Dimensionality hyper-reduction and machine learning for dynamical systems with varying parameters

Thesis (Ph.D.)--University of Washington, 2016-08

Linear non-local diffusion problems in metric measure spaces

The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ℝN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods.

Time evolution in the unimolecular master equation at low temperatures: full spectral solution with scalable iterative methods and high precision

A new method is presented to determine an accurate eigendecomposition of difficult low temperature unimolecular master equation problems. Based on a generalisation of the Nesbet method, the new method is capable of achieving complete spectral resolution of the master equation matrix with relative accuracy in the eigenvectors. The method is applied to a test case of the decomposition of ethane at 300 K from a microcanonical initial population with energy transfer modelled by both Ergodic Collision Theory and the exponential-down model. The fact that quadruple precision (16-byte) arithmetic is required irrespective of the eigensolution method used is demonstrated. (C) 2001 Elsevier Science B.V. All rights reserved.

Electrohysterogram signal component cataloging with spectral and time-frequency methods

The Electrohysterogram (EHG) is a new instrument for pregnancy monitoring. It measures the uterine muscle electrical signal, which is closely related with uterine contractions. The EHG is described as a viable alternative and a more precise instrument than the currently most widely used method for the description of uterine contractions: the external tocogram. The EHG has also been indicated as a promising tool in the assessment of preterm delivery risk. This work intends to contribute towards the EHG characterization through the inventory of its components which are: • Contractions; • Labor contractions; • Alvarez waves; • Fetal movements; • Long Duration Low Frequency Waves; The instruments used for cataloging were: Spectral Analysis, parametric and non-parametric, energy estimators, time-frequency methods and the tocogram annotated by expert physicians. The EHG and respective tocograms were obtained from the Icelandic 16-electrode Electrohysterogram Database. 288 components were classified. There is not a component database of this type available for consultation. The spectral analysis module and power estimation was added to Uterine Explorer, an EHG analysis software developed in FCT-UNL. The importance of this component database is related to the need to improve the understanding of the EHG which is a relatively complex signal, as well as contributing towards the detection of preterm birth. Preterm birth accounts for 10% of all births and is one of the most relevant obstetric conditions. Despite the technological and scientific advances in perinatal medicine, in developed countries, prematurity is the major cause of neonatal death. Although various risk factors such as previous preterm births, infection, uterine malformations, multiple gestation and short uterine cervix in second trimester, have been associated with this condition, its etiology remains unknown [1][2][3].

Impedance boundary conditions for pseudo-spectral time-domain methods in room acoustics

The pseudo-spectral time-domain (PSTD) method is an alternative time-marching method to classicalleapfrog finite difference schemes in the simulation of wave-like propagating phenomena. It is basedon the fundamentals of the Fourier transform to compute the spatial derivatives of hyperbolic differential equations. Therefore, it results in an isotropic operator that can be implemented in an efficient way for room acoustics simulations. However, one of the first issues to be solved consists on modeling wallabsorption. Unfortunately, there are no references in the technical literature concerning to that problem. In this paper, assuming real and constant locally reacting impedances, several proposals to overcome this problem are presented, validated and compared to analytical solutions in different scenarios.

Impedance boundary conditions for pseudo-spectral time-domain methods in room acoustics

The Pseudo-Spectral Time Domain (PSTD) method is an alternative time-marching method to classical leapfrog finite difference schemes inthe simulation of wave-like propagating phenomena. It is based on the fundamentals of the Fourier transform to compute the spatial derivativesof hyperbolic differential equations. Therefore, it results in an isotropic operator that can be implemented in an efficient way for room acousticssimulations. However, one of the first issues to be solved consists on modeling wall absorption. Unfortunately, there are no references in thetechnical literature concerning to that problem. In this paper, assuming real and constant locally reacting impedances, several proposals toovercome this problem are presented, validated and compared to analytical solutions in different scenarios.

Introduction of sensor spectral response into image fusion methods. Application to wavelet-based methods

Usual image fusion methods inject features from a high spatial resolution panchromatic sensor into every low spatial resolution multispectral band trying to preserve spectral signatures and improve spatial resolution to that of the panchromatic sensor. The objective is to obtain the image that would be observed by a sensor with the same spectral response (i.e., spectral sensitivity and quantum efficiency) as the multispectral sensors and the spatial resolution of the panchromatic sensor. But in these methods, features from electromagnetic spectrum regions not covered by multispectral sensors are injected into them, and physical spectral responses of the sensors are not considered during this process. This produces some undesirable effects, such as resolution overinjection images and slightly modified spectral signatures in some features. The authors present a technique which takes into account the physical electromagnetic spectrum responses of sensors during the fusion process, which produces images closer to the image obtained by the ideal sensor than those obtained by usual wavelet-based image fusion methods. This technique is used to define a new wavelet-based image fusion method.

