The main topic of the thesis is the synthesis of control laws for interacting agent-based dynamics and their mean-field limit. In particular, after a general introduction, in the second chapter a linearization-based approach is used for the computation of sub-optimal feedback laws obtained from the solution of differential matrix Riccati equations. Quantification of dynamic performance of such control laws leads to theoretical estimates on suitable linearization points of the nonlinear dynamics. Subsequently, the feedback laws are embedded into a nonlinear model predictive control framework where the control is updated adaptively in time according to dynamic information on moments of linear mean-field dynamics. The performance and robustness of the proposed methodology is assessed through different numerical experiments in collective dynamics. In the other chapters of the thesis I present some related projects, robustness of systems with uncertainties, a proximal gradient approach for sparse control and an application in crowd evacuation dynamics.

Mathematical models represent a powerful tool for investigating the dynamics of human infection diseases, providing useful predictions about the spread of a disease and the effectiveness of possible control measures. One of the central aspects to understand the dynamics of human infection is the heterogeneity in behavioral patters adopted by the host population. Beyond control measures imposed by public authorities, human behavioral changes can be triggered by uncoordinated responses driven by the diffusion of fear in the general population or by the risk perception. In order to assess how and when behavioral changes can affect the spread of an epidemic, spontaneous social distancing - e.g. produced by avoiding crowded environments, using face masks or limiting travels - is investigated. Moreover, in order to assess whether vaccine preventable diseases can be eliminated through not compulsory vaccination programs, vaccination choices are investigated as well. The proposed models are based on an evolutionary game theory framework. Considering dynamical games allows explicitly modeling the coupled dynamics of disease transmission and human behavioral changes. Specifically, the information diffusion is modeled through an imitation process in which the convenience of different behaviors depends on the perceived risk of infection and vaccine side effects. The proposed models allow the investigation of the effects of misperception of risks induced by partial, delayed or incorrect information (either concerning the state of the epidemic or vaccine side effects) as well. The performed investigation highlights that a small reduction in the number of potentially infectious contacts in response to an epidemic and an initial misperception of the risk of infection can remarkably affect the spread of infection. On the other hand, the analysis of vaccination choices showed that concerns about proclaimed risks of vaccine side effects can result in widespread refusal of vaccination which in turn leads to drops in vaccine uptake and suboptimal vaccination coverage.

IIn this work we present a new class of well-balanced, arbitrary high order accurate semi-implicit discontinuous Galerkin methods for the solution of the shallow water and incompressible Navier-Stokes equations on staggered unstructured curved meshes. Isoparametric finite elements are used to take into account curved domain boundaries. Regarding two-dimensional shallow water equations, the discrete free surface elevation is defined on a primal triangular grid, while the discrete total height and the discrete velocity field are defined on an edge-based staggered dual grid. Similarly, for the two-dimensional incompressible Navier-Stokes case, the discrete pressure is defined on the main triangular grid and the velocity field is defined on the edge-based staggered grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier-Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. High order (better than second order) in time can be achieved by using a space-time finite element framework, where the basis and test functions are piecewise polynomials in both space and time. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse system for the scalar pressure involving only the direct neighbor elements, so that it becomes a block four-point system in 2D and a block five-point system for 3D tetrahedral meshes. The resulting linear system is conveniently solved with a matrix-free GMRES algorithm. Note that the same space-time DG scheme on a collocated grid would lead to ten non-zero blocks per element in 2D and seventeen non-zero blocks in 3D, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is well-behaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier-Stokes equations. The special case of high order in space low order in time allows us to recover further regularity about the main linear system for the pressure, such as the symmetry and the positive semi-definiteness in the general case. This allows us to use a very fast linear solver such as the conjugate gradient (CG) method. The flexibility and accuracy of high order space-time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time. We will further extend the previous method to three-dimensional incompressible Navier-Stokes system using a tetrahedral main grid and a corresponding face-based hexaxedral dual grid. The resulting dual mesh consists in non-standard 5-vertex hexahedral elements that cannot be represented using tensor products of one dimensional basis functions. Indeed a modal polynomial basis will be used for the dual mesh. This new family of numerical schemes is verified by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data. Furthermore, the comparison with available experimental results will be presented for incompressible Navier-Stokes equations.

