Associate Professor in Numerical Analysis (MAT/08)
Mathematics Department "Guido Castelnuovo", Sapienza University of Rome
Email: simone.cacace (at) uniroma1.it
Address: Dipartimento di Matematica "Guido Castelnuovo", Sapienza Università di Roma,
Piazzale Aldo Moro 5, 00185 Rome, Italy
Room 102, first floor
Affiliated to INdAM-GNCS (National Group for Scientific Computing) since 2011
Research Interests
Numerical methods for Hamilton-Jacobi 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 Hamilton-Jacobi 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 full-grid fixed-point 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 Gauss-Seidel like fashion, a relevant reduction of the iterations needed for the convergence of the algorithms (Fast-Marching, Fast-Sweeping, Fast-Iterative 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 (including networks and stratified domains), to solve problems on structured/unstructured meshes via finite-difference/semi-Lagrangian schemes, and also for more general equations related to controlled fractional SIS models and 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 real-world 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 semi-Lagrangian 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 Hamilton-Jacobi equation for the optimal control of the agents, one Fokker-Planck 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 Gauss-Newton type, which provides a relevant speedup over the existing methods. I also have extended the method in different directions: to the case of multi-population 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 (K-means, Expectation-Maximization), which are able to extract relevant features from scattered big-data, 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 Expectation-Maximization algorithm, given by mixtures of probability distributions, coincide with the solutions of a multi-population 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 Hamilton-Jacobi 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 soft-robots, 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 bio-inspired 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 tentacle-like soft manipulator in two dimensions, which results in a generalization of the nonlinear Euler-Bernoulli 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 finite-differences in space combined with a velocity Verlet time integrator, while the numerical optimization employs augmented-Lagrangian methods. For optimal reachability problems, I also have investigated the dynamic setting, using adjoint-based gradient descent methods for the numerical solution of the related optimality systems.