Novel Regression Methods For Spectral Data

Singular Value Decomposition (SVD), Principal Component Analysis (PCA) and Multiple Linear Regression (MLR) are some of the mathematical pre- liminaries that are discussed prior to explaining PLS and PCR models. Both PLS and PCR are applied to real spectral data and their di erences and similarities are discussed in this thesis. The challenge lies in establishing the optimum number of components to be included in either of the models but this has been overcome by using various diagnostic tools suggested in this thesis. Correspondence analysis (CA) and PLS were applied to ecological data. The idea of CA was to correlate the macrophytes species and lakes. The di erences between PLS model for ecological data and PLS for spectral data are noted and explained in this thesis. i

Precision of yule-walker methods for the arma spectral model

In this work a new method is proposed of separated estimation for the ARMA spectral model based on the modified Yule-Walker equations and on the least squares method. The proposal of the new method consists of performing an AR filtering in the random process generated obtaining a new random estimate, which will reestimate the ARMA model parameters, given a better spectrum estimate. Some numerical examples will be presented in order to ilustrate the performance of the method proposed, which is evaluated by the relative error and the average variation coefficient.

Adaptation Strategies for Discontinuous Galerkin Spectral Element Methods by means of Truncation Error Estimations

In this work a p-adaptation (modification of the polynomial order) strategy based on the minimization of the truncation error is developed for high order discontinuous Galerkin methods. The truncation error is approximated by means of a truncation error estimation procedure and enables the identification of mesh regions that require adaptation. Three truncation error estimation approaches are developed and termed a posteriori, quasi-a priori and quasi-a priori corrected. Fine solutions, which are obtained by enriching the polynomial order, are required to solve the numerical problem with adequate accuracy. For the three truncation error estimation methods the former needs time converged solutions, while the last two rely on non-converged solutions, which lead to faster computations. Based on these truncation error estimation methods, algorithms for mesh adaptation were designed and tested. Firstly, an isotropic adaptation approach is presented, which leads to equally distributed polynomial orders in different coordinate directions. This first implementation is improved by incorporating a method to extrapolate the truncation error. This results in a significant reduction of computational cost. Secondly, the employed high order method permits the spatial decoupling of the estimated errors and enables anisotropic p-adaptation. The incorporation of anisotropic features leads to meshes with different polynomial orders in the different coordinate directions such that flow-features related to the geometry are resolved in a better manner. These adaptations result in a significant reduction of degrees of freedom and computational cost, while the amount of improvement depends on the test-case. Finally, this anisotropic approach is extended by using error extrapolation which leads to an even higher reduction in computational cost. These strategies are verified and compared in terms of accuracy and computational cost for the Euler and the compressible Navier-Stokes equations. The main result is that the two quasi-a priori methods achieve a significant reduction in computational cost when compared to a uniform polynomial enrichment. Namely, for a viscous boundary layer flow, we obtain a speedup of a factor of 6.6 and 7.6 for the quasi-a priori and quasi-a priori corrected approaches, respectively. RESUMEN En este trabajo se ha desarrollado una estrategia de adaptación-p (modificación del orden polinómico) para métodos Galerkin discontinuo de alto orden basada en la minimización del error de truncación. El error de truncación se estima utilizando el método tau-estimation. El estimador permite la identificación de zonas de la malla que requieren adaptación. Se distinguen tres técnicas de estimación: a posteriori, quasi a priori y quasi a priori con correción. Todas las estrategias requieren una solución obtenida en una malla fina, la cual es obtenida aumentando de manera uniforme el orden polinómico. Sin embargo, mientras que el primero requiere que esta solución esté convergida temporalmente, el resto utiliza soluciones no convergidas, lo que se traduce en un menor coste computacional. En este trabajo se han diseñado y probado algoritmos de adaptación de malla basados en métodos tau-estimation. En primer lugar, se presenta un algoritmo de adaptacin isótropo, que conduce a discretizaciones con el mismo orden polinómico en todas las direcciones espaciales. Esta primera implementación se mejora incluyendo un método para extrapolar el error de truncación. Esto resulta en una reducción significativa del coste computacional. En segundo lugar, el método de alto orden permite el desacoplamiento espacial de los errores estimados, permitiendo la adaptación anisotropica. Las mallas obtenidas mediante esta técnica tienen distintos órdenes polinómicos en cada una de las direcciones espaciales. La malla final tiene una distribución óptima de órdenes polinómicos, los cuales guardan relación con las características del flujo que, a su vez, depenen de la geometría. Estas técnicas de adaptación reducen de manera significativa los grados de libertad y el coste computacional. Por último, esta aproximación anisotropica se extiende usando extrapolación del error de truncación, lo que conlleva un coste computational aún menor. Las estrategias se verifican y se comparan en téminors de precisión y coste computacional utilizando las ecuaciones de Euler y Navier Stokes. Los dos métodos quasi a priori consiguen una reducción significativa del coste computacional en comparación con aumento uniforme del orden polinómico. En concreto, para una capa límite viscosa, obtenemos una mejora en tiempo de computación de 6.6 y 7.6 respectivamente, para las aproximaciones quasi-a priori y quasi-a priori con corrección.

