The investigation is initially a continuation of Neuberger's work on linear boundary value problems. A very general iterative procedure for solution of these problems is described. The alternating-projection theorem of von Neumann is the mathematical starting point for this study. Later theorems demonstrate the validity of numerical approximation for Neuberger's method under certain conditions. A sampling of differential equations within the scope of our iterative method is given. The numerical evidence is that the procedure works well on neutral-state equations, for which no software is written now.

To make energy production sustainable and reduce carbon dioxide emissions it is necessary to stop using fossil fuels as our primary energy source. The Accelerator Driven Subcritical Reactor (ADSR) could provide safe nuclear power. It uses thorium as fuel, which is more abundant than uranium, and produces less long lived waste. An ADSR uses neutron spallation, caused by a high power proton beam impacting a metal target, to drive and control the reaction. The beam needs to have an energy of around 1 Ge V and a current of 10 mA with a very high reliability, the combination of which is beyond the capabilities of existing particle accelerators. Cyclotrons and synchrotrons both have trouble producing such a beam, while a suitable linac would be several hundred metres long, and expensive. A more compact accelerator design would allow multiple accelerators to be combined to improve reliability. This thesis examines the use of a Fixed-Field Alternating-Gradient (FFAG) accelerator as the proton driver. FFAGs are compact, and can simultaneously achieve higher energies than a cyclotron at higher repetition rates than a synchrotron. However, it is still a challenge to reach the high currents required. A 35 to 400 MeV non-scaling FFAG was designed to demonstrate issues encountered at high currents. Two methods were investigated in order to increase the number of particle bunches that could be simultaneously accelerated. One uses multiple solutions to the harmonic conditions for acceleration, and the second injects bunches after the acceleration has started. Neither was found to give significant practical improvement in current. Space charge is a destructive force at high currents. Software was developed to simulate the effect of space charge in an FFAG using several models. Space charge tune shifts were measured for a range of energies and currents, and peak currents of above 1 A were found to be unstable. In order to provide 10 mA of average current, acceleration would need to occur in around 100 turns, which will require a very rapid RF sweep.

We study stockpiling problems under uncertain economic and physical factors, and investigate the valuation and optimisation of storage systems where the availability and spot price of the underlying are both subject to stochasticity. Following a Real Options valuation approach, we first study financial derivatives linked to Asian options. A comprehensive set of boundary conditions is compiled, and an alternative (and novel) similarity reduction for fixed-strike Asian options is derived. Hybrid semi-Lagrangian methods for numerically solving the related partial differential equations (PDEs) are implemented, and we assess the accuracy of the valuations thus obtained with respect to results from classical finite-difference valuation methods and with respect to high precision calculations for valuing Asian options with spectral expansion theory techniques. Next we derive a PDE model for valuing the storage of electricity from a wind farm, with an attached back-up battery, that operates by trading electricity in a volatile market in order to meet a contracted fixed rate of energy generation; this system comprises two diffusive-type (stochastic) variables, namely the energy production and the electricity spot price, and two time-like (deterministic) variables, specifically the battery state and time itself. An efficient and novel semi-Lagrangian alternating-direction implicit (SLADI) methodology for numerically solving advection-diffusion problems is developed: here a semi-Lagrangian approach for hyperbolic problems of advection is combined with an alternating-direction implicit method for parabolic problems involving diffusion. Efficiency is obtained by solving (just) tridiagonal systems of equations at every time step. The results are compared to more standard semi-Lagrangian Crank-Nicolson (SLCN) and semi-Lagrangian fully implicit (SLFI) methods. Once he have established our PDE model for a storage-upgraded wind farm, a system that depends heavily on the highly stochastic nature of wind and the volatile market where electricity is sold, we derive a Hamilton-Jacobi-Bellman (HJB) equation for optimally controlling charging and discharging rates of the battery in time, and we assess a series of operation regimes. The solution of the related PDE models is approached numerically using our SLADI methodology to efficiently treat this mixed advection and diffusion problem in four dimensions. Extensive numerical experimentation confirms our SLADI methodology to be robust and yields highly accurate solutions and efficient computations, we also explore effects from correlation between stochastic electricity generation and random prices of electricity as well as effects from a seasonal electricity spot price. Ultimately, the objective of approximating optimal storage policies for a system under uncertain economic and physical factors is accomplished. Finally we examine the steady-state solution of a stochastic storage problem under uncertain electricity market prices and fixed demand. We use a HJB formulation for optimally controlling charging and discharging rates of the storage device with respect to the electricity spot price. A projected successive over-relaxation coupled with the semi-Lagrangian method is implemented, and we explore the use of boundary-fitted coordinates techniques.