The main object of our research is the notion of "intrinsic regular surfaces" introduced and studied by Franchi, Serapioni, Serra Cassano in a Carnot group G. More precisely, an intrinsic regular hypersurface (i.e. a topological codimension 1 surface) S is a subset of G which is locally defined as a non critical level set of a C^1 intrinsic function. In a similar way, a k-codimensional intrinsic regular surface is locally defined as a non critical level set of a C^1 intrinsic vector function. Through Implicit Function Theorem, S can be locally represented as an intrinsic graph by a function phi. Here the intrinsic graph is defined as follows: let V and W be complementary subgroups of G, then the intrinsic graph of phi defined from W to V is the set { A \cdot phi(A) | A belongs to W}, where \cdot indicates the group operation in G. A fine characterization of intrinsic regular surfaces in Heisenberg groups (examples of Carnot groups) as suitable 1-codimensional intrinsic graphs has been established in [1]. We extend this result in a general Carnot group introducing an appropriate notion of differentiability, denoted uniformly intrinsic differentiability, for maps acting between complementary subgroups of G. Finally we provide a characterization of intrinsic regular surfaces in terms of existence and continuity of suitable "derivatives" of phi introduced by Serra Cassano et al. in the context of Heisenberg groups. All the results have been obtained in collaboration with Serapioni. [1] L.Ambrosio, F. Serra Cassano, D. Vittone, \emph{Intrinsic regular hypersurfaces in Heisenberg groups}, J. Geom. Anal. 16, (2006), 187-232.

In this thesis we study intrinsic Lipschitz functions. In particular we provide a regular approximation result and a Poincarè type inequality for this class of functions. Moreover we study the obstacle problem in the Heisenberg group and we prove a geometric Poincarè inequality for a class of semilinear equations in the Engel group.

In this work we develop a new class of high order accurate Arbitrary-Lagrangian-Eulerian (ALE) one-step finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral meshes, respectively. As usual for finite volume schemes, data are represented within each control volume by piecewise constant values that evolve in time, hence implying the use of some strategies to improve the order of accuracy of the algorithm. In our approach high order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space-time finite element predictor, which performs a one-step time discretization. Specifically, we adopt either the continuous Galerkin (CG) predictor, which does not allow discontinuities in time and is suitable for smooth time evolutions, or the discontinuous Galerkin (DG) predictor which can handle stiff source terms that might produce jumps in the local space-time solution. Since we are dealing with moving meshes the elements deform while the solution is evolving in time, hence making the use of a reference system very convenient. Therefore, within the space-time predictor, the physical element is mapped onto a reference element using a high order isoparametric approach, where the space-time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space-time nodes. The computational mesh continuously changes its configuration in time, following as closely as possible the flow motion. The entire mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. In order to obtain a continuous mesh configuration at any time level, the mesh motion is evaluated by assigning each node of the computational mesh with a unique velocity vector at each timestep. The node solver algorithm preforms the Lagrangian stage, while we rely on a rezoning algorithm to improve the mesh quality when the flow motion becomes very complex, hence producing highly deformed computational elements. A so-called relaxation algorithm is finally employed to partially recover the optimal Lagrangian accuracy where the computational elements are not distorted too much. We underline that our scheme is supposed to be an ALE algorithm, where the local mesh velocity can be chosen independently from the local fluid velocity. Once the vertex velocity and thus the new node location has been determined, the old element configuration is connected with the new one at the future time level with straight edges to represent the local mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space-time conservation formulation of the governing system of hyperbolic balance laws. The nonlinear system is reformulated more compactly using a space-time divergence operator and is then integrated on a moving space-time control volume. We adopt a linear parametrization of the space-time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. In our algorithm either a simple and robust Rusanov-type numerical flux or a more sophisticated and less dissipative Osher-type numerical flux is employed. We apply the new high order direct ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical and relativistic magneto-hydrodynamics (MHD) equations and the non-conservative seven-equation Baer-Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes. Furthermore we focus on the following issues to improve the algorithm efficiency: the time evolution, the numerical flux computation across element boundaries and the high order WENO reconstruction procedure. First, a time-accurate local time stepping (LTS) algorithm for unstructured triangular meshes is derived and presented, where each element can run at its own optimal time step, given by a local CFL stability condition. Then, we propose a new and efficient quadrature-free formulation for the flux computation, in which the space-time boundaries of each element are split into simplex sub-elements. This leads to space-time normal vectors as well as Jacobian matrices that are constant within each sub-element, hence allowing the flux integrals to be evaluated on the space-time reference control volume once and for all analytically during a preprocessing step. Finally, we consider the very new a posteriori MOOD paradigm, recently proposed for the Eulerian framework, to overcome the expensive WENO approach on moving meshes. The MOOD technique requires the use of only one central reconstruction stencil because the limiting procedure is carried out a posteriori instead of a priori, as done in the WENO formulation.

