MODEL ORDER REDUCTION OF NONLINEAR DYNAMIC SYSTEMS USING MULTIPLE PROJECTION BASES AND OPTIMIZED STATE-SPACE SAMPLING

Model order reduction (MOR) is a very powerful technique that is used to deal with the increasing complexity of dynamic systems. It is a mature and well understood field of study that has been applied to large linear dynamic systems with great success. However, the continued scaling of integrated micro-systems, the use of new technologies, and aggressive mixed-signal design has forced designers to consider nonlinear effects for more accurate model representations. This has created the need for a methodology to generate compact models from nonlinear systems of high dimensionality, since only such a solution will give an accurate description for current and future complex systems.The goal of this research is to develop a methodology for the model order reduction of large multidimensional nonlinear systems. To address a broad range of nonlinear systems, which makes the task of generalizing a reduction technique difficult, we use the concept of transforming the nonlinear representation into a composite structure of well defined basic functions from multiple projection bases.We build upon the concept of a training phase from the trajectory piecewise-linear (TPWL) methodology as a practical strategy to reduce the state exploration required for a large nonlinear system. We improve upon this methodology in two important ways: First, with a new strategy for the use of multiple projection bases in the reduction process and their coalescence into a unified base that better captures the behavior of the overall system; and second, with a novel strategy for the optimization of the state locations chosen during training. This optimization technique is based on using the Hessian of the system as an error bound metric.Finally, in order to treat the overall linear/nonlinear reduction task, we introduce a hierarchical approach using a block projection base. These three strategies together offer us a new perspective to the problem of model order reduction of nonlinear systems and the tracking or preservation of physical parameters in the final compact model

Momenten-Abgleich-Verfahren in der Modellreduktion von elektromagnetischen Problemstellungen

In this thesis, the application of moment matching based model order reduction techniques to first- and second-order model problems of Maxwell's equations in semiconductor structures is considered. Apart from preserving the specific structure of Maxwell's equations in the reduced order model, we provide a new Greedy-type expansion point selection strategy based on the adaptive-order rational Arnoldi method. Moreover, we give an overview on the appropriate treatment of the discrete divergence conditions for moment matching based model order reduction. With respect to the offline stage of model order reduction, we introduce a specific framework of moment matching methods allowing for the efficient computation of a reduced order model. In detail, we consider a modification of the adaptive-order rational Arnoldi method avoiding the complete recomputation of the orthonormal vector sequences for subsequently computed reduced order models. Apart from employing an algebraic two-level approach for solving sequences of shifted linear systems, we have also discussed the application of the recycling SQMR method in moment matching based model order reduction. In the latter case, we typically benefit from exploiting the fact that the discretized first- and second-order Maxwell's equations offer a specific J-symmetry.Die zugrundeliegende Arbeit beinhaltet die Anwendung der Modellreduktion mittels Momenten-Abgleich-Verfahren auf Maxwell-Gleichungen erster bzw. zweiter Ordnung aus dem Anwendungsgebiet der Halbleiterstrukturen. Abgesehen von der Erhaltung der speziellen Struktur der Maxwell-Gleichungen im reduzierten Modell, wird eine neue Greedy-artige Entwicklungspunktauswahl basierend auf dem adaptiven rationalen Arnoldi-Verfahren eingeführt. Darüber hinaus geben wir einen Überblick über die geeignete Behandlung der diskreten Divergenz-Bedingungen für Momenten-Abgleich-Verfahren in der Modellreduktion. Im Hinblick auf die Offline-Phase der Modellreduktion, werden wir im weiteren Verlauf ein effizientes Framework für Momenten-Abgleich-Verfahren einführen, die eine effiziente Berechnung einer Folge reduzierter Modelle erlaubt. Insbesondere werden wir dabei eine Modifikation des adaptiven rationalen Arnoldi-Verfahrens vorstellen, die eine vollständige, wiederholte Berechnung der Sequenzen orthonormaler Vektoren für aufeinanderfolgende reduzierte Modelle vermeidet. Abgesehen von der Anwendung eines algebraischen Zwei-Level-Verfahrens für die Lösung geshifteter linearer Gleichungssysteme, haben wir darüber hinaus die Anwendung des recycling SQMR Verfahrens innerhalb der Modellreduktion mittels Momenten-Abgleich-Verfahren betrachtet. Im letzteren Fall profitieren wir in der Regel von der Tatsache, dass die diskretisierten Maxwell-Gleichungen erster bzw. zweiter Ordnung eine spezielle J-Symmetrie aufweisen

