Aspects of parameter identification in semilinear reaction-diffusion systems

This thesis provides an approach for parameter identification in general semilinear parabolic partial differential equations. The problem of of parameter identification in these equations is investigated from two different angles. On one hand, Tikhonov regularization is proposed to deal with possible non continuous dependence of the parameters on the data and on the other hand the uniqueness of a solution of the parameter identification problem is discussed. Additionally, numerical results are shown that support the theoretical findings

Suurikokoiset päällekkäiset kelat, uusi lähestymistapa monikanavaisen transkraniaalisen magneettistimulaatiolaitteen rakentamiseen

Transcranial magnetic stimulation (TMS) allows for studying the functionality of the brain. Present TMS devices have one or two separate stimulation coils. More stimulation coils would allow new types of stimulation sequences, and thus they could be used to reveal more about brain functionality. However, due to the dimensions of the existing TMS coils, having multiple separate coils is a very limited approach. Rather, the coils should be combined into a single multichannel (mTMS) device. The purpose of this Thesis is to make mTMS more feasible. In order to realize this purpose, a new coil design paradigm is introduced which employs large thin overlapping coils. This paradigm requires a new coil design method and a new coil-former design method, which are developed and tested in this Thesis. This Thesis solves two problems that appear with existing mTMS designs and is a significant step towards successful mTMS.Transkraniaalinen magneettistimulaatio (TMS) mahdollistaa aivotoiminnan tutkimisen. Nykyisissä TMS-laitteissa on yleensä yksi tai joissain tapauksissa kaksi erillistä stimulaatiokelaa. Suurempi kelamäärä mahdollistaisi uudentyyppisiä stimulaatiosekvenssejä, jotka mahdollistaisivat monipuolisemman aivotoiminnan tutkimisen. Koska TMS-kelat ovat verrattain suurikokoisia, ei tätä tavoitetta kuitenkaan pystytä saavuttamaan yhdistämällä monta erillistä TMS-kelaa. Sen sijaan tarvittaisiin yksi monikanavainen (mTMS) laite, jossa eri kanavien kelat on yhdistetty yhdeksi suuremmaksi kokonaisuudeksi. Tämän diplomityön tarkoitus on edistää osaltaan mTMS-laitteen suunnittelua. Tätä varten esitellään uusi mTMS-rakenne, jossa mTMS-kela koostuu suurikokoisista ohuista päällekkäisistä keloista. Tässä diplomityössä kehitetään ja testataan yksittäisten kelojen suunnittelumenetelmä tämäntyyppistä mTMS-kelaa varten. Diplomityössä esiteltävä rakenne ratkaisee kaksi nykyisissä mTMS-kelarakennesuunnitelmissa esiintyvää ongelmaa

Audio source separation for music in low-latency and high-latency scenarios

Aquesta tesi proposa mètodes per tractar les limitacions de les tècniques existents de separació de fonts musicals en condicions de baixa i alta latència. En primer lloc, ens centrem en els mètodes amb un baix cost computacional i baixa latència. Proposem l'ús de la regularització de Tikhonov com a mètode de descomposició de l'espectre en el context de baixa latència. El comparem amb les tècniques existents en tasques d'estimació i seguiment dels tons, que són passos crucials en molts mètodes de separació. A continuació utilitzem i avaluem el mètode de descomposició de l'espectre en tasques de separació de veu cantada, baix i percussió. En segon lloc, proposem diversos mètodes d'alta latència que milloren la separació de la veu cantada, gràcies al modelatge de components específics, com la respiració i les consonants. Finalment, explorem l'ús de correlacions temporals i anotacions manuals per millorar la separació dels instruments de percussió i dels senyals musicals polifònics complexes.Esta tesis propone métodos para tratar las limitaciones de las técnicas existentes de separación de fuentes musicales en condiciones de baja y alta latencia. En primer lugar, nos centramos en los métodos con un bajo coste computacional y baja latencia. Proponemos el uso de la regularización de Tikhonov como método de descomposición del espectro en el contexto de baja latencia. Lo comparamos con las técnicas existentes en tareas de estimación y seguimiento de los tonos, que son pasos cruciales en muchos métodos de separación. A continuación utilizamos y evaluamos el método de descomposición del espectro en tareas de separación de voz cantada, bajo y percusión. En segundo lugar, proponemos varios métodos de alta latencia que mejoran la separación de la voz cantada, gracias al modelado de componentes que a menudo no se toman en cuenta, como la respiración y las consonantes. Finalmente, exploramos el uso de correlaciones temporales y anotaciones manuales para mejorar la separación de los instrumentos de percusión y señales musicales polifónicas complejas.This thesis proposes specific methods to address the limitations of current music source separation methods in low-latency and high-latency scenarios. First, we focus on methods with low computational cost and low latency. We propose the use of Tikhonov regularization as a method for spectrum decomposition in the low-latency context. We compare it to existing techniques in pitch estimation and tracking tasks, crucial steps in many separation methods. We then use the proposed spectrum decomposition method in low-latency separation tasks targeting singing voice, bass and drums. Second, we propose several high-latency methods that improve the separation of singing voice by modeling components that are often not accounted for, such as breathiness and consonants. Finally, we explore using temporal correlations and human annotations to enhance the separation of drums and complex polyphonic music signals

