Overview of Constrained PARAFAC Models

In this paper, we present an overview of constrained PARAFAC models where the constraints model linear dependencies among columns of the factor matrices of the tensor decomposition, or alternatively, the pattern of interactions between different modes of the tensor which are captured by the equivalent core tensor. Some tensor prerequisites with a particular emphasis on mode combination using Kronecker products of canonical vectors that makes easier matricization operations, are first introduced. This Kronecker product based approach is also formulated in terms of the index notation, which provides an original and concise formalism for both matricizing tensors and writing tensor models. Then, after a brief reminder of PARAFAC and Tucker models, two families of constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models, are described in a unified framework, for $N^{th}$ order tensors. New tensor models, called nested Tucker models and block PARALIND/CONFAC models, are also introduced. A link between PARATUCK models and constrained PARAFAC models is then established. Finally, new uniqueness properties of PARATUCK models are deduced from sufficient conditions for essential uniqueness of their associated constrained PARAFAC models

Seismic sparse-spike deconvolution via Toeplitz-sparse matrix factorization

We have developed a new sparse-spike deconvolution (SSD) method based on Toeplitz-sparse matrix factorization (TSMF), a bilinear decomposition of a matrix into the product of a Toeplitz matrix and a sparse matrix, to address the problems of lateral continuity, effects of noise, and wavelet estimation error in SSD. Assuming the convolution model, a constant source wavelet, and the sparse reflectivity, a seismic profile can be considered as a matrix that is the product of a Toeplitz wavelet matrix and a sparse reflectivity matrix. Thus, we have developed an algorithm of TSMF to simultaneously deconvolve the seismic matrix into a wavelet matrix and a reflectivity matrix by alternatively solving two inversion subproblems related to the Toeplitz wavelet matrix and sparse reflectivity matrix, respectively. Because the seismic wavelet is usually compact and smooth, the fused Lasso was used to constrain the elements in the Toeplitz wavelet matrix. Moreover, due to the limitations of computer memory, large seismic data sets were divided into blocks, and the average of the source wavelets deconvolved from these blocks via TSMF-based SSD was used as the final estimation of the source wavelet for all blocks to deconvolve the reflectivity; thus, the lateral continuity of the seismic data can be maintained. The advantages of the proposed deconvolution method include using multiple traces to reduce the effect of random noise, tolerance to errors in the initial wavelet estimation, and the ability to preserve the complex structure of the seismic data without using any lateral constraints. Our tests on the synthetic seismic data from the Marmousi2 model and a section of field seismic data demonstrate that the proposed method can effectively derive the wavelet and reflectivity simultaneously from band-limited data with appropriate lateral coherence, even when the seismic data are contaminated by noise and the initial wavelet estimation is inaccurate

Contributions à l’analyse des systèmes en réseau

La dernière décennie a vu l’émergence des travaux autour des systèmes dynamiques interconnectés (systèmes en réseaux ou systèmes cyberphysiques). Dans cette habilitation à diriger des recherches, je donne un aperçu des contributions qui ont été les miennes durant la dernière décennie sur : l’analyse des systèmes en réseaux (problème de consensus, observabilité et application à la préservation de la vie privée), le traitement des données de grandes dimensions (analyse tensorielle pour l’identification des systèmes non-linéaires, décomposition distribuée de tenseurs de grandes dimensions), et l’application à la mobilité intelligente (navigation en milieu urbain, prédiction et estimation de trafic, estimation d’attitude pour la navigation pédestre). Une prospective est ensuite développée autour de la sécurité des systèmes en réseaux, en se basant sur la théorie des systèmes, et sur l’analyse des données de grandes dimensions organisées dans des tenseurs de données avec des applications sur la mobilité intelligente

Operational modal analysis - Theory and aspects of application in civil engineering

