Fiedler matrices: numerical and structural properties

The first and second Frobenius companion matrices appear frequently in numerical application, but it is well known that they possess many properties that are undesirable numerically, which limit their use in applications. Fiedler companion matrices, or Fiedler matrices for brevity, introduced in 2003, is a family of matrices which includes the two Frobenius matrices. The main goal of this work is to study whether or not Fiedler companion matrices can be used with more reliability than the Frobenius ones in the numerical applications where Frobenius matrices are used. For this reason, in this work we present a thorough study of Fiedler matrices: their structure and numerical properties, where we mean by numerical properties those properties that are interesting for applying these matrices in numerical computations, and some of their applications in the field on numerical linear algebra. The introduction of Fiedler companion matrices is an example of a simple idea that has been very influential in the development of several lines of research in the numerical linear algebra field. This family of matrices has important connections with a number of topics of current interest, including: polynomial root finding algorithms, linearizations of matrix polynomials, unitary Hessenberg matrices, CMV matrices, Green’s matrices, orthogonal polynomials, rank structured matrices, quasiseparable and semiseparable matrices, etc.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Paul Van Dooren.- Secretario: Juan Bernardo Zaballa Tejada.- Vocal: Françoise Tisseu

Efficient numerical methods to solve some reaction-diffusion problems arising in biology

Philosophiae Doctor - PhDIn this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with those obtained by other researchers

Non-invasive ultrasound monitoring of regional carotid wall structure and deformation in atherosclerosis

Thesis (Ph. D.)--Harvard--Massachusetts Institute of Technology Division of Health Sciences and Technology, 2001.Includes bibliographical references (p. 223-242).Atherosclerosis is characterized by local remodeling of arterial structure and distensibility. Developing lesions either progress gradually to compromise tissue perfusion or rupture suddenly to cause catastrophic myocardial infarction or stroke. Reliable measurement of changes in arterial structure and composition is required for assessment of disease progression. Non-invasive carotid ultrasound can image the heterogeneity of wall structure and distensibility caused by atherosclerosis. However, this capability has not been utilized for clinical monitoring because of speckle noise and other artifacts. Clinical measures focus instead on average wall thickness and diameter distension in the distal common carotid to reduce sensitivity to noise. The goal of our research was to develop an effective system for reliable regional structure and deformation measurements since these are more sensitive indicators of disease progression. We constructed a system for freehand ultrasound scanning based on custom software which simultaneously acquires real-time image sequences and 3D frame localization data from an electromagnetic spatial localizer. With finite element modeling, we evaluated candidate measures of regional wall deformation.(cont.) Finally, we developed a multi-step scheme for robust estimation of local wall structure and deformation. This new strategy is based on a directionally-sensitive segmentation functional and a motion-region-of-interest constrained optical flow algorithm. We validated this estimator with simulated images and clinical ultrasound data. The results show structure estimates that are accurate and precise, with inter- and intra-observer reproducibility surpassing existing methods. Estimates of wall velocity and deformation likewise show good overall accuracy and precision. We present results from a proof-of-principle evaluation conducted in a pilot study of normal subjects and clinical patients. For one example, we demonstrate the combination of 2D image processing with 3D frame localization for visualization of the carotid volume. With slice localization, estimates of carotid wall structure and deformation can be derived for all axial positions along the carotid artery. The elements developed here provide the tools necessary for reliable quantification of regional wall structure and composition changes which result from atherosclerosis.by Raymond C. Chan.Ph.D