This dissertation consists of an introduction and four papers. The papers deal with several problems of non-Riemannian metric spaces, such as sub-Riemannian Carnot groups and homogeneous metric spaces. The research has been carried out between the University of Trento (Italy) and the University of Jyväskylä (Finland) under the supervision of prof. F. Serra Cassano and E. Le Donne, respectively. In the following we present the abstracts of the four papers. 1) REGULARITY PROPERTIES OF SPHERES IN HOMOGENEOUS GROUPS E. Le Donne AND S. Nicolussi Golo We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an algebraic criterium in terms of the dilating automorphisms, which for example makes us conclude the regularity of every homogeneous distance on the Heisenberg group. In such a group, we analyze in more details the geometry of metric spheres. We also provide examples of homogeneous groups where spheres present cusps. 2) ASYMPTOTIC BEHAVIOR OF THE RIEMANNIAN HEISENBERG GROUP AND ITS HOROBOUNDARY E. Le Donne, S. Nicolussi Golo, AND A. Sambusetti The paper is devoted to the large scale geometry of the Heisenberg group H equipped with left-invariant Riemannian metrics. We prove that two such metrics have bounded difference if and only if they are asymptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant Riemannian metric d on H there is a unique sub-Riemanniann metric d' for which d − d' goes to zero at infinity, and we estimate the rate of convergence. As a first immediate consequence we get that the Riemannian Heisenberg group is at bounded distance from its asymptotic cone. The second consequence, which was our aim, is the explicit description of the horoboundary of the Riemannian Heisenberg group. 3) FROM HOMOGENEOUS METRIC SPACES TO LIE GROUPS M. G. Cowling, V. Kivioja, E. Le Donne, S. Nicolussi Golo, AND A. Ottazzi We study connected, locally compact metric spaces with transitive isometry groups. For all $\epsilon\in R_+$, each such space is $(1,\epsilon)$- quasi-isometric to a Lie group equipped with a left-invariant metric. Further, every metric Lie group is $(1,C)$-quasi-isometric to a solvable Lie group, and every simply connected metric Lie group is $(1,C)$-quasi-isometrically homeomorphic to a solvable-by-compact metric Lie group. While any contractible Lie group may be made isometric to a solvable group, only those that are solvable and of type (R) may be made isometric to a nilpotent Lie group, in which case the nilpotent group is the nilshadow of the group. Finally, we give a complete metric characterisation of metric Lie groups for which there exists an automorphic dilation. These coincide with the metric spaces that are locally compact, connected, homogeneous, and admit a metric dilation. 4) SOME REMARKS ON CONTACT VARIATIONS IN THE FIRST HEISENBERG GROUP S. Nicolussi Golo We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, we show that if $f : R^2 \rightarrow R^2$ is a $C^1$-intrinsic function, and $\nabla^f\nabla^ff = 0$, then the first contact variation of the sub-Riemannian area of its intrinsic graph is zero and the second contact variation is positive.

