Some Inverse Problems in Analysis and Geometry

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

A domain decomposition based algorithm for non-linear 2D inverse heat conduction problems

Inverse heat conduction problems (IHCPs) appear in many important scientific and technological fields. Hence analysis, design, implementation and testing of inverse algorithms are also of great scientific and technological interest. The numerical simulation of 2-D and –D inverse (or even direct) problems involves a considerable amount of computation. Therefore, the investigation and exploitation of parallel properties of such algorithms are equally becoming very important. Domain decomposition (DD) methods are widely used to solve large scale engineering problems and to exploit their inherent ability for the solution of such problems.

The Inverse Jacobian Control Method Applied to a Four Degree of Freedom Confined Space Robot

Thesis (Master's)--University of Washington, 2016-06

Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

This thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems

We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.

A comprehensive study of multiplicative attribute graph model

Graphs are powerful tools to describe social, technological and biological networks, with nodes representing agents (people, websites, gene, etc.) and edges (or links) representing relations (or interactions) between agents. Examples of real-world networks include social networks, the World Wide Web, collaboration networks, protein networks, etc. Researchers often model these networks as random graphs. In this dissertation, we study a recently introduced social network model, named the Multiplicative Attribute Graph model (MAG), which takes into account the randomness of nodal attributes in the process of link formation (i.e., the probability of a link existing between two nodes depends on their attributes). Kim and Lesckovec, who defined the model, have claimed that this model exhibit some of the properties a real world social network is expected to have. Focusing on a homogeneous version of this model, we investigate the existence of zero-one laws for graph properties, e.g., the absence of isolated nodes, graph connectivity and the emergence of triangles. We obtain conditions on the parameters of the model, so that these properties occur with high or vanishingly probability as the number of nodes becomes unboundedly large. In that regime, we also investigate the property of triadic closure and the nodal degree distribution.

Utilisation d’un modèle de co-culture cellulaire pour l’évaluation in vitro du transport inverse du cholestérol

De nouveaux modèles cellulaires in vitro par transfert de milieu et par coculture ont été mis au point afin d’évaluer la capacité des HDL à éliminer l’excès de cholestérol des tissus périphériques et de le transporter vers le foie afin d’être excrété par le foie, un processus nommé le transport inverse du cholestérol (TIC). Le système cellulaire par transfert in vitro où des macrophages J774 sont gorgés de LDL acétylées et marqués au 3H-cholestérol a été préalablement établi afin de mesurer par scintillation l’efflux de cholestérol marqué vers le milieu de culture contenant des accepteurs de cholestérol. Ce milieu conditionné est transféré sur des cellules HepG2 afin d’étudier l’influx du cholestérol marqué. Ce dernier nous permet d’observer un transport de cholestérol de 25 % hors des J774 et un transport de 39 000 cpm dans les HepG2 en utilisant un milieu contenant 2 % de sérums humains mis en commun. Une stimulation des cellules J774 par l’AMPc augmente l’efflux et l’influx d’environ 45 %. Des tests de preuve de concept ont été effectués sur le système cellulaire par co-culture qui utilise des chambres de Boyden où les J774 sont localisées au fond d’un puits et les HepG2 dans un insert, et où le milieu est partagé entre les deux types cellulaires. On a déterminé qu’une confluence densité de 60 000 cellules/cm2 sur un insert constitué d’une membrane de polyester avec des pores de 3,0 μm, sans autre revêtement, permet d’observer un influx spécifique au sérum d’environ 6 000 cpm associés aux cellules HepG2, où 50 % des comptes radioactifs sont dans les cellules et l’autre moitié présente à la surface cellulaire.

Inverse identification of the bending stiffness of a braided polyethylene twine subject to large deformation: application to the identification of the mesh opening rigidity of fishing nets.

