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Nonlinear parametric inverse problems appear in many applications in science and engineering. We focus on diffuse optical tomography (DOT) in medical imaging. DOT aims to recover an unknown image of interest, such as the absorption coefficient in tissue to locate tumors in the body. Using a mathematical (forward) model to predict measurements given a parametrization of the tissue, we minimize the misfit between predicted and actual measurements up to a given noise level. The main computational bottleneck in such inverse problems is the repeated evaluation of this large-scale forward model, which corresponds to solving large linear systems for each source and frequency at each optimization step. Moreover, to efficiently compute derivative information, we need to solve, repeatedly, linear systems with the adjoint for each detector and frequency. As rapid advances in technology allow for large numbers of sources and detectors, these problems become computationally prohibitive. In this thesis, we introduce two methods to drastically reduce this cost. To efficiently implement Newton methods, we extend the use of simultaneous random sources to reduce the number of linear system solves to include simultaneous random detectors. Moreover, we combine simultaneous random sources and detectors with optimized ones that lead to faster convergence and more accurate solutions. We can use reduced order models (ROM) to drastically reduce the size of the linear systems to be solved in each optimization step while still solving the inverse problem accurately. However, the construction of the ROM bases still incurs a substantial cost. We propose to use randomization to drastically reduce the number of large linear solves needed for constructing the global ROM bases without degrading the accuracy of the solution to the inversion problem. We demonstrate the efficiency of these approaches with 2-dimensional and 3-dimensional examples from DOT; however, our methods have the potential to be useful for other applications as well. / Ph. D.

A computationally efficient, discrete model is presented for transmission line analysis and passive model order reduction of high-speed interconnect systems. The development of this model was motivated by the on-going efforts in chip/package co-design to route a major portion of the on-chip clock and high-speed data buses through the package in order to overcome the bandwidth reduction and delay caused by the high ohmic loss of on-chip wiring. The geometric complexity of the resulting interconnections is such that model order reduction is essential for rapid and accurate signal integrity assessment to support pre-layout design iteration and optimization. The modal network theory of the skin effect in conjunction with the theory of compact differences is used for the development of discrete models for dispersive, multi-conductor interconnects compatible with passive model order reduction algorithms. The passive reduced-order interconnect modeling algorithm, PRIMA, is then used on the resulting discrete model to generate a low-order, multi-port macromodel for interconnect networks. Numerical examples are used to demonstrate the validity and efficiency of the proposed model.

Le domaine de l'interaction fluide-structure regroupe l'étude de tous les phénomènes présentant le couplage du mouvement d'une structure avec celui d'un fluide. La gamme des phénomènes étudiés est très étendue, allant de l'étude de cylindres vibrants dans des écoulements comme c'est le cas dans l'industrie nucléaire, à des structures vibrantes dans des écoulements turbulents, en passant par des phénomènes de surface libre dans des réservoirs. Cependant, la complexité des phénomènes étudiés se répercute par des coûts de calculs prohibitifs, ce qui nous amène à rechercher des modèles réduits dont le temps de calcul serait plus réaliste. Dans cette thèse, on va présenter les différents modèles d'interaction fluide-structure et on va mettre en avant le modèle adopté dans notre étude. La réduction du modèle ainsi que l'optimisation des structures vont être introduites dans un contexte de couplage. En introduisant les incertitudes, l'étude fiabiliste de même qu'une approche d'optimisation basée fiabilité vont être proposées. Les différentes méthodologies adoptées vont être validées numériquement et comparées expérimentalement / The domain of the fluid-structure interaction includes the study of all phenomena presenting the coupling of the motion of a structure with the one of a fluid. The range of the phenomena being studied is very extensive, going from the study of vibrating cylinders in the flow as is the case in the nuclear industry, to vibrating structures in turbulent flows, through the free surface phenomena in reservoirs. However, the complexity of the phenomena studied is reflected by the cost of the prohibitive calculations, which leads us to look for models with the computation time would be more realistic. In this thesis, we will present different models of fluid-structure interaction and we will put forward the model adopted in our study. Reducing the model as well as the optimization of the structures will be introduced into a coupling setting. By introducing uncertainties, the reliability study as well as an optimization based reliability approach will be proposed. The different methodologies adopted will be validated numerically and experimentally compared

Two methods using lumped element (lumped parameter) methods to model flexible bodies have been presented. The methods are based on the concept of using a Guyan reduced stiffness matrix to describe the elasticity of a body. The component to be modeled has been divided into two parts using FE software and the mass and inertia tensor for the respective part of the component have been retrieved. The first method has been based on including the elements from the stiffness matrix in compliant constraints. The compliant constraints have been derived and a prototype has been implemented in MATLAB. It has been shown that using compliant constraints and stiffness parameters from a Guyan reduced stiffness matrix it is possible, with highly accurate results, to describe the deformation of a flexible body in multibody simulations. The second method is based on springs and dampers and has been implemented in the simulation environment Dymola. The springs and dampers have been constructed to include coupling elements from a Guyan reduced stiffness matrix. It has been shown that using the proposed method it is possible, with highly accurate results, to describe the static deformation of a flexible body. Further, using dynamic simulations of a full robot manipulator model, it has been shown that it is possible to use the spring-damper model to capture the deformation of the links of a manipulator in dynamic simulations with large translations and rotations.

