Berstel and Reutenauer stated the iteration theorem for recognizable formal power series on trees over fields and vector spaces. The key idea of its proof is the existence of pseudo-regular matrices in matrix-products. This theorem is generalized to integral domains and modules over integral domains in this thesis. It only requires the reader to have basic knowledge in linear algebra. Concepts from the advanced linear algebra and abstract algebra are introduced in the preliminary chapter.:1. Introduction 2. Preliminaries 3. Long products of matrices 4. Formal power series on trees 5. The generalized iteration theorem 6. Conclusion

In this thesis, we explore a novel approach to solving the Helmholtz equation,the WaveHolz iteration method. This method aims to overcome some ofthe difficulties with solving the Helmholtz equation by providing a highlyparallelizable iterative method based on solving the time-dependent Waveequation. If this method proves reliable and computationally feasible it wouldhave great value for future application. Therefore, it is of interest to evaluatethe performance and properties of this method. To fully evaluate this method,the method was implemented and conclusions were based on results fromsimulations of the method. The method was able to solve problems in threedimensions and it seems that the method is very well suited for parallelized computations. To replicate real-world scenarios simulations of problems ininfinite and curvilinear domains were conducted. Based on the result presentedhere it is possible to further refine the method, especially regarding the setupof the domain and the implementation of boundary conditions for infinitedomains. / I detta examensarbete presenteras en ny metod för att lösa Helmholtz ekvation, WaveHoltz iterativa metod. Målet med denna metod är att undkomma vissa problem som uppstår med andra metoder för att lösa Helmholtz ekvation genom att tillhandahålla iterativ metod som baseras på lösningar av den tidsberoende vågekvationen samt kan parallelliseras effektivt. Om denna metod visar sig vara stabil och effektiv beräkningsmässigt skulle detta medföra stor potential för framtida tillämpningar. Av denna anledning undersöks metoden och dess egenskaper. För att fullt ut kunna evaluera denna method implementerades den vartefter simuleringar genomfördes och slutsatser drogs. Med metoden var att det var möjligt att lösa problem i tre dimensioner och metoden visade sig vara lämplig för parallella beräkningar. För att återskapa verklighetstrogna scenarion beräknades problem i oändliga och kroklinjiga domäner. Baserat på resultaten som presenteras i denna rapport är det möjligt att förfina metoden, speciellt vid konstruktionen av komplicerade beräkningsnät och randvillkoren för de oändliga problemen.

This thesis considers a deep learning approach to a dynamic portfolio optimization problem. A proposed deep learning algorithm is tested on a simplified version of the problem with promising results, which suggest continued testing of the algorithm, on a larger scale for the original problem. First the dynamics and objective function of the problem are presented, and the existence of a no-trade-region is explained via the Hamilton-Jacobi-Bellman equation. The no-trade-region dictates the optimal trading strategy. Solving the Hamilton-Jacobi-Bellman equation to find the no-trade-region is not computationally feasible in high dimension with a classic finite difference approach. Therefore a new algorithm to iteratively update and improve an estimation of the no-trade-region is derived. This is a deep learning algorithm that utilizes neural network function approximation. The algorithm is tested on the one-dimensional version of the problem for which the true solution is known. While testing in one dimension only does not assess whether this algorithm scales better than a finite difference approach to higher dimensions, the learnt solution comes fairly close to true solution with a relative score of 0.72, why it is suggested that continued research of this algorithm is performed for the multidimensional version of the problem. / Den här uppsatsen undersöker en djupinlärningsmetod for att lösa ett dynamiskt portföljoptimeringsproblem. En föreslagen djupinlärningsalgoritm testas på en föreklad version av problemet, med lovande resultat. Därför föreslås det vidare att algoritmens prestanda testas i större skala även för det urpsrungliga problemet. Först presenteras dynamiken och målfunktionen för problemet. Det förklaras via Hamilton-Jacobi-Bellman-ekvationen varför det finns en handelsstoppregion. Handelsstoppregionen bestämmer den optimala handelsstrategin. Att lösa Hamilton-Jacobi-Bellman-ekvationen för att hitta handelsstoppregionen är inte beräkningspratiskt möjligt i hög dimension om ett traditionellt tillvägagångssätt med finita differenser används. Därför härleds en ny algoritm som iterativt uppdaterar och förbättrar en skattning av handelsstoppregionen. Det är en djupinlärningsalgoritm som utnyttjar funktionsapproximation med neurala nätverk. Algoritmen testas på den endimensionella verisonen av problemet, för vilken den sanna lösningen är känd. Tester i det endimensionella fallet kan naturligtvis inte ge svar på frågan om den nya algoritmen skalar bättre än en finit differensmetod till högre dimensioner. Men det är i alla fall klart att den inlärda lösningen kommer tämligen nära den sanna med relativ poäng 0.72, och därför föreslås fortsatt forskning kring algoritmen i förhållande till den flerdimensionella versionen av problemet.

