114 research outputs found
Harnessing the Power of Sample Abundance: Theoretical Guarantees and Algorithms for Accelerated One-Bit Sensing
One-bit quantization with time-varying sampling thresholds (also known as
random dithering) has recently found significant utilization potential in
statistical signal processing applications due to its relatively low power
consumption and low implementation cost. In addition to such advantages, an
attractive feature of one-bit analog-to-digital converters (ADCs) is their
superior sampling rates as compared to their conventional multi-bit
counterparts. This characteristic endows one-bit signal processing frameworks
with what one may refer to as sample abundance. We show that sample abundance
plays a pivotal role in many signal recovery and optimization problems that are
formulated as (possibly non-convex) quadratic programs with linear feasibility
constraints. Of particular interest to our work are low-rank matrix recovery
and compressed sensing applications that take advantage of one-bit
quantization. We demonstrate that the sample abundance paradigm allows for the
transformation of such problems to merely linear feasibility problems by
forming large-scale overdetermined linear systems -- thus removing the need for
handling costly optimization constraints and objectives. To make the proposed
computational cost savings achievable, we offer enhanced randomized Kaczmarz
algorithms to solve these highly overdetermined feasibility problems and
provide theoretical guarantees in terms of their convergence, sample size
requirements, and overall performance. Several numerical results are presented
to illustrate the effectiveness of the proposed methodologies.Comment: arXiv admin note: text overlap with arXiv:2301.0346
Foundations of the Geometric Mechanics Udwadia-Kalaba Framework for Rigid Body Constrained Motion Analysis
Presented herein are multiple tools for constrained motion analysis extended to different dynamical frameworks. The Udwadia-Kalaba (UK) formalism for the constrained motion analysis of a point mass is a well-documented and applied methodology. Here, UK formulation is generalized to the dynamics of rigid bodies on nonlinear manifolds in the geometric mechanics framework. This approach simultaneously treats rotational and translational motion in a unified method without encountering singularites or non-uniqueness, issues that would arise were attitude parameterization sets used. The viability of this geometric mechanics UK formalism is demonstrated for the cases of fully and underconstrained systems. The nominal UK formalism requires the complete knowledge of the system dynamics. In the presence of unmodeled dynamics or uncertainties in the system, the stability of the system cannot be assessed using the nominal UK formulation. Therefore, a controller is presented that stabilizes the system under unmodeled dynamics and external perturbations. In addition, the UK formulation has been historically applied to systems with equality constraints. However, it has not been formulated for usage with inequality constraints. Here, the implementation of slack and excess variables to treat this class of constraints is presented for usage within the UK formulation for the point mass constrained motion with inequality constraints. Also contained within is an extension of pre-existing work which models the gravitational force acting on a rigid body from a nonuniform gravitational field that holds for any degree and order of spherical harmonics
Extrinsic and Intrinsic Control of Integrative Processes in Neural Systems
At the simplest dynamical level, neurons can be understood as integrators. That is, neurons accumulate excitation from afferent neurons until, eventually, a threshold is reached and they produce a spike. Here, we consider the control of integrative processes in neural circuits in two contexts. First, we consider the problem of extrinsic neurocontrol, or modulating the spiking activity of neural circuits using stimulation, as is desired in a wide range of neural engineering applications. From a control-theoretic standpoint, such a problem presents several interesting nuances, including discontinuity in the dynamics due to the spiking process, and the technological limitations associated with underactuation (i.e., many neurons controlled by the same stimulation input). We consider these factors in a canonical problem of selective spiking, wherein a particular integrative neuron is controlled to a spike, while other neurons remain below threshold. This problem is solved in an optimal control framework, wherein several new geometric phenomena associated with the aforementioned nuances are revealed. Further, in an effort to enable scaling to large populations, we develop relaxations and alternative approaches, including the use of statistical models within the control design framework. Following this treatment of extrinsic control, we turn attention to a scientifically-driven question pertaining to intrinsic control, i.e., how neurons in the brain may themselves be controlling higher-level perceptual processes. We specifically postulate that neural activity is decoded, or “read-out” in terms of a drift-diffusion process, so that spiking activity drives a latent state towards a detection/perception threshold. Under this premise, we optimize the neural spiking trajectories according to several empirical cost functions and show that the optimal responses are physiologically plausible. In this vein, we also examine the nature of \u27optimal evidence\u27 for the general class of threshold-based integrative decision problems
Sharp recovery bounds for convex demixing, with applications
Demixing refers to the challenge of identifying two structured signals given
only the sum of the two signals and prior information about their structures.
Examples include the problem of separating a signal that is sparse with respect
to one basis from a signal that is sparse with respect to a second basis, and
the problem of decomposing an observed matrix into a low-rank matrix plus a
sparse matrix. This paper describes and analyzes a framework, based on convex
optimization, for solving these demixing problems, and many others. This work
introduces a randomized signal model which ensures that the two structures are
incoherent, i.e., generically oriented. For an observation from this model,
this approach identifies a summary statistic that reflects the complexity of a
particular signal. The difficulty of separating two structured, incoherent
signals depends only on the total complexity of the two structures. Some
applications include (i) demixing two signals that are sparse in mutually
incoherent bases; (ii) decoding spread-spectrum transmissions in the presence
of impulsive errors; and (iii) removing sparse corruptions from a low-rank
matrix. In each case, the theoretical analysis of the convex demixing method
closely matches its empirical behavior.Comment: 51 pages, 13 figures, 2 tables. This version accepted to J. Found.
