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

In this digital age, it is more important than ever to have good methods for processing images. We focus on the removal of blur from a captured image, which is called the image deblurring problem. In particular, we make no assumptions about the blur itself, which is called a blind deconvolution. We approach the problem by miniming an energy functional that utilizes total variation norm and a fidelity constraint. In particular, we extend the work of Chan and Wong to use a reference image in the computation. Using the shock filter as a reference image, we produce a superior result compared to existing methods. We are able to produce good results on non-black background images and images where the blurring function is not centro-symmetric. We consider using a general Lp norm for the fidelity term and compare different values for p. Using an analysis similar to Strong and Chan, we derive an adaptive scale method for the recovery of the blurring function. We also consider two numerical methods in this disseration. The first method is an extension of Picards method for PDEs in the discrete case. We compare the results to the analytical Picard method, showing the only difference is the use of the approximation versus exact derivatives. We relate the method to existing finite difference schemes, including the Lax-Wendroff method. We derive the stability constraints for several linear problems and illustrate the stability region is increasing. We conclude by showing several examples of the method and how the computational savings is substantial. The second method we consider is a black-box implementation of a method for solving the generalized eigenvalue problem. By utilizing the work of Golub and Ye, we implement a routine which is robust against existing methods. We compare this routine against JDQZ and LOBPCG and show this method performs well in numerical testing.

L'évolution des turbines à gaz vers des régimes de combustion en mélange pauvre augmente la sensibilité de la flamme aux perturbations de l'écoulement. Plus particulièrement, cela augmente le risque que des instabilités de combustion apparaissent. Comme ces oscillations peuvent affecter le processus de combustion, il est très important d'être capable de prédire ce comportement au niveau de la conception.L'objectif du travail présenté est de développer un solveur numérique qui permet de décrire ces instabilités, et d'évaluer les effets du nombre de Mach de l'écoulement moyen sur ce phénomène. L'approche choisie consiste à résoudre les équations d'Euler linéarisées, qui sont écrites dans le domaine fréquentiel sous la forme d'un problème aux valeurs propres. Ce système d'équations permets de prendre en compte la vitesse moyenne de l'écoulement, et donc d'évaluer les effets causés par la convection et leur impact sur la stabilité des modes. Parmi les mécanismes qui peuvent être étudiés se trouve notamment l'effet des ondes d'entropie convectées, ce qui est particulièrement intéressant dans le contexte des chambres de combustions. Afin de déterminer l'effet des termes liés à la vitesse de l'écoulement moyen sur la stabilité des modes, une analyse de l'énergie contenue dans les perturbations est effectuée. Finalement, l'aspect de la non-orthogonalité des modes propres, qui permet une croissance d'énergie transitoire dans un système linéairement stable, est abordé. / The development of gas turbines towards lean combustion increases the susceptibility of the flame to flow perturbations, and leads more particularly to a higher risk of combustion instability. As these self-sustained oscillations may affect the performance of the combustion device, it is very important to be able to predict them at the design level. At present, several methods are used to describe combustion instabilities, ranging from complex LES and DNS calculations to low-order network models. An intermediate method consists in solving a set of equations describing the acoustic field using a finite volume technique, which is the approach used in the present study.This thesis discusses the impact of a non zero Mach number mean flow field on thermoacoustic instability. The study is based on the linearized Euler equations, which are stated in the frequency domain in the form of an eigenvalue problem. Using the linearized Euler equations rather than the Helmholtz equation avoids making the commonly used assumption of the mean flow being at rest, and allows to take into account convection effects and their impact on the stability of the system. Among the mechanisms that can be studied using the present approach is namely the impact of convected entropy waves, which is especially interesting in combustion applications.For this study, a 1D and a 2D numerical solver have been developed and are presented in this thesis. In order to asses the effect of the mean flow terms on the modes' stability, an analysis of the disturbance energy budget is performed. Finally, the aspect of the eigenmodes being non-orthogonal and thus allowing for transient growth in linearly stable systems is adressed.

In vibration-based damage detection in structures, often changes in the dynamic properties such as natural frequencies, modeshapes, and derivatives of modeshapes are used to identify the damaged elements. If only a partial list of natural frequencies is known, optimization methods may need to be used to identify the damage. In this research, the algorithm proposed by Podlevskyi & Yaroshko (2013) is used to determine the stiffness distribution in shear building models. The lateral load resisting elements are presented as a single equivalent spring, and masses are lumped at floor levels. The proposed method calculates stiffness values directly, i.e., without optimization, from the known partial list of natural frequency data and mass distribution. It is shown that if the number of stories with reduced stiffness is smaller than the number of known natural frequencies, the stories with reduced stiffnesses can be identified. Numerical studies on building models with two stories and four stories are used to illustrate the solution method. Effect of error or noise in given natural frequencies on stiffness estimates and, conversely, sensitivity of natural frequencies to changes in stiffness are studied using 7-, 15-, 30-, and 50-story numerical models. From the studies, it is learnt that as the number of stories increases, the natural frequencies become less sensitive to stiffness changes. Additionally, eight laboratory experiments were conducted on a five-story aluminum structural model. Ten slender columns were used in each story of the specimen. Damage was simulated by removing columns in one, two, or three stories. The method can locate and quantify the damage in cases presented in the experimental studies. It is also applied to a 1/3 scaled 18-story steel moment frame building tested on an earthquake simulator (Suita et al., 2015) to identify the reduction in the stiffness due to fractures of beam flanges. Only the first two natural frequencies are used to determine the reductions in the stiffness since the third mode of the tower is torsional and no reasonable planar spring-mass model can be developed to present all of the translational modes. The method produced possible cases of the softening when the damage was assumed to occur at a single story.