The evaluation of the mesh opening stiffness of fishing nets is an important issue in assessing the selectivity of trawls. It appeared that a larger bending rigidity of twines decreases the mesh opening and could reduce the escapement of fish. Nevertheless, netting structure is complex. A netting is made up of braided twines made of polyethylene or polyamide. These twines are tied with non-symmetrical knots. Thus, these assemblies develop contact-friction interactions. Moreover, the netting can be subject to large deformation. In this study, we investigate the responses of netting samples to different types of solicitations. Samples are loaded and unloaded with creep and relaxation stages, with different boundary conditions. Then, two models have been developed: an analytical model and a finite element model. The last one was used to assess, with an inverse identification algorithm, the bending stiffness of twines. In this paper, experimental results and a model for netting structures made up of braided twines are presented. During dry forming of a composite, for example, the matrix is not present or not active, and relative sliding can occur between constitutive fibres. So an accurate modelling of the mechanical behaviour of fibrous material is necessary. This study offers experimental data which could permit to improve current models of contact-friction interactions [4], to validate models for large deformation analysis of fibrous materials [1] on a new experimental case, then to improve the evaluation of the mesh opening stiffness of a fishing net

Inverse kinematics for upper limb compound movement estimation in exoskeleton-assisted rehabilitation

Robot-Assisted Rehabilitation (RAR) is relevant for treating patients affected by nervous system injuries (e.g., stroke and spinal cord injury) -- The accurate estimation of the joint angles of the patient limbs in RAR is critical to assess the patient improvement -- The economical prevalent method to estimate the patient posture in Exoskeleton-based RAR is to approximate the limb joint angles with the ones of the Exoskeleton -- This approximation is rough since their kinematic structures differ -- Motion capture systems (MOCAPs) can improve the estimations, at the expenses of a considerable overload of the therapy setup -- Alternatively, the Extended Inverse Kinematics Posture Estimation (EIKPE) computational method models the limb and Exoskeleton as differing parallel kinematic chains -- EIKPE has been tested with single DOFmovements of the wrist and elbow joints -- This paper presents the assessment of EIKPEwith elbow-shoulder compoundmovements (i.e., object prehension) -- Ground-truth for estimation assessment is obtained from an optical MOCAP (not intended for the treatment stage) -- The assessment shows EIKPE rendering a good numerical approximation of the actual posture during the compoundmovement execution, especially for the shoulder joint angles -- This work opens the horizon for clinical studies with patient groups, Exoskeleton models, and movements types --

Geological realism in hydrogeological and geophysical inverse modeling: A review

Scientific curiosity, exploration of georesources and environmental concerns are pushing the geoscientific research community toward subsurface investigations of ever-increasing complexity. This review explores various approaches to formulate and solve inverse problems in ways that effectively integrate geological concepts with geophysical and hydrogeological data. Modern geostatistical simulation algorithms can produce multiple subsurface realizations that are in agreement with conceptual geological models and statistical rock physics can be used to map these realizations into physical properties that are sensed by the geophysical or hydrogeological data. The inverse problem consists of finding one or an ensemble of such subsurface realizations that are in agreement with the data. The most general inversion frameworks are presently often computationally intractable when applied to large-scale problems and it is necessary to better understand the implications of simplifying (1) the conceptual geological model (e.g., using model compression); (2) the physical forward problem (e.g., using proxy models); and (3) the algorithm used to solve the inverse problem (e.g., Markov chain Monte Carlo or local optimization methods) to reach practical and robust solutions given today's computer resources and knowledge. We also highlight the need to not only use geophysical and hydrogeological data for parameter estimation purposes, but also to use them to falsify or corroborate alternative geological scenarios.

Effect of hamstring injuries on kicking in soccer using inverse dynamics

Las lesiones musculares del muslo tienen una gran incidencia en el fútbol. El objetivo del estudio ha sido desarrollar un nuevo procedimiento para evaluar el efecto de las lesiones de isquiotibiales en los golpeos con el pie s en el fútbol utilizando los principios de la dinámica inversa. El trabajo se ha centrado en la evaluación de la diferencia entre sujetos que habían sufrido la lesión en los últimos 5 años y los que no. Se analizaron 17 jugadores de fútbol profesionales realizando cinco tiros con el empeine y cinco con el interior del pie. Los movimientos se registraron mediante una plataforma de fuerza y un sistema de captura de movimiento Vicon funcionando a 500Hz. Los participantes también tomaron parte en una prueba de isocinético en la que se midió el torque isocinético en 60 º/s y 120 º/s. Se observaron diferencias significativas en los parámetros cinemáticos y cinéticos entre los dos grupos (lesionados y no lesionados) en la fase posterior del golpeo y en el instante de máxima flexión de cadera. No se encontraron diferencias significativas entre los dos grupos en la prueba isocinética tradicional. Estos resultados indican que el procedimiento empleado probablemente podría ser muy útil en la evaluación del efecto de las lesiones de isquiotibiales en el fútbol.