Efficient wideband electromagnetic scattering computation for frequency dependent lossy dielectrics using WCAWE

This paper presents a model order reduction algorithm for the volume electric field integral equation (EFIE) formulation, that achieves fast and accurate frequency sweep calculations of electromagnetic wave scattering. An inhomogeneous, two-dimensional, lossy dielectric object whose material is characterized by a complex permittivity which varies with frequency is considered. The variation in the dielectric properties of the ceramic BaxLa4Ti 2+xO 12+3x in the <1 GHz frequency range is investigated for various values of x in a frequency sweep analysis. We apply the well-conditioned asymptotic waveform evaluation (WCAWE) method to circumvent the computational complexity associated with the numerical solution of such formulations. A multipoint automatic WCAWE method is also demonstrated which can produce an accurate solution over a much broader bandwidth. Several numerical examples are given on order to illustrate the accuracy and robustness of the proposed methods

Model order reduction of time-delay systems using a laguerre expansion technique

The demands for miniature sized circuits with higher operating speeds have increased the complexity of the circuit, while at high frequencies it is known that effects such as crosstalk, attenuation and delay can have adverse effects on signal integrity. To capture these high speed effects a very large number of system equations is normally required and hence model order reduction techniques are required to make the simulation of the circuits computationally feasible. This paper proposes a higher order Krylov subspace algorithm for model order reduction of time-delay systems based on a Laguerre expansion technique. The proposed technique consists of three sections i.e., first the delays are approximated using the recursive relation of Laguerre polynomials, then in the second part, the reduced order is estimated for the time-delay system using a delay truncation in the Laguerre domain and in the third part, a higher order Krylov technique using Laguerre expansion is computed for obtaining the reduced order time-delay system. The proposed technique is validated by means of real world numerical examples

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

Efficient positive-real balanced truncation of symmetric systems via cross-riccati equations

We present a highly efficient approach for realizing a positive-real balanced truncation (PRBT) of symmetric systems. The solution of a pair of dual algebraic Riccati equations in conventional PRBT, whose cost constrains practical large-scale deployment, is reduced to the solution of one cross-Riccati equation (XRE). The cross-Riccatian nature of the solution then allows a simple construction of PRBT projection matrices, using a Schur decomposition, without actual balancing. An invariant subspace method and a modified quadratic alternating-direction-implicit iteration scheme are proposed to efficiently solve the XRE. A low-rank variant of the latter is shown to offer a remarkably fast PRBT speed over the conventional implementations. The XRE-based framework can be applied to a large class of linear passive networks, and its effectiveness is demonstrated through numerical examples. © 2008 IEEE.published_or_final_versio

Matemaattisten mallien dimension redusointi neurotieteessä

Dimensionality reduction is a commonly used method in engineering sciences, such as control theory, for improving computational efficiency of simulations of complex nonlinear mathematical models. Additionally, it is a way of surfacing the most important factors that drive the dynamics of the system. In the field of neuroscience, there is a great demand to incorporate molecular and cellular level detail in large-scale models of the brain in order to produce phenomena such as learning and behavior. This cannot be achieved with the computing power available today, since the detailed models are unsuitable for large-scale network or system level simulations. In this thesis, methods for mathematical model reduction are reviewed. In the field of systems biology, models are typically simplified by completely eliminating variables, such as molecules, from the system, and making assumptions of the system behavior, for example regarding the steady state of the chemical reactions. However, this approach is not meaningful in neuroscience since comprehensive models are needed in order to increase understanding of the target systems. This information loss problem is solved by mathematical reduction methods that strive to approximate the entire system with a smaller number of dimensions compared to the original system. In this study, mathematical model reduction is applied in the context of an experimentally verified signaling pathway model of plasticity. The chosen biophysical model is one of the most comprehensive models out of those that are currently able to explain aspects of plasticity on the molecular level with chemical interactions and the law of mass action. The employed reduction method is Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD+DEIM), a subspace projection method for reducing the dimensionality of nonlinear systems. By applying these methods, the simulation time of the plasticity model was radically shortened although approximation errors are present if the model is reviewed on large time scales. It is up to the final application of the model whether some error or none at all is tolerated. Based on these promising results, subspace projection methods are recommended for dimensionality reduction in computational neuroscience

Representaciones racionales de series matriciales con aplicación a la especificación de modelos multivariantes

Se caracterizan funciones racionales matriciales de dimensión arbitraria, a partir de series formales de potencias cuyos coeficientes son matrices, trasladando posteriormente los resultados a la especificación de modelos racionales de series temporales, en particular a modelos Varma. Los aspectos de minimalidad y unicidad de representación o, en terminología de series temporales, la intercambiabilidad de modelos y la identificabilidad de los parámetros han sido considerados también desde la aproximación de Padé matricial para su tratamiento y posterior aplicación a series temporales. Al mismo tiempo se aportan resultados sobre la estructura de la tabla de Padé en el caso de funciones matriciales de dimensión arbitraria. por otro lado, la forma que se indica para estudiar los parámetros nulos y/o redundantes de una representación racional responde también a ciertos problemas de sobreparametrización en la estimación de modelos racionales de series temporales

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques

Transient simulation of complex electronic circuits and systems operating at ultra high frequencies

The electronics industry worldwide faces increasingly difficult challenges in a bid to produce ultra-fast, reliable and inexpensive electronic devices. Electronic manufacturers rely on the Electronic Design Automation (EDA) industry to produce consistent Computer A id e d Design (CAD) simulation tools that w ill enable the design of new high-performance integrated circuits (IC), the key component of a modem electronic device. However, the continuing trend towards increasing operational frequencies and shrinking device sizes raises the question of the capability of existing circuit simulators to accurately and efficiently estimate circuit behaviour. The principle objective of this thesis is to advance the state-of-art in the transient simulation of complex electronic circuits and systems operating at ultra high frequencies. Given a set of excitations and initial conditions, the research problem involves the determination of the transient response o f a high-frequency complex electronic system consisting of linear (interconnects) and non-linear (discrete elements) parts with greatly improved efficien cy compared to existing methods and with the potential for very high accuracy in a way that permits an effective trade-off between accuracy and computational complexity. High-frequency interconnect effects are a major cause of the signal degradation encountered b y a signal propagating through linear interconnect networks in the modem IC. Therefore, the development of an interconnect model that can accurately and efficiently take into account frequency-dependent parameters of modem non-uniform interconnect is of paramount importance for state-of-art circuit simulators. Analytical models and models based on a set of tabulated data are investigated in this thesis. Two novel, h igh ly accurate and efficient interconnect simulation techniques are developed. These techniques combine model order reduction methods with either an analytical resonant model or an interconnect model generated from frequency-dependent sparameters derived from measurements or rigorous full-wave simulation. The latter part o f the thesis is concerned with envelope simulation. The complex mixture of profoundly different analog/digital parts in a modern IC gives rise to multitime signals, where a fast changing signal arising from the digital section is modulated by a slower-changing envelope signal related to the analog part. A transient analysis of such a circuit is in general very time-consuming. Therefore, specialised methods that take into account the multi-time nature o f the signal are required. To address this issue, a novel envelope simulation technique is developed. This technique combines a wavelet-based collocation method with a multi-time approach to result in a novel simulation technique that enables the desired trade-off between the required accuracy and computational efficiency in a simple and intuitive way. Furthermore, this new technique has the potential to greatly reduce the overall design cycle