The Faber-Krahn theorem states that among all bounded domains with the same volume in Rn (with the standard Euclidean metric), a ball that has lowest first Dirichlet eigenvalue. Recently it has been shown that a similar result holds for (semi-)regular trees. In this article we show that such a theorem also hold for other classes of (not necessarily non-regular) trees. However, for these new results no couterparts in the world of the Laplace-Beltrami-operator on manifolds are known. / Series: Preprint Series / Department of Applied Statistics and Data Processing

Many methods for computing eigenvalues of a large sparse matrix involve shift-invert transformations which require the solution of a shifted linear system at each step. This thesis deals with shift-invert iterative techniques for solving eigenvalue problems where the arising linear systems are solved inexactly using a second iterative technique. This approach leads to an inner-outer type algorithm. We provide convergence results for the outer iterative eigenvalue computation as well as techniques for efficient inner solves. In particular eigenvalue computations using inexact inverse iteration, the Jacobi-Davidson method without subspace expansion and the shift-invert Arnoldi method as a subspace method are investigated in detail. A general convergence result for inexact inverse iteration for the non-Hermitian generalised eigenvalue problem is given, using only minimal assumptions. This convergence result is obtained in two different ways; on the one hand, we use an equivalence result between inexact inverse iteration applied to the generalised eigenproblem and modified Newton's method; on the other hand, a splitting method is used which generalises the idea of orthogonal decomposition. Both approaches also include an analysis for the convergence theory of a version of inexact Jacobi-Davidson method, where equivalences between Newton's method, inverse iteration and the Jacobi-Davidson method are exploited. To improve the efficiency of the inner iterative solves we introduce a new tuning strategy which can be applied to any standard preconditioner. We give a detailed analysis on this new preconditioning idea and show how the number of iterations for the inner iterative method and hence the total number of iterations can be reduced significantly by the application of this tuning strategy. The analysis of the tuned preconditioner is carried out for both Hermitian and non-Hermitian eigenproblems. We show how the preconditioner can be implemented efficiently and illustrate its performance using various numerical examples. An equivalence result between the preconditioned simplified Jacobi-Davidson method and inexact inverse iteration with the tuned preconditioner is given. Finally, we discuss the shift-invert Arnoldi method both in the standard and restarted fashion. First, existing relaxation strategies for the outer iterative solves are extended to implicitly restarted Arnoldi's method. Second, we apply the idea of tuning the preconditioner to the inner iterative solve. As for inexact inverse iteration the tuned preconditioner for inexact Arnoldi's method is shown to provide significant savings in the number of inner solves. The theory in this thesis is supported by many numerical examples.

Aeroelastic behavior prediction is often confined to analytical or highly computational methods, so I developed a low degree of freedom computational method using structural finite elements and unsteady loading to cover a gap in the literature. Finite elements are readily suitable for determination of the free vibration characteristics of eccentric, elastic structures, and the free vibration characteristics fundamentally determine the aeroelastic behavior. I used Theodorsen’s unsteady strip loading formulation to model the aerodynamic loading on linear elastic structures assuming harmonic motion. I applied Hassig’s ‘p-k’ method to predict the flutter boundary of nonsymmetric, aeroelastic systems. I investigated the application of a quintic interpolation assumed displacement shape to accurately predict higher order characteristic effects compared to linear analytical results. I show that quintic interpolation is especially accurate over cubic interpolation when multi-modal interactions are considered in low degree of freedom flutter behavior for high aspect ratio HALE aircraft wings.

In the last years some results for the Laplacian on manifolds have been shown to hold also for the graph Laplacian, e.g. Courant's nodal domain theorem or Cheeger's inequality. Friedman (Some geometric aspects of graphs and their eigenfunctions, Duke Math. J. 69 (3), pp. 487-525, 1993) described the idea of a ``graph with boundary". With this concept it is possible to formulate Dirichlet and Neumann eigenvalue problems. Friedman also conjectured another ``classical" result for manifolds, the Faber-Krahn theorem, for regular bounded trees with boundary. The Faber-Krahn theorem states that among all bounded domains $D \subset R^n$ with fixed volume, a ball has lowest first Dirichlet eigenvalue. In this paper we show such a result for regular trees by using a rearrangement technique. We give restrictive conditions for trees with boundary where the first Dirichlet eigenvalue is minimized for a given "volume". Amazingly Friedman's conjecture is false, i.e. in general these trees are not ``balls". But we will show that these are similar to ``balls". (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

The variety of principal minors of nxn symmetric matrices, denoted Zn, can be described naturally as a projection from the Lagrangian Grassmannian. Moreover, Zn is invariant under the action of a group G C GL(2n) isomorphic to (SL(2)xn) x Sn. One may use this symmetry to study the defining ideal of Zn as a G-module via a coupling of classical representation theory and geometry. The need for the equations in the defining ideal comes from applications in matrix theory, probability theory, spectral graph theory and statistical physics. I describe an irreducible G-module of degree 4 polynomials called the hyperdeterminantal module (which is constructed as the span of the G-orbit of Cayley's hyperdeterminant of format 2 x 2 x 2) and show that it that cuts out Zn set theoretically. This result solves the set-theoretic version of a conjecture of Holtz and Sturmfels and gives a collection of necessary and sufficient conditions for when it is possible for a given vector of length 2n to be the principal minors of a symmetric n x n matrix. In addition to solving the Holtz and Sturmfels conjecture, I study Zn as a prototypical G-variety. As a result, I exhibit the use of and further develop techniques from classical representation theory and geometry for studying G-varieties.