A comparison of different choices for the regularization parameter in inverse electrocardiography models

Calculating the potentials on the heart’s epicardial surface from the body surface potentials constitutes one form of inverse problems in electrocardiography (ECG). Since these problems are ill-posed, one approach is to use zero-order Tikhonov regularization, where the squared norms of both the residual and the solution are minimized, with a relative weight determined by the regularization parameter. In this paper, we used three different methods to choose the regularization parameter in the inverse solutions of ECG. The three methods include the L-curve, the generalized cross validation (GCV) and the discrepancy principle (DP). Among them, the GCV method has received less attention in solutions to ECG inverse problems than the other methods. Since the DP approach needs knowledge of norm of noises, we used a model function to estimate the noise. The performance of various methods was compared using a concentric sphere model and a real geometry heart-torso model with a distribution of current dipoles placed inside the heart model as the source. Gaussian measurement noises were added to the body surface potentials. The results show that the three methods all produce good inverse solutions with little noise; but, as the noise increases, the DP approach produces better results than the L-curve and GCV methods, particularly in the real geometry model. Both the GCV and L-curve methods perform well in low to medium noise situations.

Generalized mean field approximation for parallel dynamics of the Ising model

The dynamics of the non-equilibrium Ising model with parallel updates is investigated using a generalized mean field approximation that incorporates multiple two-site correlations at any two time steps, which can be obtained recursively. The proposed method shows significant improvement in predicting local system properties compared to other mean field approximation techniques, particularly in systems with symmetric interactions. Results are also evaluated against those obtained from Monte Carlo simulations. The method is also employed to obtain parameter values for the kinetic inverse Ising modeling problem, where couplings and local field values of a fully connected spin system are inferred from data. © 2014 IOP Publishing Ltd and SISSA Medialab srl.

Generalized Fractional Evolution Equation

2000 Mathematics Subject Classification: Primary 46F25, 26A33; Secondary: 46G20

Direct and Converse Theorems for Generalized Bernstein-Type Operators

2000 Mathematics Subject Classification: 41A25, 41A27, 41A36.

The Direct and Inverse Spectral Problems for some Banded Matrices

2000 Mathematics Subject Classification: 15A29.

Design and implementation of generalized topologies of time-interleaved variable bandpass Σ−Δ modulators

In this thesis, novel analog-to-digital and digital-to-analog generalized time-interleaved variable bandpass sigma-delta modulators are designed, analysed, evaluated and implemented that are suitable for high performance data conversion for a broad-spectrum of applications. These generalized time-interleaved variable bandpass sigma-delta modulators can perform noise-shaping for any centre frequency from DC to Nyquist. The proposed topologies are well-suited for Butterworth, Chebyshev, inverse-Chebyshev and elliptical filters, where designers have the flexibility of specifying the centre frequency, bandwidth as well as the passband and stopband attenuation parameters. The application of the time-interleaving approach, in combination with these bandpass loop-filters, not only overcomes the limitations that are associated with conventional and mid-band resonator-based bandpass sigma-delta modulators, but also offers an elegant means to increase the conversion bandwidth, thereby relaxing the need to use faster or higher-order sigma-delta modulators. A step-by-step design technique has been developed for the design of time-interleaved variable bandpass sigma-delta modulators. Using this technique, an assortment of lower- and higher-order single- and multi-path generalized A/D variable bandpass sigma-delta modulators were designed, evaluated and compared in terms of their signal-to-noise ratios, hardware complexity, stability, tonality and sensitivity for ideal and non-ideal topologies. Extensive behavioural-level simulations verified that one of the proposed topologies not only used fewer coefficients but also exhibited greater robustness to non-idealties. Furthermore, second-, fourth- and sixth-order single- and multi-path digital variable bandpass digital sigma-delta modulators are designed using this technique. The mathematical modelling and evaluation of tones caused by the finite wordlengths of these digital multi-path sigmadelta modulators, when excited by sinusoidal input signals, are also derived from first principles and verified using simulation and experimental results. The fourth-order digital variable-band sigma-delta modulator topologies are implemented in VHDL and synthesized on Xilinx® SpartanTM-3 Development Kit using fixed-point arithmetic. Circuit outputs were taken via RS232 connection provided on the FPGA board and evaluated using MATLAB routines developed by the author. These routines included the decimation process as well. The experiments undertaken by the author further validated the design methodology presented in the work. In addition, a novel tunable and reconfigurable second-order variable bandpass sigma-delta modulator has been designed and evaluated at the behavioural-level. This topology offers a flexible set of choices for designers and can operate either in single- or dual-mode enabling multi-band implementations on a single digital variable bandpass sigma-delta modulator. This work is also supported by a novel user-friendly design and evaluation tool that has been developed in MATLAB/Simulink that can speed-up the design, evaluation and comparison of analog and digital single-stage and time-interleaved variable bandpass sigma-delta modulators. This tool enables the user to specify the conversion type, topology, loop-filter type, path number and oversampling ratio.