This study proposes an approach for solving density-based multi-material topology optimization of cracked structures using Peridynamics. The alternating active-phase algorithm is utilized to transform the multi-material problem into a series of binary phase topology optimization sub-problems. Instead of the conventional mesh-based methods, the Peridynamics theory (PD) is used as a tool to model the behaviour of the materials and solve for the displacement field. The most significant advantage of PD is its ability to model discontinuities in a relatively straightforward manner. Thus, in the present work, the effect of cracks as a discontinuity is investigated on the optimal multi-material topologies. The Solid Isotropic Material with Penalty (SIMP) method is utilized to define the material properties as a function of the design variables. Also, the optimization problem is solved through the Optimality Criteria (OC) approach. The proposed method is compared to the results reported in the literature by executing three numerical examples that investigate the effect of the direction of an interior crack on the optimal topologies. Moreover, the efficiency of the proposed approach is verified by solving several examples where we aim at minimizing the compliance of the structure with and without initial cracks. / Graduate

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21536号 / 理博第4443号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 大槻 知忠, 教授 向井 茂, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22862号 / 情博第741号 / 新制||情||127(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 山下 信雄, 教授 太田 快人, 教授 鹿島 久嗣 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Cancer prevails as one of the most devastating diseases being at the top of death causes for adults despite continuous development and innovation in cancer therapy. Nanotechnology may be used to achieve therapeutic dosing, establish sustained-release drug profiles, and increase the half-life of drugs. In this context, magnetic nanowires (NWs) have shown a good biocompatibility and cellular internalization with a low cytotoxic effect. In this thesis, I induced cancer cell death by combining the chemotherapeutic effect of iron NWs functionalized with Doxorubicin (DOX) with mechanical disturbance under a low frequency alternating magnetic field. Two different agents, APTES and BSA, were separately used for coating NWs permitting further functionalization with DOX. Internalization was qualitatively and quantitatively assessed for both formulations by confocal reflection microscopy and inductively coupled plasma-mass spectrometry. From confocal reflection analysis, BSA formulations demonstrate to have a higher internalization degree and a broader distribution within the cells in comparison to APTES formulations. Both groups of functionalized NWs generated a comparable cytotoxic effect in MDA-MB-231 breast cancer cells in a DOX concentration-dependent manner, (~60% at the highest concentration tested) that was significantly different from the effect produced by the free DOX (~95% at the same concentration) and non-functionalized NWs formulations (~10% at the same NWs concentration). A synergistic cytotoxic effect is obtained when a low frequency magnetic field (1 mT, 10 Hz) is applied to cells treated with the two formulations that is again comparable (~70% at the highest concentration). Furthermore, the cytotoxic effect of both groups of coated NWs without the drug increased notoriously when the field is applied (~25% at the highest concentration tested). Here, a novel bimodal method for cancer cell destruction was developed by the conjugation of the magneto-mechanical properties of the iron NWs coupled with the chemotoxic effect of an anticancer drug. Moreover, it was demonstrated that iron nanowires possess an outstanding biocompatibility and showed high efficacy as drug delivery agents coupled to a high degree of cell internalization. Finally, the proposed method benefits from the low power fields applied during treatment. This poses much less safety risks and allows using cheaper and simpler equipment.

