Qualified for the calls in the role of Associate Professor, ASN 20162018 session V, SC 01/A5  Numerical Analysis, SSD MAT/08 from 31/08/2018 to 31/08/2027
Affiliated to INdAMGNCS (National Group for Scientific Computing) since 2011
Research Interests

Numerical methods for HamiltonJacobi equations, optimal control of ODEs,
differential games.
In optimal control theory of ODEs, the application of Bellman’s dynamic programming principle
allows one to obtain partial differential equations of HamiltonJacobi type, satisfied by the value
functions of the underlying control problems. Such equations are in general nonlinear and they do
not admit explicit solutions. My research focuses on the efficient numerical solution of these
equations.
The main difficulty is the typically high dimension of the problems, which makes unfeasible the
application of standard solvers based on fullgrid fixedpoint iterations. The idea is then to
exploit the hyperbolic nature of the equations, i.e. the fact that the relevant information
propagates along characteristics (causality), to reduce as much as possible the number of computations.
In this direction, one can mimic the causality at the discrete level, by reordering the grid nodes
in a suitable way, so to produce, in a GaussSeidel like fashion, a relevant reduction of the
iterations needed for the convergence of the algorithms (FastMarching, FastSweeping, FastIterative methods).
On the other hand, one can employ domain decomposition techniques, to split the CPU load of the
algorithms among parallel processors. Using again the causality, I have been able to build a smart
decomposition, by cutting the domain along the characteristics. In this way, the resulting partition
is composed of almost independent patches, which implies a relevant reduction of the synchronization
between the parallel processors. I have studied and implemented these methods in very different settings,
to solve problems on structured/unstructured meshes via finitedifference/semiLagrangian schemes,
and also for more general equations related to stochastic differential games for hybrid systems.
More recently, part of my research is devoted to the implementation of these methods in CUDA on GPUs,
in order to attack some interesting realworld problems.
Finally, I am also working on the implementation of fast algorithms to localize points in an arbitrary mesh,
a crucial step for the efficiency of semiLagrangian schemes running on unstructured meshes.

Numerical methods for Mean Field Games, applications to Cluster Analysis.
Mean Field Games (MFG) theory is a generalization of the theory of stochastic differential games
to the case of a huge (formally infinite) number of players (or agents), which applies to very
complex scenarios, e.g. in the study of financial markets, population dynamics and communication
networks. The main assumptions are the fact that all the agents are indistinguishable (i.e. they have
the same goal) and that the actions of a single agent, based only on a “mean” information of all the
agents, have a negligible impact on the evolution of the whole system. This leads to a system of
strongly coupled nonlinear partial differential equations, one HamiltonJacobi equation for the
optimal control of the agents, one FokkerPlanck equation for the probability distribution of the agents.
My research focuses on the convergence and implementation of numerical methods for the solution of MFG systems.
For stationary problems, I have introduced a new algorithm, based on iterations of GaussNewton type,
which provides a relevant speedup over the existing methods. I also have extended the method in
different directions: to the case of multipopulation Mean Field Games, in which each population is described
by a MFG system, and all the systems are coupled by some interaction terms; to the case in which the state
space for the game is a network, namely a collection of nodes connected by continuous arcs. In this case,
the MFG system is much more complicated, due to the transmission conditions of Kirchhoff type at the nodes.
More recently, part of my research is devoted to the study of clustering algorithms in Machine Learning
(Kmeans, ExpectationMaximization), which are able to extract relevant features from scattered bigdata,
organizing them in groups (clusters), according to some (more or less specified) membership criterion.
In this direction, I have proposed a connection between Mean Field Games and Cluster Analysis,
namely an attempt to provide a mathematical model in a mainstream research field which is, nowadays,
still far from being well understood. In particular, I have proved that the clusters obtained by the
classical ExpectationMaximization algorithm, given by mixtures of probability distributions, coincide
with the solutions of a multipopulation Mean Field Game, in which each population represents a cluster
and the coupling terms are suitably chosen according to the type of distributions (Gaussian, Bernoulli, Categorical).
Finally, I have studied and proved the convergence of a “policy iteration method” for Mean Field Games,
an iterative technique initially proposed in the literature for HamiltonJacobi equations, which allows
to decouple the non linear MFG system into a sequence of linear problems, and results in a relevant
acceleration for the convergence to the solution of the problem.

Modeling and control of softrobots, optimal control of PDEs, numerical methods for
constrained optimization.
Soft Robotics is an emerging branch of classical Robotics. Differently from standard manipulators
widely employed in industrial applications, mainly composed of rigid joints with a relatively small
number of degrees of freedom, soft robots are realized using soft materials and they can perform
very complex tasks thanks to a potentially very large number of (pneumatic, piezoelectric) actuators.
The aim is to create a new generation of machines that can operate in synergy with humans, ensuring
safe interactions, also in unknown environments. Some futuristic applications: microsurgery, rehabilitation,
exploration and rescue.
My research focuses on the mathematical modeling and control of bioinspired soft manipulators, using
the tools of optimal control theory of PDEs and numerical techniques for constrained optimization.
I have proposed a model for an octopus tentaclelike soft manipulator in two dimensions, which results
in a generalization of the nonlinear EulerBernoulli beam model, namely a system of controlled, fourth order,
partial differential algebraic equations. The model accounts for inextensibility and curvature constraints
on the manipulator, plus the actuators, represented by distributed controls on the local curvature of the
device symmetry axis. I have addressed several optimal control problems in a stationary setting, including
low energy consumption, reachability and grasping, also in presence of obstacles and in case of mechanical
breakdowns. Discretization is performed by finitedifferences in space combined with a velocity Verlet time
integrator, while the numerical optimization employs augmentedLagrangian methods. For optimal reachability
problems, I also have investigated the dynamic setting, using adjointbased gradient descent methods for the
numerical solution of the related optimality systems.
