Compressed Sensing of Memoryless Sources:A Deterministic Hadamard Construction

Compressed sensing is a new trend in signal processing for efficient sampling and signal acquisition. The idea is that most real-world signals have a sparse representation in an appropriate basis and this can be exploited to capture the sparse signal by taking only a few linear projections. The recovery is possible by running appropriate low-complexity algorithms that exploit the sparsity (prior information) to reconstruct the signal from the linear projections (posterior information). The main benefit is that the required number of measurements is much smaller than the dimension of the signal. This results in a huge gain in sensor cost (in measurement devices) or a dramatic saving in data acquisition time. However, some difficulties naturally arise in applying the compressed sensing to real-world applications such as robustness issues in taking the linear projections and computational complexity of the recovery algorithm. In this thesis, we design structured matrices for compressed sensing. In particular, we claim that some of the practical difficulties can be reasonably solved by imposing some structure on the measurement matrices. The thesis evolves around the Hadamard matrices which are $\{+1,-1\}$-valued matrices with many applications in signal processing, coding, optics and mathematics. As the title of the thesis implies, there are two main ingredients to this thesis. First, we use a memoryless assumption for the source, i.e., we assume that the nonzero components of the sparse signal are independently generated by a given probability distribution and their position is completely random. This allows us to use tools from probability, information theory and coding theory to rigorously assess the achievable performance. Second, using the mathematical properties of the Hadamard matrices, we design measurement matrices by selecting specific rows of a Hadamard matrix according to a deterministic criterion. We call the resulting matrices ``partial Hadamard matrices''. We design partial Hadamard matrices for three signal models: memoryless discrete signals and sparse signals with linear or sub-linear sparsity. A signal has linear sparsity if the number $k$ of its nonzero components is proportional to $n$, the dimension of signal, whereas it has a sub-linear sparsity if $k$ scales like $O(n^\alpha)$ for some $\alpha \in (0,1)$. We develop tools to rigorously analyze the performance of the proposed constructions by borrowing ideas from information theory and coding theory. We also extend our construction to distributed (multi-terminal) signals. Distributed compressed sensing is a ubiquitous problem in distributed data acquisition systems such as ad-hoc sensor networks. From both a theoretical and an engineering point of view, it is important to know how many measurement per dimension are necessary from different terminals in order to have a reliable estimate of the distributed data. We theoretically analyze this problem for a very simple setup where the components of the distributed signal are generated by a joint probability distribution which captures the spatial correlation among different terminals. We give an information-theoretic characterization of the measurements-rate region that results in a negligible recovery distortion. We also propose a low-complexity distributed message passing algorithm to achieve the theoretical limits

Stochastic spreading on complex networks

Complex interacting systems are ubiquitous in nature and society. Computational modeling of these systems is, therefore, of great relevance for science and engineering. Complex networks are common representations of these systems (e.g., friendship networks or road networks). Dynamical processes (e.g., virus spreading, traffic jams) that evolve on these networks are shaped and constrained by the underlying connectivity. This thesis provides numerical methods to study stochastic spreading processes on complex networks. We consider the processes as inherently probabilistic and analyze their behavior through the lens of probability theory. While powerful theoretical frameworks (like the SIS-epidemic model and continuous-time Markov chains) already exist, their analysis is computationally challenging. A key contribution of the thesis is to ease the computational burden of these methods. Particularly, we provide novel methods for the efficient stochastic simulation of these processes. Based on different simulation studies, we investigate techniques for optimal vaccine distribution and critically address the usage of mathematical models during the Covid-19 pandemic. We also provide model-reduction techniques that translate complicated models into simpler ones that can be solved without resorting to simulations. Lastly, we show how to infer the underlying contact data from node-level observations.Komplexe, interagierende Systeme sind in Natur und Gesellschaft allgegenwärtig. Die computergestützte Modellierung dieser Systeme ist daher von immenser Bedeutung für Wissenschaft und Technik. Netzwerke sind eine gängige Art, diese Systeme zu repräsentieren (z. B. Freundschaftsnetzwerke, Straßennetze). Dynamische Prozesse (z. B. Epidemien, Staus), die sich auf diesen Netzwerken ausbreiten, werden durch die spezifische Konnektivität geformt. In dieser Arbeit werden numerische Methoden zur Untersuchung stochastischer Ausbreitungsprozesse in komplexen Netzwerken entwickelt. Wir betrachten die Prozesse als inhärent probabilistisch und analysieren ihr Verhalten nach wahrscheinlichkeitstheoretischen Fragestellungen. Zwar gibt es bereits theoretische Grundlagen und Paradigmen (wie das SIS-Epidemiemodell und zeitkontinuierliche Markov-Ketten), aber ihre Analyse ist rechnerisch aufwändig. Ein wesentlicher Beitrag dieser Arbeit besteht darin, die Rechenlast dieser Methoden zu verringern. Wir erforschen Methoden zur effizienten Simulation dieser Prozesse. Anhand von Simulationsstudien untersuchen wir außerdem Techniken für optimale Impfstoffverteilung und setzen uns kritisch mit der Verwendung mathematischer Modelle bei der Covid-19-Pandemie auseinander. Des Weiteren führen wir Modellreduktionen ein, mit denen komplizierte Modelle in einfachere umgewandelt werden können. Abschließend zeigen wir, wie man von Beobachtungen einzelner Knoten auf die zugrunde liegende Netzwerkstruktur schließt

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Advanced Sensing, Fault Diagnostics, and Structural Health Management

Advanced sensing, fault diagnosis, and structural health management are important parts of the maintenance strategy of modern industries. With the advancement of science and technology, modern structural and mechanical systems are becoming more and more complex. Due to the continuous nature of operation and utilization, modern systems are heavily susceptible to faults. Hence, the operational reliability and safety of the systems can be greatly enhanced by using the multifaced strategy of designing novel sensing technologies and advanced intelligent algorithms and constructing modern data acquisition systems and structural health monitoring techniques. As a result, this research domain has been receiving a significant amount of attention from researchers in recent years. Furthermore, the research findings have been successfully applied in a wide range of fields such as aerospace, manufacturing, transportation and processes

Annual report 2015

Accessions considered in the study. Overview of the material considered in this study. For all materials, the GenBank identifier, the accession and species name as used in this study (Species) as well as their species synonyms used in the donor seed banks or in the NCBI GenBank (Material source/Reference) are provided. The genome symbol, and the country of origin, where the material was originally collected are given. The ploidy level measured in the scope of this study and the information if a herbarium voucher could be deposited in the herbarium of IPK Gatersleben (GAT) is given. Genomic formulas of tetraploids and hexploids are given as “female x male parent”. The genomes of Aegilops taxa follow Kilian et al. [74] and Li et al. [84]. Genome denominations for Hordeum follow Blattner [107] and Bernhardt [12] for the remaining taxa. (XLS 84 kb