Numerical techniques for linear systems arising from discretization of partial differential equations are nowadays essential for understanding the physical world. Among these techniques, iterative methods and the accompanying preconditioning techniques have become increasingly popular due to their great potential on large scale computation. In this work, we present preconditioning techniques for linear systems built with tensor product basis functions. Efficient algorithms are designed for various problems by exploiting the Kronecker product structure in the matrices, inherited from tensor product basis functions. Specifically, we design preconditioners for mass matrices to remove the complexity from the basis functions used in isogeometric analysis, obtaining numerical performance independent of mesh size, polynomial order and continuity order; we also present a compound iteration preconditioner for stiffness matrices in two dimensions, obtaining fast convergence speed; lastly, for the Helmholtz problem, we present a strategy to `hide' its indefiniteness from Krylov subspace methods by eliminating the part of initial error that corresponds to those negative generalized eigenvalues. For all three cases, the Kronecker product structure in the matrices is exploited to achieve high computational efficiency.

Due to the increase in the advances in wireless communication, there has been an increase in the use of multi-agents systems to complete any given task. In various applications, multi-agent systems are required to solve an underlying optimization problem to obtain the best possible solution within a feasible region. Solving such multi-agent optimization problems in a distributed framework preferable over centralized frameworks as the former ensures scalability, robustness, and security. Further distributed optimization problem becomes challenging when the decision variables of the individual agents are coupled. In this thesis, a distributed optimization problem with coupled constraints is considered, where a network of agents aims to cooperatively minimize the sum of their local objective functions, subject to individual constraints. This problem setup is relevant to many practical applications like formation flying, sensor fusion, smart grids, etc. For practical scenarios, where agents can solve their local optimal solution efficiently and require fewer assumptions on objective functions, the Alternating Direction Method of Multipliers(ADMM)-based approaches are preferred over gradient-based approaches. For such a constraint coupled problem, several distributed ADMM algorithms are present that guarantee convergence to optimality but they do not discuss the complete analysis for the rate of convergence. Thus, the primary goal of this work is to improve upon the convergence rate of the existing state-of-the-art Tracking-ADMM (TADMM) algorithm to solve the above-distributed optimization problem. Moreover, the current analysis in literature does not discuss the convergence in the case of a time-varying communication network. The first part of the thesis focuses on improving the convergence rate of the Tracking-ADMM algorithm to solve the above-distributed optimization problem more efficiently. To this end, an upper bound on the convergence rate of the TADMM algorithm is derived in terms of the weight matrix of the network. To achieve faster convergence, the optimal weight matrix is computed using a semi-definite programming (SDP) formulation. The improved convergence rate of this Fast-TADMM (F-TADMM) is demonstrated with a simple yet illustrative, coupled constraint optimization problem. Then, the applicability of F-TADMM is demonstrated to the problem of distributed optimal control for trajectory generation of aircraft in formation flight. In the second part of the thesis, the convergence analysis for TADMM is extended while considering a time-varying communication network. The modified algorithm is named as Time-Varying Tracking (TV-TADMM). The formal guarantees on asymptotic convergence are provided with the help of control system analysis of a dynamical system that uses Lyapunov-like theory. The convergence of this TV-TADMM is demonstrated on a simple yet illustrative, coupled constraint optimization problem with switching topology and is compared with the fixed topology setting.

The purpose of this study was to determine if by alternating certain classes biennially the number of different teaching preparations could be reduced thus decreasing the teachers' load. The study also attempted to ascertain the effect this type of scheduling had upon the attitudes of teachers and students toward these classes. A further attempt was made to determine if mixing students from two grade levels resulted in the younger student being penalized with respect to his cumulative grade point average. The significant conclusions that can be drawn from the results follow: Teachers, generally, thought well of the project and desires to participate in it. Although some teachers expressed reservations about the project there were none that could not be removed by correcting the problems expressed. At the conclusion of the study most were in favor of continuing the project. By alternating classes biennially which were normally taught annually the class preparation load for teachers participating in the project was reduced and this basic quest of the project was, in fact, met. Students were not aroused unduly by the experimental nature of the project but sensed the need for long-range planning and increased guidance in setting up their schedules. They did not object to being combined with other grade levels in the project classes. Contrary to administrative expectations, however, the younger students did tend to receive lower marks in the project classes. Achievement progress as measured by the use of standardized tests revealed no significant difference between students in the experimental school and those students in the control school.