Temporal and spectral nonlinear pulse shaping methods in optical fibers

The combination of the third-order optical nonlinearity with chromatic dispersion in optical fibers offers an extremely rich variety of possibilities for tailoring the temporal and spectral content of a light signal, depending on the regime of dispersion that is used. Here, we review recent progress on the use of third-order nonlinear processes in optical fibers for pulse shaping in the temporal and spectral domains. Various examples of practical significance will be discussed, spanning fields from the generation of specialized temporal waveforms to the generation of ultrashort pulses, and to stable continuum generation.

Petroleomics By Electrospray Ionization Ft-icr Mass Spectrometry Coupled To Partial Least Squares With Variable Selection Methods: Prediction Of The Total Acid Number Of Crude Oils.

Negative-ion mode electrospray ionization, ESI(-), with Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR MS) was coupled to a Partial Least Squares (PLS) regression and variable selection methods to estimate the total acid number (TAN) of Brazilian crude oil samples. Generally, ESI(-)-FT-ICR mass spectra present a power of resolution of ca. 500,000 and a mass accuracy less than 1 ppm, producing a data matrix containing over 5700 variables per sample. These variables correspond to heteroatom-containing species detected as deprotonated molecules, [M - H](-) ions, which are identified primarily as naphthenic acids, phenols and carbazole analog species. The TAN values for all samples ranged from 0.06 to 3.61 mg of KOH g(-1). To facilitate the spectral interpretation, three methods of variable selection were studied: variable importance in the projection (VIP), interval partial least squares (iPLS) and elimination of uninformative variables (UVE). The UVE method seems to be more appropriate for selecting important variables, reducing the dimension of the variables to 183 and producing a root mean square error of prediction of 0.32 mg of KOH g(-1). By reducing the size of the data, it was possible to relate the selected variables with their corresponding molecular formulas, thus identifying the main chemical species responsible for the TAN values.

Effects Of Misalignments In The Retinal Nerve Fiber Layer Thickness Measurements With Spectral Domain Optical Coherence Tomography.

Purpose. To investigate misalignments (MAs) on retinal nerve fiber layer thickness (RNFLT) measurements obtained with Cirrus(©) SD-OCT. Methods. This was a retrospective, observational, cross-sectional study. Twenty-seven healthy and 29 glaucomatous eyes of 56 individuals with one normal exam and another showing MA were included. MAs were defined as an improper alignment of vertical vessels in the en face image. MAs were classified in complete MA (CMA) and partial MA (PMA), according to their site: 1 (superior, outside the measurement ring (MR)), 2 (superior, within MR), 3 (inferior, within MR), and 4 (inferior, outside MR). We compared RNFLT measurements of aligned versus misaligned exams in all 4 sectors, in the superior area (sectors 1 + 2), inferior area (sectors 3 + 4), and within the measurement ring (sectors 2 + 3). Results. RNFLT measurements at 12 clock-hour of eyes with MAs in the superior area (sectors 1 + 2) were significantly lower than those obtained in the same eyes without MAs (P = 0.043). No significant difference was found in other areas (sectors 1 + 2 + 3 + 4, sectors 3 + 4, and sectors 2 + 3). Conclusion. SD-OCT scans with superior MAs may present lower superior RNFLT measurements compared to aligned exams.