PDE-betinga optimering : prekondisjonerarar og metodar for diffuse domene

This thesis is mainly concerned with the efficient numerical solution of optimization problems subject to linear PDE-constraints, with particular focus on robust preconditioners and diffuse domain methods. Associated with such constrained optimization problems are the famous first-order KarushKuhn-Tucker (KKT) conditions. For certain minimization problems, the functions satisfying the KKT conditions are also optimal solutions of the original optimization problem, implying that we can solve the KKT system to obtain the optimum; the so-called “all-at-once” approach. We propose and analyze preconditioners for the different KKT systems we derive in this thesis.Denne avhandlinga ser i hovudsak på effektive numeriske løysingar av PDE-betinga optimeringsproblem, med eit særskilt fokus på robuste prekondisjonerar og “diffuse domain”-metodar. Assosiert med slike optimeringsproblem er dei velkjende Karush-Kuhn-Tucker (KKT)-føresetnadane. For mange betinga optimeringsproblem, vil funksjonar som tilfredstillar KKT-vilkåra samstundes vere ei optimal løysing på det opprinnelege optimeringsproblemet. Dette impliserar at vi kan løyse KKT-likningane for å finne optimum. Vi konstruerar og analyserar prekondisjonerar for dei forskjellige KKT-systema vi utleiar i denne avhandlinga

Adaptive Wavelet Methods for Inverse Problems: Acceleration Strategies, Adaptive Rothe Method and Generalized Tensor Wavelets

In general, inverse problems can be described as the task of inferring conclusions about the cause u from given observations y of its effect. This can be described as the inversion of an operator equation K(u) = y, which is assumed to be ill-posed or ill-conditioned. To arrive at a meaningful solution in this setting, regularization schemes need to be applied. One of the most important regularization methods is the so called Tikhonov regularization. As an approximation to the unknown truth u it is possible to consider the minimizer v of the sum of the data error K(v)-y (in a certain norm) and a weighted penalty term F(v). The development of efficient schemes for the computation of the minimizers is a field of ongoing research and a central Task in this thesis. Most computation schemes for v are based on some generalized gradient descent approach. For problems with weighted lp-norm penalty terms this typically leads to iterated soft shrinkage methods. Without additional assumptions the convergence of these iterations is only guaranteed for subsequences, and even then only to stationary points. In general, stationary points of the minimization problem do not have any regularization properties. Also, the basic iterated soft shrinkage algorithm is known to converge very poorly in practice. This is critical as each iteration step includes the application of the nonlinear operator K and the adjoint of its derivative. This in itself may already be numerically demanding. This thesis is concerned with the development of strategies for the fast computation of the solution of inverse problems with provable convergence rates. In particular, the application and generalization of efficient numerical schemes for the treatment of the arising nonlinear operator equations is considered. The first result of this thesis is a general acceleration strategy for the iterated soft thresholding iteration to compute the solution of the inverse problem. It is based on a decreasing strategy for the weights of the penalty term. The new method converges with linear rate to a global minimizer. A very important class of inverse problems are parameter identification problems for partial differential equations. As a prototype for this class of problems the identification of parameters in a specific parabolic partial differential equation is investigated. The arising operators are analyzed, the applicability of Tikhonov Regularization is proven and the parameters in a simplified test equation are reconstructed. The parabolic differential equations are solved by means of the so called horizontal method of lines, also known as Rothes method. Here the parabolic problem is interpreted as an abstract Cauchy problem. It is discretized in time by means of an implicit scheme. This is combined with a discretization of the resulting system of spatial problems. In this thesis the application of adaptive discretization schemes to solve the spatial subproblems is investigated. Such methods realize highly nonuniform discretizations. Therefore, they tend to require much less degrees of freedom than classical discretization schemes. To ensure the convergence of the resulting inexact Rothe method, a rigorous convergence proof is given. In particular, the application of implementable asymptotically optimal adaptive methods, based on wavelet bases, is considered. An upper bound for the degrees of freedom of the overall scheme that are needed to adaptively approximate the solution up to a prescribed tolerance is derived. As an important case study, the complexity of the approximate solution of the heat equation is investigated. To this end a regularity result for the spatial equations that arise in the Rothe method is proven. The rate of convergence of asymptotically optimal adaptive methods deteriorates with the spatial dimension of the problem. This is often called the curse of dimensionality. One way to avoid this problem is to consider tensor wavelet discretizations. Such discretizations lead to dimension independent convergence rates. However, the classical tensor wavelet construction is limited to domains with simple product geometry. Therefor, in this thesis, a generalized tensor wavelet basis is constructed. It spans a range of Sobolev spaces over a domain with a fairly general geometry. The construction is based on the application of extension operators to appropriate local bases on subdomains that form a non-overlapping domain decomposition. The best m-term approximation of functions with the new generalized tensor product basis converges with a rate that is independent of the spatial dimension of the domain. For two- and three-dimensional polytopes it is shown that the solution of Poisson type problems satisfies the required regularity condition. Numerical tests show that the dimension independent rate is indeed realized in practice

