We have proposed a method by which the topology of a network might be discovered through an algorithm like the distributed Bellman-Ford algorithm. We have explored the inner workings of two methods to automate power distribution network reconfiguration, the ILP Solver and the Heuristic Solver. We have seen how networks of different shapes can be translated into a flattened topology, which is necessary preprocessing to find a power assignment solution for a network. We have also seen some experimental results comparing the performance of the ILP Solver and the Heuristic Solver. The Heuristic Solver is a very fast, efficient algorithm to reconfigure power distribution, which is important in the case of an emergency. It performs consistently with near perfect results at a speed that is orders of magnitude quicker than the ILP Solver in almost all cases. In an application where a network is small and time is not an important constraint, the ILP Solver could possibly be preferable, but in any context where time is sensitive and near-perfect results are as acceptable as perfect results, the Heuristic Solver is much preferable. There is always room for improvement. Future tests should perhaps allow for non-integer capacity units, or loads that require other values than unit capacity. Optimizing each algorithm by rewriting them in C could give more optimized tests, though this may not be necessary to make judgments about implementing one or the other. There may be some ways to improve the Heuristic Solver, such as arranging the ordered_links in some way that could be more optimal. The algorithm could also be improved by taking advantage of the fact that once there are no more sources with capacity to provide any loads, the process of trying to assign loads to them for power supply can cease. Perhaps this method could be combined with other methods that do not presently account for load priorities or place as much value on fast execution.

I mitt arbete "Att gestalta en text i ett ackompanjemang" har jag valt att utifrån Fredmans epistel no 6 "Hör klockorna med ängsligt dån..." av C.M Bellman fördjupa mig i hur jag som instrumentalist kan lyfta fram en text i ett ackompanjemang och på så sätt bidra till ett kammarmusikaliskt samarbete mellan sångare och instrumentalist, där båda samverkar för att berätta sångens handling. Arbetet består av en överblick över C.M Bellmans liv och verk, en historisk bakgrund till "Hör klockorna med ängsligt dån..." samt en redogörelse för hur jag utifrån dessa två har arrangerat en gitarrstämma som enligt mig understryker vistextens handling. I min slutreflektion för jag ett resonemang kring orden samspel och samverkan samt på vilket sätt jag har utgått ifrån dessa när jag har arrangerat "Hör klockorna med ängsligt dån..." Min slutsats är att det kanske inte alltid är nödvändigt att som instrumentalist arrangera ett ackompanjemang ”ord-för-ord” såsom jag har gjort. Djupast sett handlar det snarare om en grundinställning hos instrumentalisten att vilja sätta sig in i en text och aktivt fundera kring sin egen berättarroll.

A presente dissertação expõe o ambiente em que o Problema de Merton é construído e, baseando-se na bibliografia apresentada, constrói exemplos em softwares cujas especificidades podem colaborar na clareza da resolução. O software Matlab engloba as soluções numéricas, enquanto o software Maple é responsável pela solução de equações diferenciais ordinárias e parciais de forma simbólica. Apresenta-se modificações do Problema de Merton original como exercícios para melhor esclarecer os diferentes parâmetros abordados. Na seção final é apresentada a solução de viscosidade, uma alternativa quando a função valor não apresenta características desejáveis para a análise apresentada. / This dissertation explicit the environment which Merton’s problem is built, according to the presented bibliography, exemples are built in softwares whose specificity might help to clarify the solution. The Matlab software embraces numeric solutions, while Maple software is appropriate to solve ordinary and parcial differential equations in symbolic form. Some modifications are presented to Merton’s Problem as exercise to improve understanding on the variations adopted. On final section, viscosity solutions are presented as an alternative solution for when the value function does not possess the desirables properties that allow the analysis on focus.

A presente dissertação expõe o ambiente em que o Problema de Merton é construído e, baseando-se na bibliografia apresentada, constrói exemplos em softwares cujas especificidades podem colaborar na clareza da resolução. O software Matlab engloba as soluções numéricas, enquanto o software Maple é responsável pela solução de equações diferenciais ordinárias e parciais de forma simbólica. Apresenta-se modificações do Problema de Merton original como exercícios para melhor esclarecer os diferentes parâmetros abordados. Na seção final é apresentada a solução de viscosidade, uma alternativa quando a função valor não apresenta características desejáveis para a análise apresentada. / This dissertation explicit the environment which Merton’s problem is built, according to the presented bibliography, exemples are built in softwares whose specificity might help to clarify the solution. The Matlab software embraces numeric solutions, while Maple software is appropriate to solve ordinary and parcial differential equations in symbolic form. Some modifications are presented to Merton’s Problem as exercise to improve understanding on the variations adopted. On final section, viscosity solutions are presented as an alternative solution for when the value function does not possess the desirables properties that allow the analysis on focus.