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對於線性過歸模式之參數，一般均採最小平方估計量來估計。最小平方估計量，在適 當條件下有強一致性（Strong consistency），而本文所探討之主題，乃是針對本特 性探討其數歛速率。 本文共分五章，第一章為緒論，含研究動機、研究目的及本文結構，第二章摘介本文 所探討之模式與最小平方估計量之收歛性，第三章為引理，介紹本文所需要之預備知 識及本文所使用之方法，第四章為主要定理及證明，第五章結論。全文共計一冊，約 一萬餘字。

As our world ramps up the speed of its connections, our identities merge with increasing speed and angles of confluence. Not only are new identities created, but also the more fringe social and cultural elements of our world are exposed to mainstream consciousness. My work is a product of my own fringe background (namely its sheer breadth of experiences not normally visited upon a single person's life). My aim is to add variables to our social and cultural speed of combination and new variety by creating work that is derived from my own experiences to speak to those who are of the newer combinations and newer social recognition. I am motivated to create this work both to perpetually define and redefine myself, and also to give ground to an ‘identity of no identity' on a global scale, to our artistic dialogue and catalog.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O esquema iterativo de fonte de espalhamento (SI) é tradicionalmente aplicado para a convergência da solução numérica de malha fina para problemas de transporte de nêutrons monoenergéticos na formulação de ordenadas discretas com fonte fixa. O esquema SI é muito simples de se implementar sob o ponto de vista computacional; porém, o esquema SI pode apresentar taxa de convergência muito lenta, principalmente para meios difusivos (baixa absorção) com vários livres caminhos médios de extensão. Nesta dissertação descrevemos uma técnica de aceleração baseada na melhoria da estimativa inicial para a distribuição da fonte de espalhamento no interior do domínio de solução. Em outras palavras, usamos como estimativa inicial para o fluxo escalar médio na grade de discretização de malha fina, presentes nos termos da fonte de espalhamento das equações discretizadas SN usadas nas varreduras de transporte, a solução numérica da equação da difusão de nêutrons em grade espacial de malha grossa com condições de contorno especiais, que aproximam as condições de contorno prescritas que são clássicas em cálculos SN, incluindo condições de contorno do tipo vácuo. Para aplicarmos esta solução gerada pela equação da difusão em grade de discretização de malha grossa nas equações discretizadas SN de transporte na grade de discretização de malha fina, primeiro implementamos uma reconstrução espacial dentro de cada nodo de discretização, e então determinamos o fluxo escalar médio em grade de discretização de malha fina para usá-lo nos termos da fonte de espalhamento. Consideramos um número de experimentos numéricos para ilustrar a eficiência oferecida pela presente técnica (DSA) de aceleração sintética de difusão. / The scattering source iterative (SI) scheme is traditionally applied to converge finemesh numerical solutions to fixed-source discrete ordinates neutron transport problems. The SI scheme is very simple to implement under a computational viewpoint. However, the SI scheme may show very slow convergence rate, mainly for diffusive media (low absorption) with several mean free paths in extent. In this work we describe an acceleration technique based on an improved initial guess for the scattering source distribution within the slab. In other words, we use as initial guess for the fine-mesh average scalar flux in the scattering source terms of the SN discretized equations used in the transport sweeps, the coarse-mesh solution of the neutron diffusion equation with special boundary conditions to account for the classical SN prescribed boundary conditions, including vacuum boundary conditions. To apply this coarse-mesh diffusion solution into the fine-mesh SN transport sweep discretized equations, we first perform within-node spatial reconstruction, and then we determine the fine-mesh average scalar flux for use in the scattering source terms. We consider a number of numerical experiments to illustrate the efficiency of the offered diffusion synthetic acceleration (DSA) technique.