The thesis surveys some recent results obtained in the field of nonlinear partial differential equations and calculus of variations about eigenvalues of nonlinear operators.

Model order reduction is a technique to reduce computational times of parameterized PDEs while maintaining good accuracy of the approximated solution. Reduced basis methods (RB) are the most common algorithms for reducing the complexity of parameterized PDEs and nowadays they are widely applied and very actively researched in numerous fields. We propose two ideas to further enhance model reduction: the Fundamental Order Reduction Method (FOR) and offline error estimators for RB methods. The FOR method uses nonlinear combinations of the solutions to build the reduced model and use simple affine evaluations to execute the online stage. On the other hand, offline estimators are a class of estimators that move a-posteriori operations to the offline stage, reducing in this way the load of computations in the online stage. We apply these two ideas to an EEG equation which is useful for detecting the position where an epilepsy seizure begins inside the brain. We present two known ways to solve this equation: direct approach and subtraction approach, and show theoretical and numerical results of the application of the RB and FOR methods. We prove that is not feasible to apply model reduction in the direct approach but show that it is possible in the subtraction approach. Afterwards we solve the inverse problem associated with the EEG equation using a combination of the FOR method and neural networks.

Recent advances in medical science regarding the role of the venous system in the development of neurological conditions has renewed the attention of researchers in this district of the cardiovascular system. The main goal of this thesis is to perform a theoretical study of Chronic CerebroSpinal Venous Insufficiency (CCSVI), a venous pathology that has been associated to Multiple Sclerosis. CCSVI is a condition in which main cerebral venous drainage pathways are obstructed. Its impact in cerebral hemodynamics and its connection to Multiple Sclerosis is subject of current debate in the medical community. In order to perform a credible study of the haemodynamical aspects of CCSVI, a sufficiently accurate mathematical model of the problem under investigation must be used. The venous system has not received the same attention as the arterial counterpart by the medical community. As a consequence, the mathematical modeling and numerical simulation of the venous system lies far behind that of the arterial system. The venous system is a low-pressure system, formed by very thin-walled vessels, if compared to arteries, that are likely to collapse under the action of gravitational or external forces. These properties set special requirements on the mathematical models and numerical schemes to be used. In this thesis we present a closed-loop multi-scale mathematical model of the cardiovascular system, where medium to large arteries and veins are represented as one-dimensional (1D) vessels, whereas the heart, the pulmonary circulation, capillary beds and intracranial pressure are modeled as lumped parameter models. A characteristic feature of our closed-loop model is the detailed description of head and neck veins. Due to the large inter-subject variability of the venous system, we perform a patient-specific characterization of major veins of the head and neck using MRI data collected in collaboration with the Magnetic Resonance Research Facility of the Wayne State University, Detroit (USA). Computational results are carefully validated using published data for the arterial system and most regions of the venous system. For head and neck veins validation is carried out through a detailed comparison of simulation results against patient-specific Phase-Contrast MRI flow quantification data. Regarding the development of novel numerical schemes, we construct high-order accurate, robust and efficient numerical schemes for 1D blood flow in elastic and viscoelastic vessels, as well as a solver for vessel networks. The solver is validated in the context of an in vitro network of vessels for which experimental and numerical results are available. After validation of both, the mathematical model and the numerical methodology, we use our theoretical tool to study the influence of different CCSVI patterns on cerebral hemodynamics. CCSVI patterns are defined by the medical literature as combinations of venous obstructions at different locations. Here we used two strategies. First, we take a venous configuration corresponding to a healthy control and explore the effect of different CCSVI patterns by modifying this network. Then, we characterize our venous network with the geometry of a real CCSVI patient and compare results with the ones obtained for the healthy control. The presented model provides a powerful tool to study still unresolved aspects of cerebral blood flow physiology, as well as several venous pathologies. Furthermore, it constitutes an ideal platform for improving currently used algorithms and for integrating fundamental physiological processes, such as detailed hemodynamics, regulatory mechanisms and transport of substances.