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### Abstract

A sensor is a device that detects or measures a physical property and records, indicates, or otherwise responds to it. In other words, a sensor allows us to interact with the surrounding environment, by measuring qualitatively or quantitatively a given phenomena. Biological evolution provided every living entity with a set of sensors to ease the survival to daily challenges. In addition to the biological sensors, humans developed and designed “artificial” sensors with the aim of improving our capacity of sensing the real world. Today, thanks to technological developments, sensors are ubiquitous and thus, we measure an exponentially growing amount of data. Here is the challenge—how do we process and use this data? Nowadays, it is common to design real-world sensing architectures that use the measured data to estimate certain parameters of the measured physical field. This type of problems are known in mathematics as inverse problems and finding their solution is challenging. In fact, we estimate a set of parameters of a physical field with possibly infinite degrees of freedom with only a few measurements, that are most likely corrupted by noise. Therefore, we would like to design algorithms to solve the given inverse problem, while ensuring the existence of the solution, its uniqueness and its robustness to the measurement noise. In this thesis, we tackle different inverse problems, all inspired by real-world applications. First, we propose a new regularization technique for linear inverse problems based on the sensor placement optimization of the sensor network collecting the data. We propose Frame- Sense, a greedy algorithm inspired by frame theory that finds a near-optimal sensor placement with respect to the reconstruction error of the inverse problem solution in polynomial time. We substantiate our theoretical findings with numerical simulations showing that our method improves the state of the art. In particular, we show significant improvements on two realworld applications: the thermal monitoring of many-core processors and the adaptive sampling scheduling of environmental sensor networks. Second, we introduce the dual of the sensor placement problem, namely the source placement problem. In this case, instead of regularizing the inverse problem, we enable a precise control of the physical field by means of a forward problem. For this problem, we propose a near-optimal algorithm for the noiseless case, that is when we know exactly the current state of the physical field. Third, we consider a family of physical phenomena that can be modeled by means of graphs, where the nodes represent a set of entities and the edges model the transmission delay of an information between the entities. Examples of this phenomena are the spreading of a virus within the population of a given region or the spreading of a rumor on a social network. In this scenario, we identify two new key problems: the source placement and vaccination. For the former, we would like to find a set of sources such that the spreading of the information over the network is as fast as possible. For the latter, we look for an optimal set of nodes to be “vaccinated” such that the spreading of the virus is the slowest. For both problems, we propose greedy algorithms directly optimizing the average time of infection of the network. Such algorithms out-perform the current state of the art and we evaluate their performance with a set of experiments on synthetic datasets. Then, we discuss three distinct inverse problems for physical fields characterized by a diffusive phenomena, such as temperature of solid bodies or the dispersion of pollution in the atmosphere. We first study the uniform sampling and reconstruction of diffusion fields and we show that we can exploit the kernel of the field to control and bound the aliasing error. Second, we study the source estimation of a diffusive field given a set of spatio-temporal measurements of the field and under the assumption that the sources can be modeled as a set of Dirac’s deltas. For this estimation problem, we propose an algorithm that exploits the eigenfunctions representation of the diffusion field and we show that this algorithm recovers the sources precisely. Third, we propose an algorithm for the estimation of time-varying emissions of smokestacks from the data collected in the surrounding environment by a sensor network, under the assumption that the emission rates can be modeled as signals lying on low-dimensional subspaces or with a finite rate of innovation. Last, we analyze a classic non-linear inverse problem, namely the sparse phase retrieval. In such a problem, we would like to estimate a signal from just the magnitude of its Fourier transform. Phase retrieval is of interest for many scientific applications, such as X-ray crystallography and astronomy. We assume that the signal of interest is spatially sparse, as it happens for many applications, and we model it as a linear combination of Dirac’s delta. We derive sufficient conditions for the uniqueness of the solution based on the support of the autocorrelation function of the measured sparse signal. Finally, we propose a reconstruction algorithm for the sparse phase retrieval taking advantage of the sparsity of the signal of interest.