In this thesis, we consider the inverse Toeplitz eigenvalue problem which recover a real symmetric Toeplitz with desired eigenvalues. First some lower dimensional cases are solved by algebraic methods. This gives more insight on the inverse problem. Next, we explore the geometric meaning of real symmetric Toeplitz matrices. For high dimensional cases, numerical are unavoidable. From our numerical experiments, Newton-like methods are very effective for this problem.

The central theme of this thesis deals with problems related to the question, “Can one hear the shape of a drum?” first posed formally by Mark Kac in 1966. More precisely, can one determine the shape of a membrane with fixed boundary from the spectrum of the associated differential operator? For this paper, Kac received both the Lester Ford Award and the Chauvant Prize of the Mathematical Association of America. This problem has received a great deal of attention in the past forty years and has led to similar questions in completely different contexts such as “Can one hear the shape of a graph associated with the Schrödinger operator?”, “Can you hear the shape of your throat?”, “Can you feel the shape of a manifold with Brownian motion?”, “Can one hear the crack in a beam?”, “Can one hear into the sun?”, etc. Each of these topics deals with inverse eigenvalue problems or related inverse problems. For inverse problems in general, the problem may or may not have a solution, the solution may not be unique, and the solution does not necessarily depend continuously on perturbation of the data. For example, in the case of the drum, it has been shown that the answer to Kac’s question in general is “no.” However, if we restrict the class of drums, then the answer can be yes. This is typical of inverse problems when a priori information and restriction of the class of admissible solutions and/or data are used to make the problem well-posed. This thesis provides an analysis of shapes for which the answer to Kac's question is positive and a variety of interesting questions on this problem and its variants, including cases that remain open. This thesis also provides a synopsis and perspectives of other types of “can one hear” problems mentioned above. Another part of this thesis deals with aspects of direct problems related to musical instruments.

The presence of structure in a computational problem can often be exploited and can lead to a more efficient numerical algorithm. In this dissertation, we look at structured numerical problems that arise from applications in wireless communications and machine learning that also impact other areas of scientific computing. In wireless communication system designs, certain structured matrices (frames) need to be generated. The design of such matrices is equivalent to a symmetric inverse eigenvalue problem where the values of the diagonal elements are prescribed. We present algorithms that are capable of generating a larger set of these constructions than previous algorithms. We also discuss the existence of equiangular tight frames---frames that satisfy additional structural properties. Kernel learning is an important class of problems in machine learning. It often relies on efficient numerical algorithms that solve underlying convex optimization problems. In our work, the objective functions to be minimized are the von Neumann and the LogDet Bregman matrix divergences. The algorithm that solves this optimization problem performs matrix updates based on repeated eigendecompositions of diagonal plus rank-one matrices in the case of von Neumann matrix divergence, and Cholesky updates in case of the LogDet Bregman matrix divergence. Our contribution exploits the low-rank representations and the structure of the constraint matrices, resulting in more efficient algorithms than previously known. We also present two specialized zero-finding algorithms where we exploit the structure through the shape and exact formulation of the objective function. The first zero-finding task arises during the matrix update step which is part of the above-mentioned kernel learning application. The second zero-finding problem is for the secular equation; it is equivalent to the computation of the eigenvalues of a diagonal plus rank-one matrix. The secular equation arises in various applications, the most well-known is the divide-and-conquer eigensolver. In our solutions, we build upon a somewhat forgotten zero-finding method by P. Jarratt, first described in 1966. The method employs first derivatives only and needs the same amount of evaluations as Newton's method, but converges faster. Our contributions are the more efficient specialized zero-finding algorithms. / text

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

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 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.

As a musical instrument construction material, wood is both musically and aesthetically pleasing. Easy to work and abundant, it has traditionally been the material of choice. Unfortunately, wood is also a very inconsistent material. Due to great environmental and climatic variations, wooden specimens present large variations in their mechanical properties, even within species of a similar region. Surprisingly, an industry based entirely on acoustics has done very little to account for these variations. For this reason, manufactured wooden guitars are acoustically inconsistent. Previous work has shown that varying the dimensions of a guitar soundboard brace is a good method for taking into account variations in the mechanical properties of the wooden soundboard plate. In this thesis, the effects of a scalloped-shaped brace on the natural frequencies of a brace-plate system have been studied and tools have been developed in order to calculate the dimensions of the brace required to account for variations in the mechanical properties of the plate. It has been shown that scalloped braces can be used to modify two natural frequencies of a brace-plate system simultaneously. Furthermore, the most important criteria in modifying any given frequency of a brace-plate system is the mass and stiffness properties of the brace at the antinode of the given frequency’s associated modeshape. Subsequently, designing a brace for desired system natural frequencies, by taking into account the mechanical properties of the wooden plate, is an inverse eigenvalue problem. Since few methods exist for solving the inverse eigenvalue problem of general matrices, a new method based on the generalized Cayley-Hamilton theorem was proposed in the thesis. A further method, based on the determinant of the generalized eigenvalue problem was also presented. Both methods work well, although the determinant method is shown to be more efficient for partially described systems. Finally, experimental results were obtained for the natural frequencies of simply supported wooden plates, with and without a brace, as well as the inverse eigenvalue determinant method. Good correlation was found between theoretical and experimental results.

Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in MRF(G). We call these vertices nil or nonzero vertices, respectively. If a vertex is not a nil or nonzero vertex, we call it a neutral vertex. In addition, we completely classify the vertices of trees in terms of the classifications: nil, nonzero and neutral. Next we give an example of how nil vertices can help solve the inverse inertia problem. Lastly we give results about the inverse eigenvalue problem and solve a more complex variation of the problem (the λ, µ problem) for the path on 4 vertices. We also obtain a general result for the λ, µ problem concerning the number of λ’s and µ’s that can be equal.

In this thesis three methods LMGS, TQR and GR are applied to solve an inverseeigenvalue problem. We list the numerical results and compare the accuracy of the computed Jacobi matrix $T$ and the associated orthogonal matrix $Q$, wherethe columns of $Q^T$ are the eigenvectors of $T$. In the application of this inverse eigenvalue problem, the Fourier coefficients of $h(x)=e^x$ relative to the orthonormal polynomials associatedwith $T$ are evaluated, and these values are used to compute the least squarescoefficients of $h$ relative to the Chebyshev polynomials. We list thesenumerical results and compare them as our conclusion.

In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic $p$th root of astochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of stochastic $p$th roots. Our contributions include characterization of when a real matrix hasa real $p$th root, a classification of $p$th roots of a possibly singular matrix,a sufficient condition for a $p$th root of a stochastic matrix to have unit row sums,and the identification of two classes of stochastic matrices that have stochastic $p$th roots for all $p$. We also delineate a wide variety of possible configurationsas regards existence, nature (primary or nonprimary), and number of stochastic roots,and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix. On the computational side, we emphasize finding an approximate stochastic root: perturb the principal root $A^{1/p}$ or the principal logarithm $\log(A)$ to the nearest stochastic matrix or the nearest intensity matrix, respectively, if they are not valid ones;minimize the residual $\normF{X^p-A}$ over all stochastic matrices $X$ and also over stochastic matrices that are primary functions of $A$. For the first two nearness problems, the global minimizers are found in the Frobenius norm. For the last two nonlinear programming problems, we derive explicit formulae for the gradient and Hessian of the objective function $\normF{X^p-A}^2$ and investigate Newton's method, a spectral projected gradient method (SPGM) and the sequential quadratic programming method to solve the problem as well as various matrices to start the iteration. Numerical experiments show that SPGM starting with the perturbed $A^{1/p}$to minimize $\normF{X^p-A}$ over all stochastic matrices is method of choice.Finally, a new algorithm is developed for computing arbitrary real powers $A^\a$ of a matrix $A\in\mathbb{C}^{n\times n}$. The algorithm starts with a Schur decomposition,takes $k$ square roots of the triangular factor $T$, evaluates an $[m/m]$ Pad\'e approximant of $(1-x)^\a$ at $I - T^$, and squares the result $k$ times. The parameters $k$ and $m$ are chosen to minimize the cost subject to achieving double precision accuracy in the evaluation of the Pad\'e approximant, making use of a result that bounds the error in the matrix Pad\'e approximant by the error in the scalar Pad\'e approximant with argument the norm of the matrix. The Pad\'e approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for $T^$, yielding increased accuracy. Since the basic algorithm is designed for $\a\in(-1,1)$, a criterion for reducing an arbitrary real $\a$ to this range is developed, making use of bounds for the condition number of the $A^\a$ problem. How best to compute $A^k$ for a negative integer $k$ is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives,including the use of an eigendecomposition, a method based on the Schur--Parlett\alg\ with our new algorithm applied to the diagonal blocks and approaches based on the formula $A^\a = \exp(\a\log(A))$.