In support of society's technological evolution, the study of nonlinear systems in engineering and sciences has become a vital research area. Aiming to contribute in this field, this thesis investigates the behaviour of nonlinear systems using the 'Wiener theories'. As a useful example the Duffing oscillator is investigated in this work. In many real-life applications, nonlinear systems are excited randomly so this work examines systems under white-noise excitation using the Wiener series. Equivalent Linearisation (EL) is a well-known and simple method that approximates a nonlinear system by an equivalent linear system. However, it has deficiencies which this thesis attempts to improve. Initially, the performance of EL for different types of nonlinearities will be assessed and an alternative method to enhance it is suggested. This requires the calculation of the first Wiener kernel of various system defined quantities. The first Wiener kernel, as it will be shown, is the foundation of this research and a central element of the Wiener theory. In this thesis, an analytical proof to explain the interesting behaviour of the first Wiener kernel for a system with nonlinear stiffness is included using an energy transfer approach. Furthermore, the method mentioned above to enhance EL known as the Single-Pole Fit method (SPF) is to be tested for different kinds of systems to prove its robustness and validity. Its direct application to systems with nonlinear stiffness and nonlinear damping is shown as well as its ability to perform for systems with two degrees of freedom where an extension of the SPF method is required to achieve the desired solution. Finally, an investigation to understand and replicate the complex behaviour observed by the first Wiener kernel in the early chapters is carried out. The groundwork for this investigation is done by modelling an isolated nonlinear spring with a series of linear filters and certain nonlinear operations. Subsequently, an attempt is made to relate the principles governing the successful spring model presented to the original nonlinear system. An iterative procedure is used to demonstrate the application of this method, which also enables this new modelling approach to be related to the SPF method.

Um Processo de Decisão Markoviano (MDP) pode ser usado para modelar problemas de decisão sequencial. No entanto, podem existir limitações na obtenção de probabilidades para modelagem da transição de estados ou falta de confiabilidade nas informações existentes sobre estas probabilidades. Um modelo menos restritivo e que pode resolver este problema é o Processo de Decisão Markoviano com Probabilidades Intervalares (BMDP), que permite a representação imprecisa das probabilidades de transição de estados e raciocínio sobre uma solução robusta. Para resolver BMDPs de horizonte infinito, existem os algoritmos síncronos de Iteração de Valor Intervalar e Iteração de Política Robusto, que são ineficientes quando o tamanho do espaço de estados é grande. Neste trabalho são propostos algoritmos assíncronos de Iteração de Política baseados no particionamento do espaço de estados em subconjuntos aleatórios (Robust Asynchronous Policy Iteration - RAPI) ou em componentes fortemente conexos (Robust Topological Policy Iteration - RTPI). Também são propostas formas de inicializar a função valor e a política dos algoritmos, de forma a melhorar a convergência destes. O desempenho dos algoritmos propostos é avaliado em comparação com o algoritmo de Iteração de Política Robusto para BMDPs para domínios de planejamento existentes e um novo domínio proposto. Os resultados dos experimentos realizados mostram que (i) quanto mais estruturado é o domínio, melhor é o desempenho do algoritmo RTPI; (ii) o uso de computação paralela no algoritmo RAPI possui um pequeno ganho computacional em relação à sua versão sequencial; e (iii) uma boa inicialização da função valor e política pode impactar positivamente o tempo de convergência dos algoritmos. / A Markov Decision Process (MDP) can be used to model sequential decision problems. However, there may be limitations in obtaining probabilities for state transition modeling or lack of reliability in existing information on these probabilities. A less restrictive model that can solve this problem is the Bounded-parameter Markov Decision Process (BMDP), which allows the imprecise representation of the transition probabilities and reasoning about a robust solution. To solve infinite horizon BMDPs, there are synchronous algorithms such as Interval Value Iteration and Robust Policy Iteration, which are inefficient for large state spaces. In this work, we propose new asynchronous Policy Iteration algorithms based on state space partitioning in random subsets (Robust Asynchronous Policy Iteration - RAPI) or in strongly connected components (Robust Topological Policy Iteration - RTPI). We also propose ways to initialize the value function and policy of the algorithms, in order to improve their convergence. The performance of the proposed algorithms is evaluated in comparison with the Robust Policy Iteration algorithm for BMDPs for existing planning domains and a proposed new domain. The results of the experiments show that (i) the more structured the domain, the better is the performance of the RTPI algorithm; (ii) the use of parallel computing in the RAPI algorithm has a small computational gain compared to its sequential version; and (iii) a good initialization of the value function and policy can positively impact the convergence time of the algorithms.

<p>In the last three decades we have witnessed a remarkable development in the area of integrated circuits. From small logic devices containing some hundred transistors to modern processors containing several tens of million transistors. However, power consumption has become a real problem and may very well be the limiting factor of future development. Designing for low power is therefore increasingly important. To accomplice an efficient low power design, accurate power estimation at early design stage is essential. The aim of this thesis was to set up a power estimation flow to estimate the power consumption at early design stage. The developed flow spans over both RTL- and gate-level incorporating Mentor Graphics Modelsim (RTL-level simulator), Cadence PKS (gate- level synthesizer) and own developed power estimation tools. The power consumption is calculated based on gate-level physical information and RTL- level toggle information. To achieve high estimation accuracy, real node annotations is used together with an own developed on-chip wire model to estimate node voltage swing. </p><p>Since the power estimation may be very time consuming, the flow also includes support for low power design iteration. This gives efficient power estimation speedup when concentrating on smaller sub- parts of the design.</p>