Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance

In my PhD thesis I propose a Bayesian nonparametric estimation method for structural econometric models where the functional parameter of interest describes the economic agent's behavior. The structural parameter is characterized as the solution of a functional equation, or by using more technical words, as the solution of an inverse problem that can be either ill-posed or well-posed. From a Bayesian point of view, the parameter of interest is a random function and the solution to the inference problem is the posterior distribution of this parameter. A regular version of the posterior distribution in functional spaces is characterized. However, the infinite dimension of the considered spaces causes a problem of non continuity of the solution and then a problem of inconsistency, from a frequentist point of view, of the posterior distribution (i.e. problem of ill-posedness). The contribution of this essay is to propose new methods to deal with this problem of ill-posedness. The first one consists in adopting a Tikhonov regularization scheme in the construction of the posterior distribution so that I end up with a new object that I call regularized posterior distribution and that I guess it is solution of the inverse problem. The second approach consists in specifying a prior distribution on the parameter of interest of the g-prior type. Then, I detect a class of models for which the prior distribution is able to correct for the ill-posedness also in infinite dimensional problems. I study asymptotic properties of these proposed solutions and I prove that, under some regularity condition satisfied by the true value of the parameter of interest, they are consistent in a "frequentist" sense. Once I have set the general theory, I apply my bayesian nonparametric methodology to different estimation problems. First, I apply this estimator to deconvolution and to hazard rate, density and regression estimation. Then, I consider the estimation of an Instrumental Regression that is useful in micro-econometrics when we have to deal with problems of endogeneity. Finally, I develop an application in finance: I get the bayesian estimator for the equilibrium asset pricing functional by using the Euler equation defined in the Lucas'(1978) tree-type models.

Synthesis and aplication of linear and macrocyclic nitrogen ligand

Linear and macrocyclic nitrogen ligands have been found wide application during the years. Nitrogen has a much strong association with transition-metal ions because the electron pair is partucularly available for complexing purposes. We started our investigation with the synthesis of new chiral perazamacrocycles containing four pyrrole rings. This ligand was synthesized by the [2+2]condensation of (R,R)-diaminocyclohexane and dipirranedialdehydes and was tested, after a complexation with Cu(OAc)2, in Henry reactions. The best yields (96%) and higher ee’s (96%) were obtained when the meso-substituent on the dipyrrandialdehyde was a methyl group. The positive influence of the pyrrole-containing macrocyclic structure on the efficiency/enantioselectivity of the catalytic system was demonstrated by comparison with the Henry reactions performed using analogous ligands. Henry product was obtain in good yield but only 73% of ee, when the dialdehyde unit was replaced by a triheteroaromatic dialdehye (furan-pyrrol-furan). Another well known macrocyclic ligand is calix[4]pyrrole. We decided to investigate, in collaboration with Neier’s group, the metal-coordinating properties of calix[2]pyrrole[2]pyrrolidine compounds obtained by the reduction of calix[4]pyrrole. We focused our attention on the reduction conditions, and tested different Pd supported (charcoal, grafite) catalysts at different condition. Concerning the synthesis of linear polyamine ligands. We focused our attention to the synthesis of 2-heteroaryl- and 2,5-diheteroarylpyrrolidines. The reductive amination reaction of diarylketones and aryl-substitutedketo-aldehydes with different chiral amines was exploited to prepare a small library of diastereo-enriched substituted pyrrolidines. We have also described a new synthetic route to 1,2-disubstituted 1,2,3,4-tetrahydropyrrole[1,2-a]pyrazines, which involves the diastereoselective addition of Grignard reagents to chiral oxazolidines. The best diastereoselectivity (98:2) was dependent on the nature of both the chiral auxiliary, (S)-1-phenylglycinol, and the nature of the organometallic reagent (MeMgBr).

Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Reduced-order modelling, circuit-level design and SOI fabrication of microelectromechanical resonators

This thesis deals with two important research aspects concerning radio frequency (RF) microresonators and switches. First, a new approach for compact modeling and simulation of these devices is presented. Then, a combined process flow for their simultaneous fabrication on a SOI substrate is proposed. Compact models for microresonators and switches are extracted by applying mathematical model order reduction (MOR) to the devices finite element (FE) description in ANSYS c° . The behaviour of these devices includes forms of nonlinearities. However, an approximation in the creation of the FE model is introduced, which enables the use of linear model order reduction. Microresonators are modeled with the introduction of transducer elements, which allow for direct coupling of the electrical and mechanical domain. The coupled system element matrices are linearized around an operating point and reduced. The resulting macromodel is valid for small signal analysis around the bias point, such as harmonic pre-stressed analysis. This is extremely useful for characterizing the frequency response of resonators. Compact modelling of switches preserves the nonlinearity of the device behaviour. Nonlinear reduced order models are obtained by reducing the number of nonlinearities in the system and handling them as input to the system. In this way, the system can be reduced using linear MOR techniques and nonlinearities are introduced directly in the reduced order model. The reduction of the number of system nonlinearities implies the approximation of all distributed forces in the model with lumped forces. Both for microresonators and switches, a procedure for matrices extraction has been developed so that reduced order models include the effects of electrical and mechanical pre-stress. The extraction process is fast and can be done automatically from ANSYS binary files. The method has been applied for the simulation of several devices both at devices and circuit level. Simulation results have been compared with full model simulations, and, when available, experimental data. Reduced order models have proven to conserve the accuracy of finite element method and to give a good description of the overall device behaviour, despite the introduced approximations. In addition, simulation is very fast, both at device and circuit level. A combined process-flow for the integrated fabrication of microresonators and switches has been defined. For this purpose, two processes that are optimized for the independent fabrication of these devices are merged. The major advantage of this process is the possibility to create on-chip circuit blocks that include both microresonators and switches. An application is, for example, aswitched filter bank for wireless transceiver. The process for microresonators fabrication is characterized by the use of silicon on insulator (SOI) wafers and on a deep reactive ion etching (DRIE) step for the creation of the vibrating structures in single-crystal silicon and the use of a sacrificial oxide layer for the definition of resonator to electrode distance. The fabrication of switches is characterized by the use of two different conductive layers for the definition of the actuation electrodes and by the use of a photoresist as a sacrificial layer for the creation of the suspended structure. Both processes have a gold electroplating step, for the creation of the resonators electrodes, transmission lines and suspended structures. The combined process flow is designed such that it conserves the basic properties of the original processes. Neither the performance of the resonators nor the performance of the switches results affected by the simultaneous fabrication. Moreover, common fabrication steps are shared, which allows for cheaper and faster fabrication.

A farm-level programming model to compare the atmospheric impact of conventional and organic farming

A model is developed to represent the activity of a farm using the method of linear programming. Two are the main components of the model, the balance of soil fertility and the livestock nutrition. According to the first, the farm is supposed to have a total requirement of nitrogen, which is to be accomplished either through internal sources (manure) or through external sources (fertilisers). The second component describes the animal husbandry as having a nutritional requirement which must be satisfied through the internal production of arable crops or the acquisition of feed from the market. The farmer is supposed to maximise total net income from the agricultural and the zoo-technical activities by choosing one rotation among those available for climate and acclivity. The perspective of the analysis is one of a short period: the structure of the farm is supposed to be fixed without possibility to change the allocation of permanent crops and the amount of animal husbandry. The model is integrated with an environmental module that describes the role of the farm within the carbon-nitrogen cycle. On the one hand the farm allows storing carbon through the photosynthesis of the plants and the accumulation of carbon in the soil; on the other some activities of the farm emit greenhouse gases into the atmosphere. The model is tested for some representative farms of the Emilia-Romagna region, showing to be capable to give different results for conventional and organic farming and providing first results concerning the different atmospheric impact. Relevant data about the representative farms and the feasible rotations are extracted from the FADN database, with an integration of the coefficients from the literature.

