京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第19341号 / 情博第593号 / 新制||情||103(附属図書館) / 32343 / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 中村 佳正, 教授 矢ｹ崎 一幸, 教授 西村 直志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第20739号 / 情博第653号 / 新制||情||113(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 中村 佳正, 教授 山下 信雄, 教授 西村 直志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20880号 / 理博第4332号 / 新制||理||1622(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 堤 誉志雄, 教授 上 正明, 教授 宍倉 光広 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We implement an eigenvalue solving algorithm proposed by Ng and Trench, specialized for Toeplitz(-like) matrices, utilizing root finding in conjunction with an iteratively calculated version of the characteristic polynomial. The solver also yields corresponding eigenvectors as a free bi-product. We combine the algorithm with matrix-less methods in order to yield eigenvector approximations, and examine its behavior both regarding demands for time and computational power. The algorithm is fully parallelizable, and although solving of all eigenvalues to the bi-Laplacian discretization matrix - which we used as a model matrix - is not superior to standard methods, we see promising results when using it as an eigenvector solver, using eigenvector approximations from standard solvers or a matrix-less method. We also note that an advantage of the algorithm we examine is that it can calculate singular, specific eigenvalues (and the corresponding eigenvectors), anywhere in the spectrum, whilst standard solvers often have to calculate all eigenvalues, which could be a useful feature. We conclude that - while the algorithm shows promising results - more experiments are needed, and propose a number of topics which could be studied further, e.g. different matrices (Toeplitz-like, full), and looking at even larger matrices.

Pairwise comparison matrices (PCMs) play a key role in multi-criteria decision making, especially in the analytic hierarchy process. It could be necessary for an expert to compare alternatives based on various criteria. However, for a variety of reasons, such as lack of time or insufficient knowledge, it may happen that the expert cannot provide judgments on all pairs of alternatives. In this case, an incomplete pairwise comparison matrix is formed. In the first research part, an optimization algorithm is proposed for the optimal completion of an incomplete PCM. It is intended to numerically minimize a constrained eigenvalue problem, in which the objective function is difficult to write explicitly in terms of variables. Numerical simulations are carried out to examine the performance of the algorithm. The simulation results show that the proposed algorithm is capable of solving the minimization of the constrained eigenvalue problem. In the second part, a comparative analysis of eleven completion methods is studied. The similarity of the eleven completion methods is analyzed on the basis of numerical simulations and hierarchical clustering. Numerical simulations are performed for PCMs of different orders considering various numbers of missing comparisons. The results suggest the existence of a cluster of five extremely similar methods, and a method significantly dissimilar from all the others. In the third part, the filling in patterns (arrangements of known comparisons) of incomplete PCMs based on their graph representation are investigated under given conditions: regularity, diameter and number of vertices, but without prior information. Regular and quasi-regular graphs with minimal diameter are proposed. Finally, the simulation results indicate that the proposed graphs indeed provide better weight vectors than alternative graphs with the same number of comparisons. This research problem’s contributions include a list of (quasi-)regular graphs with diameters of 2 and 3, and vertices from 5 up to 24.

Differential equations arise frequently in modeling of physical systems, often resulting in linear eigenvalue problems. However, when dealing with large physical domains, solving such problems can be computationally expensive. This thesis examines an alternative approach to solving these problems, which involves utilizing absorbing boundary conditions and a Dirichlet-to-Neumann maps to transform the large sparse linear eigenvalue problem into a smaller nonlinear eigenvalue problem (NEP). The NEP is then solved using augmented Newton’s method. The specific equation investigated in this thesis is the two-dimensional Helmholtz equation, defined on the interval (x, y) ∈ [0, 10] × [0, 1], with the absorbing boundary condition introduced at x = 1. The results show a significant reduction in computational time when using this method compared to the original linear problem, making it a valuable tool for solving large linear eigenvalue problems. Another result is that the NEP does not affect the computational error compared to solving the linear problem, which further supports the NEP as an attractive alternative method.

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