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.

A Generalized Methodology to Characterize Composite Materials for Pyrolysis Models

The predictive capabilities of computational fire models have improved in recent years such that models have become an integral part of many research efforts. Models improve the understanding of the fire risk of materials and may decrease the number of expensive experiments required to assess the fire hazard of a specific material or designed space. A critical component of a predictive fire model is the pyrolysis sub-model that provides a mathematical representation of the rate of gaseous fuel production from condensed phase fuels given a heat flux incident to the material surface. The modern, comprehensive pyrolysis sub-models that are common today require the definition of many model parameters to accurately represent the physical description of materials that are ubiquitous in the built environment. Coupled with the increase in the number of parameters required to accurately represent the pyrolysis of materials is the increasing prevalence in the built environment of engineered composite materials that have never been measured or modeled. The motivation behind this project is to develop a systematic, generalized methodology to determine the requisite parameters to generate pyrolysis models with predictive capabilities for layered composite materials that are common in industrial and commercial applications. This methodology has been applied to four common composites in this work that exhibit a range of material structures and component materials. The methodology utilizes a multi-scale experimental approach in which each test is designed to isolate and determine a specific subset of the parameters required to define a material in the model. Data collected in simultaneous thermogravimetry and differential scanning calorimetry experiments were analyzed to determine the reaction kinetics, thermodynamic properties, and energetics of decomposition for each component of the composite. Data collected in microscale combustion calorimetry experiments were analyzed to determine the heats of complete combustion of the volatiles produced in each reaction. Inverse analyses were conducted on sample temperature data collected in bench-scale tests to determine the thermal transport parameters of each component through degradation. Simulations of quasi-one-dimensional bench-scale gasification tests generated from the resultant models using the ThermaKin modeling environment were compared to experimental data to independently validate the models.

An inverse time admittance relay for fault detection in distribution networks containing DGs

Isolation of a faulted segment, from either side of a fault, in a radial feeder that has several converter interfaced DGs is a challenging task when current sensing protective devices are employed. The protective device, even if it senses a downstream fault, may not operate if fault current level is low due to the current limiting operation of converters. In this paper, a new inverse type relay is introduced based on line admittance measurement to protect a distribution network, which has several converter interfaced DGs. The basic operation of this relay, its grading and reach settings are explained. Moreover a method is proposed to compensate the fault resistance such that the relay operation under this condition is reliable. Then designed relay performances are evaluated in a radial distribution network. The results are validated through PSCAD/EMTDC simulation and MATLAB calculations.

Load during prosthetic gait : is direct measurement better than inverse dynamics?

The knee forces and moments estimated by inverse dynamics and directly measured by a multiaxial transducer were compared during the gait of a transfemoral amputee. The estimated and directly measured forces and moments were relatively close. However, 3D inverse dynamics estimated only partially the forces and moments associated with the deformation of the prosthetic foot and locking of knee mechanism.

Loading applied on prosthetic knee of transfemoral amputee: comparison of inverse dynamics and direct measurements

Inverse dynamics is the most comprehensive method that gives access to the net joint forces and moments during walking. However it is based on assumptions (i.e., rigid segments linked by ideal joints) and it is known to be sensitive to the input data (e.g., kinematic derivatives, positions of joint centres and centre of pressure, inertial parameters). Alternatively, transducers can be used to measure directly the load applied on the residuum of transfemoral amputees. So, the purpose of this study was to compare the forces and moments applied on a prosthetic knee measured directly with the ones calculated by three inverse dynamics computations - corresponding to 3 and 2 segments, and « ground reaction vector technique » - during the gait of one patient. The maximum RMSEs between the estimated and directly measured forces (i.e., 56 N) and moment (i.e., 5 N.m) were relatively small. However the dynamic outcomes of the prosthetic components (i.e., absorption of the foot, friction and limit stop of the knee) were only partially assessed with inverse dynamic methods.

Effect of inaccuracies in anthropometric data and linked-segment inverse dynamic modeling on kinetics of gait in persons with partial foot amputation

The accuracy of data derived from linked-segment models depends on how well the system has been represented. Previous investigations describing the gait of persons with partial foot amputation did not account for the unique anthropometry of the residuum or the inclusion of a prosthesis and footwear in the model and, as such, are likely to have underestimated the magnitude of the peak joint moments and powers. This investigation determined the effect of inaccuracies in the anthropometric input data on the kinetics of gait. Toward this end, a geometric model was developed and validated to estimate body segment parameters of various intact and partial feet. These data were then incorporated into customized linked-segment models, and the kinetic data were compared with that obtained from conventional models. Results indicate that accurate modeling increased the magnitude of the peak hip and knee joint moments and powers during terminal swing. Conventional inverse dynamic models are sufficiently accurate for research questions relating to stance phase. More accurate models that account for the anthropometry of the residuum, prosthesis, and footwear better reflect the work of the hip extensors and knee flexors to decelerate the limb during terminal swing phase.