Dynamics and control issues of multi-rotor platforms

This thesis deals with the analytic study of dynamics of Multi--Rotor Unmanned Aerial Vehicles. It is conceived to give a set of mathematical instruments apt to the theoretical study and design of these flying machines. The entire work is organized in analogy with classical academic texts about airplane flight dynamics. First, the non--linear equations of motion are defined and all the external actions are modeled, with particular attention to rotors aerodynamics. All the equations are provided in a form, and with personal expedients, to be directly exploitable in a simulation environment. This has requited an answer to questions like the trim of such mathematical systems. All the treatment is developed aiming at the description of different multi--rotor configurations. Then, the linearized equations of motion are derived. The computation of the stability and control derivatives of the linear model is carried out. The study of static and dynamic stability characteristics is, thus, addressed, showing the influence of the various geometric and aerodynamic parameters of the machine and in particular of the rotors. All the theoretic results are finally utilized in two interesting cases. One concerns the design of control systems for attitude stabilization. The linear model permits the tuning of linear controllers gains and the non--linear model allows the numerical testing. The other case is the study of the performances of an innovative configuration of quad--rotor aircraft. With the non--linear model the feasibility of maneuvers impossible for a traditional quad--rotor is assessed. The linear model is applied to the controllability analysis of such an aircraft in case of actuator block.

Searching and retrieving in content-based repositories of formal mathematical knowledge

In this thesis, the author presents a query language for an RDF (Resource Description Framework) database and discusses its applications in the context of the HELM project (the Hypertextual Electronic Library of Mathematics). This language aims at meeting the main requirements coming from the RDF community. in particular it includes: a human readable textual syntax and a machine-processable XML (Extensible Markup Language) syntax both for queries and for query results, a rigorously exposed formal semantics, a graph-oriented RDF data access model capable of exploring an entire RDF graph (including both RDF Models and RDF Schemata), a full set of Boolean operators to compose the query constraints, fully customizable and highly structured query results having a 4-dimensional geometry, some constructions taken from ordinary programming languages that simplify the formulation of complex queries. The HELM project aims at integrating the modern tools for the automation of formal reasoning with the most recent electronic publishing technologies, in order create and maintain a hypertextual, distributed virtual library of formal mathematical knowledge. In the spirit of the Semantic Web, the documents of this library include RDF metadata describing their structure and content in a machine-understandable form. Using the author's query engine, HELM exploits this information to implement some functionalities allowing the interactive and automatic retrieval of documents on the basis of content-aware requests that take into account the mathematical nature of these documents.

Multifarious tumor-oriented small molecules: development of synthetic methodologies and application for diagnostic platforms

Cancer represents one of the most relevant and widespread diseases in the modern age. In this context, integrin receptors are important for the interactions of cells with extracellular matrix and for the development of both inflammation and carcinogenic phenomena. There are many tricks to improve the bioactivity and receptor selectivity of exogenous ligands; one of these is to integrate the amino acid sequence into a cyclic peptide to restrict its conformational space. Another approach is to develop small peptidomimetic molecules in order to enhance the molecular stability and open the way to versatile synthetic strategies. Starting from isoxazoline-based peptidomimetic molecules we recently reported, in this thesis we are going to present the synthesis of new integrin ligands obtained by modifying or introducing appendages on already reported structures. Initially, we are going to introduce the synthesis of linear and cyclic α-dehydro-β-amino acids as scaffolds for the preparation of bioactive peptidomimetics. Subsequently, we are going to present the construction of small molecule ligands (SMLs) based delivery systems performed starting from a polyfunctionalised isoxazoline scaffold, whose potency towards αVβ3 and α5β1 integrins has already been established by our research group. In the light of these results and due to the necessity to understand the behaviour of a single enantiomer of the isoxazoline-based compounds, the research group decided to synthesise the enantiopure heterocycle using a 1,3-dipolar cycloaddiction approach. Subsequently, we are going to introduce the synthesis of a Reporting Drug Delivery System composed by a carrier, a first spacer, a linker, a self-immolative system, a second spacer and a latent fluorophore. The last part of this work will describe the results obtained during the internship abroad in Prof. Aggarwal’s laboratory at the University of Bristol. The project was focused on the Mycapolyol A synthesis.