Exact and approximate Strang-Fix conditions to reconstruct signals with finite rate of innovation from samples taken with arbitrary kernels

In the last few years, several new methods have been developed for the sampling and exact reconstruction of specific classes of non-bandlimited signals known as signals with finite rate of innovation (FRI). This is achieved by using adequate sampling kernels and reconstruction schemes. An example of valid kernels, which we use throughout the thesis, is given by the family of exponential reproducing functions. These satisfy the generalised Strang-Fix conditions, which ensure that proper linear combinations of the kernel with its shifted versions reproduce polynomials or exponentials exactly. The first contribution of the thesis is to analyse the behaviour of these kernels in the case of noisy measurements in order to provide clear guidelines on how to choose the exponential reproducing kernel that leads to the most stable reconstruction when estimating FRI signals from noisy samples. We then depart from the situation in which we can choose the sampling kernel and develop a new strategy that is universal in that it works with any kernel. We do so by noting that meeting the exact exponential reproduction condition is too stringent a constraint. We thus allow for a controlled error in the reproduction formula in order to use the exponential reproduction idea with arbitrary kernels and develop a universal reconstruction method which is stable and robust to noise. Numerical results validate the various contributions of the thesis and in particular show that the approximate exponential reproduction strategy leads to more stable and accurate reconstruction results than those obtained when using the exact recovery methods.Open Acces

Developments in entanglement theory and applications to relevant physical systems

This Thesis is devoted to the analysis of entanglement in relevant physical systems. Entanglement is the conducting theme of this research, though I do not dedicate to a single topic, but consider a wide scope of physical situations. I have followed mainly three lines of research for this Thesis, with a series of different works each, which are, Entanglement and Relativistic Quantum Theory, Continuous-variable entanglement, and Multipartite entanglement.Comment: Ph.D. Thesis, April 2007, Universidad Autonoma de Madri

Methodologies for non-linear dynamic simulations in product development

In this thesis the efficient numerical simulation of non-linear dynamic systems is addressed through the use of reduced models. The problem of reducing simulation time with marginal loss of accuracy has been studied for many decades, with the purpose of accelerating the design phase and allowing the use of more accurate virtual prototypes. The process of transforming an original model and describing a complex physical system into a less computational demanding one, is generically defined as model order reduction or model reduction. The resulting model is therefore known as reduced model. Despite decades of attempts and several successfully applied methods, this topic still represents an open point, especially for what concerns complex non-linear systems. The aim of this thesis is to develop methodologies which exploit the linear modal analysis as a reliable and consolidated tool in reducing the computational cost of non-linear systems. Formulations which retains the non-linear behaviour while exploiting well established linear analyses are sought. Non-linearities in non-linear systems can then be retained or linearised around linearisation points. After a review of the literature, in Chapter 2, both approaches are examined. First, a reduced model which dedefines the non-linearities in a cubic form is implemented (Chapter 3). Then, a novel reduction method based on the linearisation in the configurations space is proposed in Chapter 4 and 5. Chapter 4 discusses the linearisation procedure, with the use of a specific base for each linearisation point, so that the non-linear system is globally approximated by a piecewise linear system, described through a set of linear ones. Interactions between them are then used to retain the non-linear properties, with the local linearised systems named subsystems. The reduction of the model is discussed in Chapter 5, with a focus on the mode selection procedure in generating reduced linear subsystems, while in Chapter 6, after an application to a simple lumped system, two categories of examples are proposed, defining two possible interaction methods regarding the set of subsystems. In the first category a discrete interaction is used, with a subsystem replacing the previous one, while in the second category a continuous interaction is implemented, with more reduced linear subsystems evolving simultaneously. For each category single and multi-parameters examples are proposed, with an analysis of the performance included. The method discussed in Chapter 3 is implemented, developing a non-linear beam element and testing the reduction on both numerical and experimental cases. Good agreement in reproducing the reference data is proven for the considered examples. The novel method developed in Chapter 4 and 5 is described, discussed and applied to several numerical examples. This method proves effective in reducing the computational time while maintaining a good approximation. An energy-based mode selection algorithm is introduced and applied, showing positive effects on the model reduction method performance