Rendering volumetric haptic shapes in mid-air using ultrasound

We present a method for creating three-dimensional haptic shapes in mid-air using focused ultrasound. This approach applies the principles of acoustic radiation force, whereby the non-linear effects of sound produce forces on the skin which are strong enough to generate tactile sensations. This mid-air haptic feedback eliminates the need for any attachment of actuators or contact with physical devices. The user perceives a discernible haptic shape when the corresponding acoustic interference pattern is generated above a precisely controlled two-dimensional phased array of ultrasound transducers. In this paper, we outline our algorithm for controlling the volumetric distribution of the acoustic radiation force field in the form of a three-dimensional shape. We demonstrate how we create this acoustic radiation force field and how we interact with it. We then describe our implementation of the system and provide evidence from both visual and technical evaluations of its ability to render different shapes. We conclude with a subjective user evaluation to examine users’ performance for different shapes

Goal-oriented inference : theoretical foundations and application to carbon capture and storage

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2013.This electronic version was submitted and approved by the author's academic department as part of an electronic thesis pilot project. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from department-submitted PDF version of thesis.Includes bibliographical references (p. 127-132).Many important engineering problems require computation of prediction output quantities of interest that depend on unknown distributed parameters of the governing partial differential equations. Examples include prediction of concentration levels in critical areas for contamination events in urban areas and prediction of trapped volume of supercritical carbon dioxide in carbon capture and storage. In both cases the unknown parameter is a distributed quantity that is to be inferred from indirect and sparse data in order to make accurate predictions of the quantities of interest. Traditionally parameter inference involves regularization in deterministic formulations or specification of a prior probability density in Bayesian statistical formulations to resolve the ill-posedness manifested in the many possible parameters giving rise to the same observed data. Critically, the final prediction requirements are not considered in the inference process. Goal-oriented inference, on the other hand, utilizes the prediction requirements to drive the inference process. Since prediction quantities of interest are often very low-dimensional, the same ill-posedness that stymies the inference process can be exploited when inference of the parameter is undertaken solely to obtain predictions. Many parameters give rise to the same predictions; as a result, resolving the parameter is not required in order to accurately make predictions. In goal-oriented inference, we exploit this fact to obtain fast and accurate predictions from experimental data by sacrificing accuracy in parameter estimation. When the governing models for experimental data and prediction quantities of interest depend linearly on the parameter, a linear algebraic analysis reveals a dimensionally-optimal parameter subspace within which inference proceeds. Parameter estimates are inaccurate but the resulting predictions are identical to those achieved by first performing inference in the full high-dimensional parameter space and then computing predictions. The analysis required to identify the parameter subspace reveals inefficiency in experiment and sources of uncertainty in predictions, which can also be utilized in experimental design. Linear goal-oriented inference is demonstrated on a model problem in contaminant source inversion and prediction. In the nonlinear setting, we focus on the Bayesian statistical inverse problem formulation where the target of our goal-oriented inference is the posterior predictive probability density function representing the relative likelihood of predictions given the observed experimental data. In many nonlinear settings, particularly those involving nonlinear partial differential equations, distributed parameter estimation remains an unsolved problem. We circumvent estimation of the parameter by establishing a statistical model for the joint density of experimental data and predictions using either a Gaussian mixture model or kernel density estimate derived from simulated experimental data and simulated predictions based on parameter samples from the prior distribution. When experiments are conducted and data are observed, the statistical model is conditioned on the observed data, and the posterior predictive probability density is obtained. Nonlinear goal-oriented inference is applied to a realistic application in carbon capture and storage.by Chad Lieberman.Ph.D

Solving Optimal Power Flow for Distribution Networks with State Estimation Feedback

Conventional optimal power flow (OPF) solvers assume full observability of the involved system states. However, in practice, there is a lack of reliable system monitoring devices in the distribution networks. To close the gap between the theoretic algorithm design and practical implementation, this work proposes to solve the OPF problems based on the state estimation (SE) feedback for the distribution networks where only a part of the involved system states are physically measured. The SE feedback increases the observability of the under-measured system and provides more accurate system states monitoring when the measurements are noisy. We analytically investigate the convergence of the proposed algorithm. The numerical results demonstrate that the proposed approach is more robust to large pseudo measurement variability and inherent sensor noise in comparison to the other frameworks without SE feedback