Comput. Mat
International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book
The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.
This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more
Estimation and Clustering in Statistical Ill-posed Linear Inverse Problems
The main focus of the dissertation is estimation and clustering in statistical ill-posed linear inverse problems. The dissertation deals with a problem of simultaneously estimating a collection of solutions of ill-posed linear inverse problems from their noisy images under an operator that does not have a bounded inverse, when the solutions are related in a certain way. The dissertation defense consists of three parts. In the first part, the collection consists of measurements of temporal functions at various spatial locations. In particular, we study the problem of estimating a three-dimensional function based on observations of its noisy Laplace convolution. In the second part, we recover classes of similar curves when the class memberships are unknown. Problems of this kind appear in many areas of application where clustering is carried out at the pre-processing step and then the inverse problem is solved for each of the cluster averages separately. As a result, the errors of the procedures are usually examined for the estimation step only. In both parts, we construct the estimators, study their minimax optimality and evaluate their performance via a limited simulation study. In the third part, we propose a new computational platform to better understand the patterns of R-fMRI by taking into account the challenge of inevitable signal fluctuations and interpret the success of dynamic functional connectivity approaches. Towards this, we revisit an auto-regressive and vector auto-regressive signal modeling approach for estimating temporal changes of the signal in brain regions. We then generate inverse covariance matrices from the generated windows and use a non-parametric statistical approach to select significant features. Finally, we use Lasso to perform classification of the data. The effectiveness of the proposed method is evidenced in the classification of R-fMRI scan
Empirical eigenfunctions: application in unsteady aerodynamics
Mención Internacional en el título de doctorThe main aim of modal decompositions is to obtain a set of functions which can describe
in a compact way the variability contained in a set of observables/data. While this
can be easily obtained by means of the eigenfunctions of the operator from which the
observables depends, the empirical eigenfunctions allow to obtain a similar result from
a set of data, without the knowledge of the problem operator. In Fluid Mechanics and
related sciences one of the most prominent techniques to obtain empirical eigenfunctions
is referred to as Proper Orthogonal Decomposition (POD).
This thesis contains applications of the empirical eigenfunctions to (Experimental)
Aerodynamics data. The mathematical framework of the POD is introduced following
the bi-orthogonal approach by Aubry (1991). The mathematical derivation of the
POD is given, wherever possible, in its most general formulation, without bounding
it to the decomposition of a specific quantity. This choice of the author depends
on the variety of POD applications which are included in this dissertation, ranging
from signal processing problems to applications more strictly related with flow physics.
The mathematical framework includes also one of the POD extensions, the Extended
POD (EPOD), which allows to extract modes linearly correlated to the empirical
eigenfunctions of a second quantity.
The first two applications of the empirical eigenfunctions are strictly connected
with the signal treatment in experimental techniques for Fluid Mechanics. In Chapter
3, the empirical eigenfunctions are identified as an optimal basis in which perform a
"low-pass" spectral filter of experimental fluid data, such as velocity fields measured
with Particle Image Velocimetry (PIV). This filtering is extremely beneficial to reduce
the random errors contained in the PIV fields and obtain a more accurate estimate
of derivative quantities (such as, for instance, vorticity), which are more affected by
random errors. In Chapter 4 the POD is exploited for the pre-treatment of a sequence
of PIV images. The aim is to remove background and reflections, which are sources
of uncertainty in PIV measurements. In this case a "high-pass" spectral filtering is
applied to the PIV image ensemble in order to remove the highly-coherent part of the
signal corresponding to the background.
In the third and fourth applications, the POD is applied to recover the underlying
dynamics of a flow. More specifically, in Chapter 5 the POD is applied to the complex
wake of a pair of cylinders in tandem arrangement with the additional perturbation
of the wall proximity. Through this technique it is possible to track the changes in
the oscillatory behaviour of the wake instabilities ascribed to different geometrical
configurations of the cylinders. In Chapter 6 the POD and the EPOD are applied
respectively to the flow fields around an airfoil in plunging and pitching motion and
to the unsteady aerodynamic forces acting on the airfoil. The decomposition allows
to extract a reduced set of modes of the flow field which are related to the force
generation mechanism. These modes correspond to well-recognizable phenomena of
the flow which can be identified for diverse airfoil kinematics. This flow-field driven
force decomposition is analysed on the light of existing force models, enabling their
reinterpretation and driving towards possible corrections.
