das estimativas vindas dos modelos de análise individual e conjunta; iii) a relação entre tombamento da cana e a correlação residual negativa, e iv) a eficiência dos delineamentos. Para concluir, as principais informações utilizadas nos estudos de simulação foram: o número de vezes que o algoritmo convergiu; a variância na estimativa dos parâmetros; a correlação entre os EBLUPs genético direto e os efeitos genéticos reais; a assertividade de seleção ou a semelhança média, sendo semelhança medida como a porcentagem dos 30 clones com os maiores EBLUPS genético e os 30 melhores verdadeiros clones; e a estimativa da herdabilidade ou do ganho genético.

Ädelfors is situated ca 17 km east of Vetlanda, Jönköping County, in the N-S striking Trans-scandinavian igneous belt and is a part of the NE-SW striking 1.83-1.82 Ga Oskarshamn-Jönköping belt emplaced during a continental subduction towards the Svecofennian continental margin. The continental arc hosts the 1.83 Ga metasedimentary Vetlanda supergroup composed of foliated metagreywacke, metasandstone and metaconglomerate. The sequence is intercalated by mafic and felsic volcanites and hosts the Cu-Au-Fe-mines at Ädelfors. Ädelfors mining field consists of ca 330 mineralized quartz veins hosting both copper, gold and iron. The iron mines Nilsson’s iron mine (NFE) and Fe-mine (FE), the copper mine Kamelen (KM) and the gold mines Brånad’s mine (BR), Adolf Fredrik’s mine (AF), Old Kron mine (GKR), Old Kolhag’s mine (GKO), Thörn mine (TH), New Galon mine (NG), Stenborg’s mine (ST), Tysk mine (TG), Hällaskallen (HS) and Fridhem (FR) have been investigated to deduce a possible genetic relation between the veins and their origin. Sulfur isotope ratios have also been conducted on pyrite from KM, AF and FE. The veins can stucturally be divided into several groups. AF, GKR, ST, NG, TH and possibly NFE are striking 10-70° with a dip of 55-70°. BR, GKO and KM are striking 110-140° with a dip of 80-90° whereas TG and HS strike 90-110° dipping 85°. Fridhem, being distal to the other mines, strikes 70° and dips 80°. A chlorite-quartz-biotite-sericite-rich metapelite hosts the veins in all localities except; FR where a layered, beresitizised felsic volcanite rich in plagioclase, sericite, biotite and quartz hosts disseminated pyrite; and NFE, HS and NG which are hosted by a mafic tuffite. Quartz veins are mainly milky and equigranular, exceptions are FE with black pyrite-bearing quartz veins, cutting through the banded magnetite-metapelite and KM with its dynamically recrystallized quartz. Chlorite-, zeolite-, carbonate-, hematite-, amphibole-, kalifeldspar-, sericite-, biotite- and epidote alteration has been observed among the localities. The ore minerals are dominated by: fractured sub- to euhedral pyrite in cataclastic aggregates or selvage bands, interstitial chalcopyrite in pyrite, marcasite, pyrrhotite, gold and sporadic chalcopyrite diseased sphalerite and arsenopyrite. Previously not reported tetradymite, staurolite, galena and Ce-monazite have also been observed. Bismuthinite and tetradymite as inclusions in pyrite were observed in AF, GKR, FR and TG. Gold was observed in AF, BR, GKR and TG as inclusions in pyrite or quartz with a Au/Ag median of 78.41. HS distinguishes itself with Au/Ag ratios of 4.66-5.25. The trace element ratios in pyrite reveal two major types of pyrite. 1) found in FE and KM (pyrite type 1) with Co/Ni ratio of 10.94, Bi/Au of 1.79, Bi/S of 0.037, Au/Ag of 11.13, S/Se of 235.96 and As/S of 0.006. 