Uppsatsen innehåller en jämförelse mellan Fredmans Epistlar och Predikaren. Ämnet är Förgänglighetspoesi och avsikten är att se vilka typiska drag av förgänglighetspoesi som de båda verken innehåller.

We consider nonlinear elliptic Bellman systems which arise in the theory of stochastic differential games. The right hand sides of the equations (which are called Hamiltonians) may have quadratic growth with respect to the gradient of the unknowns. Under certain assumptions on Lagrangians (from which the Hamiltonians are derived), that are satisfied for many types of stochastic games, we establish the existence and uniqueness of a Nash point and develop structural conditions on the Hamiltonians. From these conditions we establish a certain version of maximum and minimum principle. This result is then used to establish the existence of a bound solution. 1

Uruguay is a pioneer in the use of renewable sources of energy and can usually satisfy its total demand from renewable sources. Control and optimization of the system is complicated by half of the installed power - wind and solar sources - be- ing non-controllable with high uncertainty and variability. In this work we present a novel optimization technique for efficient use of the production facilities. The dy- namical system is stochastic, and we deal with its non-Markovian dynamics through a Lagrangian relaxation. Continuous-time optimal control and value function are found from the solution to a sequence of Hamilton-Jacobi-Bellman partial differential equations associated with the system. We introduce a monotone scheme to avoid spurious oscillations in the numerical solution and apply the technique to a number of examples taken from the Uruguayan grid. We use parallelization and change of variables to reduce the computational times. Finally, we study the usefulness of extra system storage capacity offered by batteries.

京都大学 / 0048 / 新制・課程博士 / 博士(農学) / 甲第22488号 / 農博第2392号 / 新制||農||1076(附属図書館) / 学位論文||R2||N5268(農学部図書室) / 京都大学大学院農学研究科地域環境科学専攻 / (主査)教授 藤原 正幸, 教授 村上 章, 准教授 宇波 耕一 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

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This thesis considers classical methods to solve stochastic control problems and valuation problems from financial mathematics numerically. To this end, (linear) partial differential equations (PDEs) in non-divergence form or the optimality conditions known as the (nonlinear) Hamilton-Jacobi-Bellman (HJB) equations are solved by means of finite differences, volumes and elements. We consider all of these three approaches in detail after a thorough introduction to stochastic control problems and discuss various solution terms including classical solutions, strong solutions, weak solutions and viscosity solutions. A particular role in this thesis play degenerate problems. Here, a new model for the optimal control of an energy storage facility is developed which extends the model introduced in [Chen, Forsyth (2007)]. This four-dimensional HJB equation is solved by the classical finite difference Kushner-Dupuis scheme [Kushner, Dupuis (2001)] and a semi-Lagrangian variant which are both discussed in detail. Additionally, a convergence proof of the standard scheme in the setting of parabolic HJB equations is given. Finite volume schemes are another classical method to solve partial differential equations numerically. Sharing similarities to both finite difference and finite element schemes we develop a vertex-centered dual finite volume scheme. We discuss convergence properties and apply the scheme to the solution of HJB equations, which has not been done in such a broad context, to the best of our knowledge. Astonishingly, this is one of the first times the finite volume approach is systematically discussed for the solution of HJB equations. Furthermore, we give many examples which show advantages and disadvantages of the approach. Finally, we investigate novel tailored non-conforming finite element approximations of second-order PDEs in non-divergence form, utilizing finite-element Hessian recovery strategies to approximate second derivatives in the equation. We study approximations with both continuous and discontinuous trial functions. Of particular interest are a-priori and a-posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature. We discuss implementations of all three approaches in MATLAB (finite differences and volumes) and FEniCS (finite elements) publicly available in GitHub repositories under https://github.com/janblechschmidt. Many numerical experiments show convergence properties as well as pros and cons of the respective approach. Additionally, a new postprocessing procedure for policies obtained from numerical solutions of HJB equations is developed which improves the accuracy of control laws and their incurred values.