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In this thesis I have characterized the trace measures for particular potential spaces of functions defined on R^n, but "mollified" so that the potentials are de facto defined on the upper half-space of R^n. The potential functions are kind Riesz-Bessel. The characterization of trace measures for these spaces is a test condition on elementary sets of the upper half-space. To prove the test condition as sufficient condition for trace measures, I had give an extension to the case of upper half-space of the Muckenhoupt-Wheeden and Wolff inequalities. Finally I characterized the Carleson-trace measures for Besov spaces of discrete martingales. This is a simplified discrete model for harmonic extensions of Lipschitz-Besov spaces.

This Ph.D thesis focuses on iterative regularization methods for regularizing linear and nonlinear ill-posed problems. Regarding linear problems, three new stopping rules for the Conjugate Gradient method applied to the normal equations are proposed and tested in many numerical simulations, including some tomographic images reconstruction problems. Regarding nonlinear problems, convergence and convergence rate results are provided for a Newton-type method with a modified version of Landweber iteration as an inner iteration in a Banach space setting.

In the present thesis, we discuss the main notions of an axiomatic approach for an invariant Harnack inequality. This procedure, originated from techniques for fully nonlinear elliptic operators, has been developed by Di Fazio, Gutiérrez, and Lanconelli in the general settings of doubling Hölder quasi-metric spaces. The main tools of the approach are the so-called double ball property and critical density property: the validity of these properties implies an invariant Harnack inequality. We are mainly interested in the horizontally elliptic operators, i.e. some second order linear degenerate-elliptic operators which are elliptic with respect to the horizontal directions of a Carnot group. An invariant Harnack inequality of Krylov-Safonov type is still an open problem in this context. In the thesis we show how the double ball property is related to the solvability of a kind of exterior Dirichlet problem for these operators. More precisely, it is a consequence of the existence of some suitable interior barrier functions of Bouligand-type. By following these ideas, we prove the double ball property for a generic step two Carnot group. Regarding the critical density, we generalize to the setting of H-type groups some arguments by Gutiérrez and Tournier for the Heisenberg group. We recognize that the critical density holds true in these peculiar contexts by assuming a Cordes-Landis type condition for the coefficient matrix of the operator. By the axiomatic approach, we thus prove an invariant Harnack inequality in H-type groups which is uniform in the class of the coefficient matrices with prescribed bounds for the eigenvalues and satisfying such a Cordes-Landis condition.

Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.

In this thesis we investigate variational problems involving 1-dimensional sets (e.g., curves, networks) and variational inequalities related to obstacle-type dynamics from a twofold prospective. On one side, we provide variational approximations and convex relaxations of the relevant energies and dynamics, moving mainly within the framework of Gamma-convergence and of convex analysis. On the other side, we thoroughly investigate the numerical optimization of the corresponding approximating energies, both to recover optimal 1-dimensional structures and to accurately simulate the actual dynamics.

This work aims at presenting different approaches to epidemics modelling. It consists of two main topics, which cover both theoretical and computational approaches to the development and analysis of mathematical models of infectious diseases. The first half regards the formulation and the analysis of SAIRS (Susceptible - Asyptomatics infected - Infected symptomatic - Recovered - Susceptible) epidemic models, including the possibility of vaccination. The model is formulated as a system of ordinary differential equations (ODEs), for which we provided a complete global stability analysis, combining two different approaches: the classical Lyapunov stability theorem, and a geometric approach, which generalises the Poincaré-Bendixon theorem. Afterwards, the model has been generalised using heterogeneous networks, which may describe different groups of individuals or different cities. For this model, the global stability analysis has been developed using the graph-theoretic approach to find Lyapunov functions. The second part of the thesis covers simulations based approaches to modelling heterogeneous humans interactions in epidemics. The first example we provide is an application with synthetic data. We investigate a stochastic SIR (Susceptible - Infected symptomatic - Recovered) dynamics on a network, by using a specialised version of the Gillespie algorithm. The other two examples we show consist of real data applications. Both regard the cost-benefit analysis of the introduction of new influenza vaccines. Both analyses have been performed using a multi-group SEIR (Susceptible - Exposed - Infected - Recovered) epidemiological model divided by age classes.