2) found in NG, GKO, ST, TH, AF, NFE, HS, GKR, BR, FR, TG and as stringers in KM4 py1 pyrite type 2) with an average Co/Ni ratio of 5.26, Bi/Au of 1.95, Bi/S of 0.031, Au/Ag of 4.19, S/Se of 0 and As/S of 0. δ34S values strengthens this grouping as KM and FE has 1,3-2,6 ‰ and AF 3,6-3,8 ‰. The following geological interpretation has been concluded: The banded iron formation in FE is the earliest mineralization and was later fractured, emplacing quartz veins with pyrite of type 1. During this event, the Cu-vein in KM was also formed. A second generation of fractures, emplaced after the Småland granitoids formed, were filled with quartz and pyrite of type 2 at mesozonal depth. This is the main stage of gold mineralization and includes NG, GKO, ST, TH, AF, NFE, GKR, BR, FR and TG. During this event, pyrite of type 2 was added to KM, causing recrystallizing of the quartz. HS is possibly emplaced last or altered as it is more enriched in silver. Morphology, mineralogy, alterations, mineral chemistry and sulfur isotope signatures indicates an orogenic origin of the gold-rich quartz veins at Ädelfors as well as the copper-rich vein in KM. / Ädelfors ligger ca 17 km öster om Vetlanda, Jönköpings län, i det N-S strykande Transskandinaviska granit och porfyrbältet och är en del av det NÖ-SV strykande 1,83-1,82 Ga Oskarshamn-Jönköpingsbältet (OJB) bildad i en kontinental subduktionszon i kanten av den Svecofenniska kontinentalplattan. I denna kontinentalbåge ligger Vetlanda supergruppen som är en metasedimentär del av OJB bestående av starkt folierad 1,83 Ga metagråvacka, metasandsten och metakonglomerat med inlagringar av mafiska och felsiska vulkaniter. Ädelfors gruvfält består utav ca. 330 kvartsgångar förande mestadels guld men också koppar. Järnmineraliseringar i form av bandad järnmalm finns också i området. Geologin, mineralogin och pyritens kemiska sammansättning från järngruvorna Nilssons järngruva (NFE) och Fe-gruvan (FE), koppargruvan Kamelen (KM) och guldgruvorna Brånadsgruvan (BR), Adolf Fredriks gruva (AF), Gamla Krongruvan (GKR), Gamla Kolhagsgruvan (GKO), Thörngruvan (TH), Nya Galongruvan (NG), Stenborgs gruva (ST), Tyskgruvan (TG), Hällaskallen (HS) och Fridhem (FR) har undersökts för att finna eventuella genetiska likheter. Svavelisotopförhållande har fastställts för pyrit från AF, FE och KM. Strukturellt kan gångarna delas in i ett antal grupper. AF, GKR, ST, NG, TH och möjligtvis NFE stryker 10-70° och stupar 55-70°. BR, GKO och KM stryker 110-140° och stupar 80-90° medan TG och HS stryker 90-110° och stupar 85°. Fridhem stryker 70° och stupar 80°. En klorit-kvarts-sericit-biotitrik metapelit utgör värdbergarten i alla gruvor förutom; FR där den utgörs av en beresitiserad felsisk vulkanit rik på plagioklas, sericit, biotit och kvarts med disseminerad pyrit; och NFE, HS, NG vilka har en mafisk tuffitisk moderbergart. Kvartsgångarna är mjölkvita med undantag för FE:s svarta, pyritförande kvarts vilket uppträder som sprickfyllnad i den bandade järnmalmen och är senare bildad. Kvartsen i KM är starkt dynamiskt omkristalliserad. Svag till måttlig foliation är vanlig i sidoberget med undantag av stark foliation i TG och NFE, vilka är lokaliserade i förkastningssprickor med stark kloritförskiffring av värdbergarten. Klorit-, zeolit-, karbonat-, hematit-, amfibol-, kalifältspat-, sericit-, biotit- och epidotomvandling förekommer i majoriteten av lokalerna. Malmmineralen är dominerande sprött deformerad subhedral till euhedral pyrit som kataklastiska aggregat eller band, interstitiell kopparkis i pyrit, markasit, magnetkis, guld och sporadiskt kopparkissjuk zinkblände och arsenikkis. I det här arbetet har även tetradymit, staurolit, blyglans och Ce-monazit observerats. Bismutinit och tetradymit i form av inneslutningar i pyrit observerades i AF, GKR, FR och TG. Guld observerades i AF, BR, GKR och TG som inneslutningar i pyrit eller fritt i kvarts med Au/Ag medianvärde på 78,41, avvikande är HS med värden mellan 4,66-5,25.    