In recent years the demand on dynamic analyses of existing structures in civil engineering has remarkably increased. These analyses are mainly based on numerical models. Accordingly, the generated results depend on the quality of the used models. Therefore it is very important that the models describe the considered systems such that the behaviour of the physical structure is realistically represented. As any model is based on assumptions, there is always a certain degree of uncertainty present in the results of a simulation based on the respective numerical model. To minimise these uncertainties in the prediction of the response of a structure to a certain loading, it has become common practice to update or calibrate the parameters of a numerical model based on observations of the structural behaviour of the respective existing system. The determination of the behaviour of an existing structure requires experimental investigations. If the numerical analyses concern the dynamic response of a structure it is sensible to direct the experimental investigations towards the identification of the dynamic structural behaviour which is determined by the modal parameters of the system. In consequence, several methods for the experimental identification of modal parameters have been developed since the 1980ies. Due to various technical restraints in civil engineering which limit the possibilities to excitate a structure with economically reasonable effort, several methods have been developed that allow a modal identification form tests with an ambient excitation. The approach of identifying modal parameters only from measurements of the structural response without precise knowledge of the excitation is known as output-only or operational modal analysis. Since operational modal analysis (OMA) can be considered as a link between numerical modelling and simulation on the one hand and the dynamic behaviour of an existing structure on the other hand, the respective algorithms connect both the concepts of structural dynamics and mathematical tools applied within the processing of experimental data. Accordingly, the related theoretical topics are revised after an introduction into the topic. Several OMA methods have been developed over the last decades. The most established algorithms are presented here and their application is illustrated by means of both a small numerical and an experimental example. Since experimentally obtained results always underly manifold influences, an appropriate postprocessing of the results is necessary for a respective quality assessment. This quality assessment does not only require respective indicators but should also include the quantification of uncertainties. One special feature in modal testing is that it is common to instrument the structure in different sensor setups to improve the spacial resolution of identified mode shapes. The modal information identified from tests in several setups needs to be merged a posteriori. Algorithms to cope with this problem are also presented. Due to the fact that the amount of data generated in modal tests can become very large, manual processing can become extremely expensive or even impossible, for example in the case of a long-term continuous structural monitoring. In these situations an automated analysis and postprocessing are essential. Descriptions of respective methodologies are therefore also included in this work. Every structural system in civil engineering is unique and so also every identification of modal parameters has its specific challenges. Some aspects that can be faced in practical applications of operational modal analysis are presented and discussed in a chapter that is dedicated specific problems that an analyst may have to overcome. Case studies of systems with very close modes, with limited accessibility as well as the application of different OMA methods are described and discussed. In this context the focus is put on several types of uncertainty that may occur in the multiple stages of an operational modal analysis. In literature only very specific uncertainties at certain stages of the analysis are addressed. Here, the topic of uncertainties has been considered in a broader sense and approaches for treating respective problems are suggested. Eventually, it is concluded that the methodologies of operatinal modal analysis and related technical solutions have been well-engineered already. However, as in any discipline that includes experiments, a certain degree of uncertainty always remains in the results. From these conclusions has been derived a demand for further research and development that should be directed towards the minimisation of these uncertainties and to a respective optimisation of the steps and corresponding parameters included in an operational modal analysis

Nonlinear models and algorithms for RF systems digital calibration

Focusing on the receiving side of a communication system, the current trend in pushing the digital domain ever more closer to the antenna sets heavy constraints on the accuracy and linearity of the analog front-end and the conversion devices. Moreover, mixed-signal implementations of Systems-on-Chip using nanoscale CMOS processes result in an overall poorer analog performance and a reduced yield. To cope with the impairments of the low performance analog section in this "dirty RF" scenario, two solutions exist: designing more complex analog processing architectures or to identify the errors and correct them in the digital domain using DSP algorithms. In the latter, constraints in the analog circuits' precision can be offloaded to a digital signal processor. This thesis aims at the development of a methodology for the analysis, the modeling and the compensation of the analog impairments arising in different stages of a receiving chain using digital calibration techniques. Both single and multiple channel architectures are addressed exploiting the capability of the calibration algorithm to homogenize all the channels' responses of a multi-channel system in addition to the compensation of nonlinearities in each response. The systems targeted for the application of digital post compensation are a pipeline ADC, a digital-IF sub-sampling receiver and a 4-channel TI-ADC. The research focuses on post distortion methods using nonlinear dynamic models to approximate the post-inverse of the nonlinear system and to correct the distortions arising from static and dynamic errors. Volterra model is used due to its general approximation capabilities for the compensation of nonlinear systems with memory. Digital calibration is applied to a Sample and Hold and to a pipeline ADC simulated in the 45nm process, demonstrating high linearity improvement even with incomplete settling errors enabling the use of faster clock speeds. An extended model based on the baseband Volterra series is proposed and applied to the compensation of a digital-IF sub-sampling receiver. This architecture envisages frequency selectivity carried out at IF by an active band-pass CMOS filter causing in-band and out-of-band nonlinear distortions. The improved performance of the proposed model is demonstrated with circuital simulations of a 10th-order band pass filter, realized using a five-stage Gm-C Biquad cascade, and validated using out-of-sample sinusoidal and QAM signals. The same technique is extended to an array receiver with mismatched channels' responses showing that digital calibration can compensate the loss of directivity and enhance the overall system SFDR. An iterative backward pruning is applied to the Volterra models showing that complexity can be reduced without impacting linearity, obtaining state-of-the-art accuracy/complexity performance. Calibration of Time-Interleaved ADCs, widely used in RF-to-digital wideband receivers, is carried out developing ad hoc models because the steep discontinuities generated by the imperfect canceling of aliasing would require a huge number of terms in a polynomial approximation. A closed-form solution is derived for a 4-channel TI-ADC affected by gain errors and timing skews solving the perfect reconstruction equations. A background calibration technique is presented based on cyclo-stationary filter banks architecture. Convergence speed and accuracy of the recursive algorithm are discussed and complexity reduction techniques are applied

Methodik zur Integration von Vorwissen in die Modellbildung

This book describes how prior knowledge about dynamical systems and functions can be integrated in mathematical modelling. The first part comprises the transformation of the known properties into a mathematical model and the second part explains four approaches for solving the resulting constrained optimization problems. Numerous examples, tables and compilations complete the book

Methodik zur Integration von Vorwissen in die Modellbildung

Das Buch zeigt, wie Vorwissen über Eigenschaften dynamischer Systeme und über Funktionen in die mathematische Modellbildung integriert werden kann. Hierzu wird im ersten Teil der Arbeit das verbale Vorwissen mathematisch formuliert. Der zweite Teil beschreibt vier Zugängen, um die entstehenden restringierten Probleme zu lösen. Zahlreiche Beispiele, Tabellen und Zusammenstellungen vervollständigen das Buch