In this thesis we present a new and accurate series of computation methods for compressible multi-phase flows with capillary effects based upon the full seven-equation Baer-Nunziato model. For that reason, there are some numerical methods to obtain high accuracy solutions, which will be shown here. First, a high resolution shock capturing Total Variation Diminishing (TVD) finite volume scheme is used on both Cartesian and unstructured triangular grids. Regarding the TVD finite volume scheme on the unstructured grid, time-accurate local time stepping (LTS) is applied to compute the solutions of the governing PDE system, in which the results are also compared with time-accurate global time stepping. Second, we propose a novel high order accurate numerical method for the solution of the seven equation Baer-Nunziato model based on ADER discontinuous Galerkin (DG) finite element schemes combined with a posteriori subcell finite volume limiting and adaptive mesh refinement (AMR). In multi-phase flows, the difficulty is to design accurate numerical methods for resolving the phase interface, as well as the simulation of the phenomena occurring at the interface, such as surface tension effects, heat transfer and friction. This is because of the interactions of the fluids at the phase interface and its complex geometry. So the accurate simulation of compressible multi-phase flows with surface tension effects is currently still one of the most challenging problems in computational fluid dynamics (CFD). In this work, we present a novel path-conservative finite volume discretization of the continuum surface force method (CSF) of Brackbill et al. to account for the surface tension effect due to curvature of the phase interface. This is achieved in the context of a diffuse interface approach, based on the seven equation Baer-Nunziato model of compressible multi-phase flows. Such diffuse interface methods for compressible multi-phase flows including capillary effects have first been proposed by Perigaud and Saurel. Regarding the high order accuracy of a diffuse interface approach, the interface is captured and allowed to travel across one single possibly refined cell, and is computed in the context of multi-dimensional high accurate space/time DG schemes with AMR and a posteriori sub-cell stabilization. The surface tension terms of the CSF approach are considered as a part of the non-conservative hyperbolic system. We propose to integrate the CSF source term as a non-conservative product and not simply as a source term, following the ideas on path conservative finite volume schemes put forward by Castro and Parés. For the validation of the current numerical methods, we compare our numerical results with those published previously in the literature.

In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space-time adaptive meshes. Two main research fields can be distinguished: (1) fully explicit DG methods on collocated grids and (2) semi-implicit DG methods on edge-based staggered grids. DG methods became increasingly popular in the last twenty years mainly because of three intriguing properties: i) non-linear L2 stability has been proven; ii) arbitrary high order of accuracy can be achieved by simply increasing the polynomial order of the chosen basis functions, used for approximating the state-variables; iii) high scalability properties make DG methods suitable for large-scale simulations on general unstructured meshes. It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called ’Gibbs phenomenon’. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. Among them, a rather intriguing paradigm has been defined in the work of [71], in which the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. In the present work the main benefits of the MOOD paradigm, i.e. the computational robustness even in the presence of strong shocks, are preserved and the numerical diffusion is considerably reduced also for the limited cells by resorting to a proper sub-grid. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, a set of numerical and physical detection criteria is applied to the candidate solution, namely: positivity of pressure and density, absence of floating point errors and satisfaction of a discrete maximum principle in the sense of polynomials. Then, in those cells where at least one of these criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages on a suitable sub-grid in order to preserve the natural sub-cell resolution of the DG scheme. Then, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the sub-grid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved sub-cell averages. Moreover, handling typical multiscale problems, dynamic adaptive mesh refinement (AMR) and adaptive polynomial order methods are probably the two main ways of preserving accuracy and efficiency, and saving computational effort. The here adopted AMRapproach is the so called ’cell by cell’ refinement because of its formally very simple tree-type data structure. In the here-presented ’cell-by-cell’ AMR every single element is recursively refined, from a coarsest refinement level l0 = 0 to a prescribed finest (maximum) refinement level lmax, accordingly to a refinement-estimator function X that drives step by step the choice for recoarsening or refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. First, the Euler equations of compressible gas dynamics and the magnetohydrodynamics (MHD) equations have been treated [281]. Then, the presented method has been readily extended to the special relativistic ideal MHD equations [280], but also the the case of diffusive fluids, i.e. fluid flows in the presence of viscosity, thermal conductivity and magnetic resistivity [116]. In particular, the adopted formalism is quite general, leading to a novel family of adaptive ADER-DG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the shock-capturing capability of the news schemes are significantly enhanced within the cell-by-cell Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). The resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, from low to high Mach numbers, from low to high Reynolds regimes. In particular, concerning MHD equations, the divergence-free character of the magnetic field is taken into account through the so-called hyperbolic ’divergence-cleaning’ approach which allows to artificially transport and spread the numerical spurious ’magnetic monopoles’ out of the computational domain. A special treatment has been followed for the incompressible Navier-Stokes equations. In fact, the elliptic character of the incompressible Navier-Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semi-implicit approach has been used. The main advantage of making use of a semi-implicit discretization is that the numerical stability can be obtained for large time-steps without leading to an excessive computational demand [117]. In this context, we derived two new families of spectral semi-implicit and spectral space-time DG methods for the solution of the two and three dimensional Navier-Stokes equations on edge-based staggered Cartesian grids [115], following the ideas outlined in [97] for the shallow water equations. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor theta in [0.5, 1] for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. Moreover, a rigorous theoretical analysis of the condition number of the resulting linear systems and the design of specific preconditioners, using the theory of matrix-valued symbols and Generalized Locally Toeplitz (GLT) algebra has been successfully carried out with promising results in terms of numerical efficiency [102]. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. The new numerical method has been thoroughly validated for approximation polynomials of degree up to N = 11, using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist. Moreover, the here mentioned semi-implicit DG method has been successfully extended to a novel edge-based staggered ’cell-by-cell’ adaptive meshes [114].