Les progrès constants en miniaturisation des transistors et l’augmentation des fréquences des signaux utilisés sont les principales tendances dans l’évolution des circuits électroniques. Avec ces évolutions apparaissent de nombreux effets indésirables qui perturbent le comportement des systèmes électroniques et sont soupçonnés d’être responsables de la majorité des dégradations de signaux dans les systèmes en haute fréquence. Des retards de propagation indésirables sont ainsi introduits par la présence des interconnexions, et la diaphonie, phénomène dû aux couplages entre lignes d’interconnexions, peut éventuellement provoquer des commutations non désirées des transistors. La prise en compte des interconnexions, dès les premières phases de conception d'un système, est par conséquent devenue une nécessité ces dernières années. Mais la simulation temporelle d’un réseau d’interconnexions est très gourmande en temps de calcul, ce qui impacte la durée globale de conception. Le remplacement des modèles électriques, décrivant précisément les interconnexions, par des modèles plus simples est primordial pour limiter les coûts de calcul. Une méthode de réduction d'ordre des modèles peut alors être employée pour effectuer cette opération efficacement. Le modèle final doit en effet décrire assez précisément certains aspects importants du modèle original et conserver les propriétés importantes du réseau d'interconnexions. Cette démarche permettra aux concepteurs d’effectuer des simulations temporelles rapides et d’étudier les paramètres d’intégrité du signal tel que le retard, le temps de montée, le dépassement….L'objectif de cette thèse est d’établir un nouvel outil de réduction de complexité des modèles de réseaux d'interconnexions. Différentes descriptions initiales des systèmes d'interconnexions sont envisagées : modèles circuits (fonctions de transfert) ou mesures fréquentielles. L’approche développée repose sur l’utilisation des fonctions orthogonales de Müntz-Laguerre et de Kautz afin de décrire mathématiquement, de manière précise, le système d'origine. Un opérateur linéaire, lié à ces fonctions de base, est ensuite appliqué pour déterminer un modèle rationnel de moindre complexité. La technique proposée est comparée à d'autres méthodes de la littérature d’abord sur des exemples académiques. Tout le potentiel de la méthode est ensuite illustré par sa mise en œuvre sur des réseaux d'interconnexions. / The ongoing progress in transistor miniaturization and a continuous frequency increase are the main trends in the present day evolution of electronic circuits. A number of undesired effects are intrinsic to these developments and are suspected to be responsible for most of the flawed signals present in high frequency systems. Parasitic delays are thus introduced by the presence of interconnect lines and crosstalk due to coupling may lead to undesired switching events in transistor circuits. Accounting for the presence of interconnect lines, at a very early stage in the design flow has become unavoidable in recent years. However, time domain simulations of massively coupled interconnect networks may be computationally costly and have a tremendous impact on the overall duration of the design process. Replacing complex, high order circuit models by more compact surrogates is thus necessary. Model order reduction is an effective way to derive such surrogates. The final model must mimic certain aspects of the original model with sufficient accuracy and preserve the interconnect network’s most important properties. This approach enables designers to account for the undesired effects of interconnect lines such as, delays, rise-times and overshoots while maintaining the overall duration of time-domain simulations within acceptable limits. The aim of this thesis is to create a new model order reduction tool applicable to complex interconnect networks. Different initial representations were considered – circuit models (transfer functions) or frequency domain measurements. The proposed approach uses orthogonal basis functions such as Müntz-Laguerre and Kautz to build an accurate mathematical representation of the original system .A linear operator, related to these functions, is subsequently used to derive a simplified model. The technique is first compared to other approaches using examples available in literature, its full potential being demonstrated on coupled interconnect models.

In this paper we describe an approach to generating low-order models of spiral inductors that accurately capture the dependence on both frequency and geometry (width and spacing) parameters. The approach is based on adapting a multiparameter Krylov-subspace based moment matching method to reducing an integral equation for the three dimensional electromagnetic behavior of the spiral inductor. The approach is demonstrated on a typical on-chip rectangular inductor. / Singapore-MIT Alliance (SMA)

The full model of a double-wishbone suspension has more than 30 differential-algebraic equations which takes a remarkably long time to simulate. By contrast, the look-up table for the same suspension is simulated much faster, but may not be very accurate. Therefore, developing reduced models that approximate complex systems is necessary because model reduction decreases the simulation time in comparison with the original model, enables real time applications, and produces acceptable accuracy. In this research, we focus on model reduction techniques for vehicle systems such as suspensions and how they are approximated by models having lower degrees of freedom. First, some existing model reduction techniques, such as irreducible realization procedures, balanced truncation, and activity-based reduction, are implemented to some vehicle suspensions. Based on the application of these techniques, their disadvantages are revealed. Then, two methods of model reduction for multi-body systems are proposed. The first proposed method is 2-norm power-based model reduction (2NPR) that combines 2-norm of power and genetic algorithms to derive reduced models having lower degrees of freedom and fewer number of components. In the 2NPR, some components such as mass, damper, and spring are removed from the original system. Afterward, the values of the remaining components are adjusted by the genetic algorithms. The most important advantage of the 2NPR is keeping the topology of multi-body systems which is useful for design purposes. The second method uses proper orthogonal decomposition. First, the equations of motion for a multi-body system are converted to explicit second-order differential equations. Second, the projection matrix is obtained from simulation or experimental data by proper orthogonal decomposition. Finally, the equations of motion are transferred to a lower-dimensional state coordinate system. The implementation of the 2NPR to two double-wishbone suspensions and the comparison with other techniques such as balanced truncation and activity-based model reduction also demonstrate the efficiency of the new reduction technique.