Production and characterization of novel thermoplastic (nano)composite materials for additive manufacturing applications

The increasing environmental global regulations have directed scientific research towards more sustainable materials, even in the field of composite materials for additive manufacturing. In this context, the presented research is devoted to the development of thermoplastic composites for FDM application with a low environmental impact, focusing on the possibility to use wastes from different industrial processes as filler for the production of composite filaments for FDM 3D printing. In particular carbon fibers recycled by pyro-gasification process of CFRP scraps were used as reinforcing agent for PLA, a biobased polymeric matrix. Since the high value of CFs, the ability to re-use recycled CFs, replacing virgin ones, seems to be a promising option in terms of sustainability and circular economy. Moreover, wastes from different agricultural industries, i.e. wheat and rice production processes, were valorised and used as biofillers for the production of PLA-biocomposites. The integration of these agricultural wastes into PLA bioplastic allowed to obtain biocomposites with improved eco-sustainability, biodegradability, lightweight, and lower cost. Finally, the study of novel composites for FDM was extended towards elastomeric nanocomposite materials, in particular TPU reinforced with graphene. The research procedure of all projects involves the optimization of production methods of composite filaments with a particular attention on the possible degradation of polymeric matrices. Then, main thermal properties of 3D printed object are evaluated by TGA, DSC characterization. Additionally, specific heat capacity (CP) and Coefficient of Linear Thermal Expansion (CLTE) measurements are useful to estimate the attitude of composites for the prevention of typical FDM issues, i.e. shrinkage and warping. Finally, the mechanical properties of 3D printed composites and their anisotropy are investigated by tensile test using distinct kinds of specimens with different printing angles with respect to the testing direction.

Data assimilation for advanced cross-scale unstructured-grid ocean modeling.

The coastal ocean is a complex environment with extremely dynamic processes that require a high-resolution and cross-scale modeling approach in which all hydrodynamic fields and scales are considered integral parts of the overall system. In the last decade, unstructured-grid models have been used to advance in seamless modeling between scales. On the other hand, the data assimilation methodologies to improve the unstructured-grid models in the coastal seas have been developed only recently and need significant advancements. Here, we link the unstructured-grid ocean modeling to the variational data assimilation methods. In particular, we show results from the modeling system SANIFS based on SHYFEM fully-baroclinic unstructured-grid model interfaced with OceanVar, a state-of-art variational data assimilation scheme adopted for several systems based on a structured grid. OceanVar implements a 3DVar DA scheme. The combination of three linear operators models the background error covariance matrix. The vertical part is represented using multivariate EOFs for temperature, salinity, and sea level anomaly. The horizontal part is assumed to be Gaussian isotropic and is modeled using a first-order recursive filter algorithm designed for structured and regular grids. Here we introduced a novel recursive filter algorithm for unstructured grids. A local hydrostatic adjustment scheme models the rapidly evolving part of the background error covariance. We designed two data assimilation experiments using SANIFS implementation interfaced with OceanVar over the period 2017-2018, one with only temperature and salinity assimilation by Argo profiles and the second also including sea level anomaly. The results showed a successful implementation of the approach and the added value of the assimilation for the active tracer fields. While looking at the broad basin, no significant improvements are highlighted for the sea level, requiring future investigations. Furthermore, a Machine Learning methodology based on an LSTM network has been used to predict the model SST increments.

Models and methods for damage and quantitative risk assessment of Natech scenarios

Natural events are a widely recognized hazard for industrial sites where relevant quantities of hazardous substances are handled, due to the possible generation of cascading events resulting in severe technological accidents (Natech scenarios). Natural events may damage storage and process equipment containing hazardous substances, that may be released leading to major accident scenarios called Natech events. The need to assess the risk associated with Natech scenarios is growing and methodologies were developed to allow the quantification of Natech risk, considering both point sources and linear sources as pipelines. A key element of these procedures is the use of vulnerability models providing an estimation of the damage probability of equipment or pipeline segment as a result of the impact of the natural event. Therefore, the first aim of the PhD project was to outline the state of the art of vulnerability models for equipment and pipelines subject to natural events such as floods, earthquakes, and wind. Moreover, the present PhD project also aimed at the development of new vulnerability models in order to fill some gaps in literature. In particular, a vulnerability model for vertical equipment subject to wind and to flood were developed. Finally, in order to improve the calculation of Natech risk for linear sources an original methodology was developed for Natech quantitative risk assessment methodology for pipelines subject to earthquakes. Overall, the results obtained are a step forward in the quantitative risk assessment of Natech accidents. The tools developed open the way to the inclusion of new equipment in the analysis of Natech events, and the methodology for the assessment of linear risk sources as pipelines provides an important tool for a more accurate and comprehensive assessment of Natech risk.