Non-linear Recovery of Sparse Signal Representations with Applications to Temporal and Spatial Localization

Foundations of signal processing are heavily based on Shannon's sampling theorem for acquisition, representation and reconstruction. This theorem states that signals should not contain frequency components higher than the Nyquist rate, which is half of the sampling rate. Then, the signal can be perfectly reconstructed from its samples. Increasing evidence shows that the requirements imposed by Shannon's sampling theorem are too conservative for many naturally-occurring signals, which can be accurately characterized by sparse representations that require lower sampling rates closer to the signal's intrinsic information rates. Finite rate of innovation (FRI) is a new theory that allows to extract underlying sparse signal representations while operating at a reduced sampling rate. The goal of this PhD work is to advance reconstruction techniques for sparse signal representations from both theoretical and practical points of view. Specifically, the FRI framework is extended to deal with applications that involve temporal and spatial localization of events, including inverse source problems from radiating fields. We propose a novel reconstruction method using a model-fitting approach that is based on minimizing the fitting error subject to an underlying annihilation system given by the Prony's method. First, we showed that this is related to the problem known as structured low-rank matrix approximation as in structured total least squares problem. Then, we proposed to solve our problem under three different constraints using the iterative quadratic maximum likelihood algorithm. Our analysis and simulation results indicate that the proposed algorithms improve the robustness of the results with respect to common FRI reconstruction schemes. We have further developed the model-fitting approach to analyze spontaneous brain activity as measured by functional magnetic resonance imaging (fMRI). For this, we considered the noisy fMRI time course for every voxel as a convolution between an underlying activity inducing signal (i.e., a stream of Diracs) and the hemodynamic response function (HRF). We then validated this method using experimental fMRI data acquired during an event-related study. The results showed for the first time evidence for the practical usage of FRI for fMRI data analysis. We also addressed the problem of retrieving a sparse source distribution from the boundary measurements of a radiating field. First, based on Green's theorem, we proposed a sensing principle that allows to relate the boundary measurements to the source distribution. We focused on characterizing these sensing functions with particular attention for those that can be derived from holomorphic functions as they allow to control spatial decay of the sensing functions. With this selection, we developed an FRI-inspired non-iterative reconstruction algorithm. Finally, we developed an extension to the sensing principle (termed eigensensing) where we choose the spatial eigenfunctions of the Laplace operator as the sensing functions. With this extension, we showed that eigensensing principle allows to extract partial Fourier measurements of the source functions from boundary measurements. We considered photoacoustic tomography as a potential application of these theoretical developments

Image Registration Workshop Proceedings

Automatic image registration has often been considered as a preliminary step for higher-level processing, such as object recognition or data fusion. But with the unprecedented amounts of data which are being and will continue to be generated by newly developed sensors, the very topic of automatic image registration has become and important research topic. This workshop presents a collection of very high quality work which has been grouped in four main areas: (1) theoretical aspects of image registration; (2) applications to satellite imagery; (3) applications to medical imagery; and (4) image registration for computer vision research

Suboptimal selecting subspace for biorthonormal signal representation

Biorthogonal representations have been widely used in signal processing. Gabor expansion and wavelet transform are two popular ones. When the signal, expressed using orthonormal representation, is to be approximated by partial set of the basis, the criterion to obtain LSE approximation is to discard those basis vectors corresponding to smaller coefficients of the representation. However, such a simple strategy for discarding/retaining basis vectors does not hold for biorthonormal cases. The re-calculation for coefficients is needed in order to avoid large distortions. In this paper, we propose and investigate into several algorithms for finding the suboptimal subspace to represent the signals, and present the error analysis pertaining to biorthonormal signal representation