The final application is devoted to overcome the low temporal resolution of typical
flow field measurements, such as PIV, by proposing a robust estimation of turbulent
flows dynamics. The method employs a modified version of the EPOD to identify the
correlation between a non-time-resolved field measurement and a time-resolved point
measurement. The estimation of the time-resolved flow fields is obtained exploiting
the correlation of the flow fields with the temporal information contained in the point
measurements.El objetivo principal de las descomposiciones modales es obtener un conjunto de
funciones que sean capaces de describir de una manera compacta la variabilidad
contenida en un conjunto de observables/datos. Si bien este objetivo puede ser
fácilmente realizado mediante el uso de las autofunciones del operador del cual los
observables dependen, las autofunciones empíricas permiten obtener un resultado
similar partiendo de un conjunto de datos sin la necesidad de conocer el operador del
problema. En Mecánica de Fluidos y en ciencias relacionadas con esta disciplina, una
de las técnicas más relevantes para obtener autofunciones empíricas es la conocida
como Descomposición Modal Ortogonal (Proper Orthogonal Decomposition, POD).
Esta tesis contiene diversas aplicaciones de las autofunciones empíricas en datos de
Aerodinámica (Experimental). La base matemática de la POD es introducida siguiendo
la aproximación biortogonal realizada por Aubry (1991). La formulación matemática
de la POD es expresada siempre que es posible en el marco más general posible,
sin condicionarla a la descomposición de una variable en concreto. La elección del
autor dependerá de las diferentes aplicaciones de la POD, todas ellas descritas en
la presente tesis, las cuales abarcan desde problemas de procesado de señales hasta
aplicaciones más estrictamente relacionadas con el análisis de la física del flujo. La
formulación matemática incluye también uno de las extensiones de la POD, la POD
Extendida (EPOD), la cual permite extraer modos linealmente correlacionados con las
autofunciones empíricas de una segunda variable. Las dos primeras aplicaciones de las
autofunciones empíricas están estrictamente relacionadas con el tratamiento de señales
en técnicas experimentales de Mecánica de Fluidos. En el Capítulo 3, las autofunciones
empíricas son identificadas como una base optima, la cual se puede utilizar para realizar
un filtro pasa bajos espectral para datos experimentales de flujos, tales como campos
de velocidad obtenidos mediante la técnica de Velocimetría por Imágenes de Partículas,
(Particle Image Velocimetry, PIV). Este tipo de filtro es muy beneficioso para reducir
los errores de carácter aleatorio contenidos en los campos de PIV y por tanto obtener
una estimación más precisa en las cantidades que precisan del uso de derivadas (por
ejemplo, la vorticidad), ya que están más afectadas por este tipo de errores. En el Capítulo 4, la POD es utilizada para el pretratamiento de una secuencia de imágenes
de PIV. El objetivo es reducir el fondo de la imagen y las reflexiones, ambas fuentes
de incertidumbre en las medidas de PIV. En este caso, un filtro pasa altos espectral
es aplicado al conjunto de imágenes de PIV para poder quitar la parte mayormente
correlacionada de la señal, la cual corresponde con el fondo de la imagen. En la tercera
y cuarta aplicación de la POD, está técnica es utilizada para reconstruir las dinámicas
fundamentales de un flujo. Concretamente, en el Capítulo 5 la POD es utilizada para
analizar la estela compleja que se produce en una pareja de cilindros en tándem con la
perturbación adicional de una pared próxima a ellos. A través de esta técnica, es posible
poder estudiar los cambios en el comportamiento oscilatorio de las inestabilidades de
la estela, las cuales están relacionadas con las diferentes configuraciones geométricas
de los cilindros. En el capítulo 6, la POD y la EPOD son aplicadas respectivamente
a campos fluidos y fuerzas aerodinámicas producidos por un perfil aerodinámico en
movimiento (de rotación y desplazamiento vertical) no estacionario. La técnica de
descomposición permite extraer un conjunto reducido de modos del campo fluido que
están relacionados con el mecanismo que genera las fuerzas aerodinámicas. Estos modos
corresponden con fenómenos característicos del flujo que pueden ser identificados para
diferentes cinemáticas de perfiles aerodinámicos. Estas dinámicas del flujo que están
conectadas con las fuerzas aerodinámicas son analizadas teniendo en cuenta los modelos
ya existentes en la literatura que describen las fuerzas aerodinámicas, permitiendo su
reinterpretación e incluso pudiendo añadir posibles correcciones. La última aplicación
propuesta está destinada a subsanar la baja resolución temporal típica de las medidas
de campo fluido, como en aquellas realizadas utilizando PIV, mediante una estimación
robusta de las dinámicas del flujo turbulento. El método propuesto emplea una versión
modificada de la EPOD para identificar para correlación entre un campo fluido medido
que no está resuelto en el tiempo y una medida puntual que sí que está resulta en el
tiempo. La estimación del campo fluido resuelto en el tiempo es obtenida mediante la
correlación de los campos de flujo con la información temporal contenida en la medida
puntual.This work has been partially supported by the Grant TRA2013-41103-P of the
Spanish Ministry of Economy and Competitiveness, which includes FEDER funding,
and by the Grant DPI2016-79401-R, funded by the Spanish State Research Agency
(SRA) and European Regional Development Fund (ERDF).Programa Oficial de Doctorado en Mecánica de FluidosPresidente: Bharathram Ganapathisubramani.- Secretario: Francisco Javier Rodríguez Rodríguez.- Vocal: Francisco J. Huera-Huart
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