Förhållanden mellan spårelement i pyrit indikerar två typer av pyrit. Typ 1 funnen i FE och KM har följande värden: Co/Ni = 10,94, Bi/Au = 1,79, Bi/S = 0,037, Au/Ag = 11,13, S/Se = 235,96 och As/S = 0,006. Typ 2 funnen i NG, GKO, ST, TH, AF, NFE, HS, GKR, BR, FR, TG och som sliror i KM4 py1 har följande värden Co/Ni = 5,26, Bi/Au = 1,95, Bi/S = 0,031, Au/Ag = 4,19, S/Se = 0 and As/S = 0. δ34S värden styrker denna uppdelning där KM och FE har värdena 1,3-2,6 ‰ och AF 3,6-3,8 ‰. Den geologiska utvecklingen av fältet har tolkats som följande: FE-gruvans bandade järnmalm är den tidigaste mineraliseringen vilket följs utav uppsprickning och läkning av kvarts med pyrit typ 1 som också bildar kopparmineraliseringen KM. Senare sprickzoner efter Smålandsgraniternas intrusion läks av kvarts med pyrit typ 2 på mesozonalt djup vilket bildar NG, GKO, ST, TH, AF, NFE, GKR, BR, FR, TG och omkristalliserar och introducerar nya pyritsliror i kvartsen i KM. HS bildas möjligtvis sist eller har blivit omvandlad eftersom den är anrikad på silver. Morfologi, omvandlingar och svavelisotop-signaturer tyder på ett orogent ursprung för Ädelfors guldrika kvartsådror samt den kopparrika kvartsådern i KM.

Cette thèse vise à la conception de nouveaux algorithmes rapides en calcul numérique via les matrices de Toeplitz. Tout d'abord, nous avons introduit un algorithme rapide sur le calcul de l'inverse d'une matrice triangulaire de Toeplitz en se basant sur des notions d'interpolation polynomiale. Cet algorithme nécessitant uniquement deux FFT(2n) est manifestement efficace par rapport à ses prédécésseurs. ensuite, nous avons introduit un algorithme rapide pour la résolution d'un système linéaire de Toeplitz bande. Cette approche est basée sur l'extension de la matrice donnée par plusieurs lignes en dessus, de plusieurs colonnes à droite et d'attribuer des zéros et des constantes non nulles dans chacune de ces lignes et de ces colonnes de telle façon que la matrice augmentée à la structure d'une matrice triangulaire inférieure de Toeplitz. La stabilité de l'algorithme a été discutée et son efficacité a été aussi justifiée. Finalement, nous avons abordé la résolution d'un système de Toeplitz bandes par blocs bandes de Toeplitz. Ceci étant primordial pour établir la connexion de nos algorithmes à des applications en restauration d'images, un domaine phare en mathématiques appliquées. / This thesis aims to design new fast algorithms for numerical computation via the Toeplitz matrices. First, we introduced a fast algorithm to compute the inverse of a triangular Toeplitz matrix with real and/or complex numbers based on polynomial interpolation techniques. This algorithm requires only two FFT (2n) is clearly effective compared to predecessors. A numerical accuracy and error analysis is also considered. Numerical examples are given to illustrate the effectiveness of our method. In addition, we introduced a fast algorithm for solving a linear banded Toeplitz system. This new approach is based on extending the given matrix with several rows on the top and several columns on the right and to assign zeros and some nonzero constants in each of these rows and columns in such a way that the augmented matrix has a lower triangular Toeplitz structure. Stability of the algorithm is discussed and its performance is showed by numerical experiments. This is essential to connect our algorithms to applications such as image restoration applications, a key area in applied mathematics.

The content of this work was to carry out construction-technical survey of timber structures of the existing building in order to design an appropriate procedure for their reconstruction. The survey showed construction in an unsatisfactory to disrepair, so further work also includes the static calculation of new structures proposed instead of the current. Timber structure in the building has form of staircase, roof and ceiling construction. Newly designed the following types of stairs and roof trusses from which the lower belts are used as ceiling beams. The new proposal also includes possible roofing, roofing options, developed one of them. The work also includes drawings, bill of materials and cost estimate staircase structure and truss. Specialization in the field of civil engineering is represented by drawings of the new truss and passport (view of current state) whole object on parcel ST.74 / 1, land Tišnov (okr.Brno-venkov), the content of which was the basis for this work.