Dipartimento Matematica
Università Salerno

Mario Annunziato

Dipartimento di Matematica
Università degli Studi di Salerno

Mario Annunziato is researcher in Mathematics, in the field of Numerical Analysis, at the "Università degli Studi di Salerno" since the year 2004. He is also member of the "Gruppo Nazionale di Calcolo Scientifico, Istituto Nazionale di Alta Matematica".
His interests focuse to the numerical solution of Partial Differential Equations (PDE) and Integral Equations, related to stochastic processes and stochastic optimal control. The goals of the research are:
i) to find the probability density function (PDF) of a stochastic processes by numerically solving PDEs of parabolic and hyperbolic type, or Volterra Integral equations, by ensuring that the discrete PDF be positive (or monotone) and conservative.
ii) to develop numerical schemes for optimization problems with PDE constraints, related to the optimal control of stochastic processes.
Since the 2004 he has taught to courses and exercises of Numerical Analysis at the Science Faculty of the University.

Address: Dr. Mario Annunziato, Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132 - 84084 Fisciano (SA), ITALY
Office n. 8, e-mail: mannunzi[at]unisa.it, Tel: +39 089 96 3372, Fax: +39 089 96 3303.

Official Projects, financial support and partecipations


Numerical methods for Piecewise Deterministic Processes

"Piecewise Deterministic Processes" (PDP) are a general model for stochastic point processes where a noise affects the motion of a state function, only at some random point epochs. They have applications to queue systems, reliability analysis and stochastic hybrid systems.
A stochastic processes can be almost completely described by its probability density (transition) function (PDF). The Chapman-Kolmogorov equation is an abstract equation for the PDF. In the case of PDP the CK equation can take the form of system of first order hyperbolic PDEs [5,7] or Volterra integral equations [9,11,12]. These equations sometime are named Liouville Master Equation or generalized Fokker-Planck equations.
The system of hyperbolic PDEs has initial Cauchy conditions and the evolutory PDF solution must be non-negative and total probability conservative in time. Further, depending on the type of point process, the PDEs can have special non-local boundary conditions [5,7,12], where the boundary conditions are not assigned functions, like classical Dirichelet or Neumann, but depend on the unknown PDF by integrals over the interior of the domain.
PDEs with this problem formulation are very little investigated. The development of stable, positive and mass preserving numerical methods is the aim of this research task.
Refs. 12, 11, 9, 7, 6, 5.


Optimal control of stochastic processes

The possibility to control a stochastic process is a very interesting and stimulating research subject, and have potentially many applications in science, engineering and finance. In the optimal control theory the state of a system is controlled by minimizing (maxmizing) an objective function of the state. Mathematically it is a constrained minimization problem.
For stochastic models, in the current scientific leterature, the problem is formulated with an average of the cost functional of the stochastic state. The solution of this optimal control problem can be found by solving an Hamilton-Jacobi-Bellman equation.
In this research we propose to use the PDF as representative of the state of the system, define the objective as a functional of the PDF, and use the Fokker-Planck equation as constraint of the optimization problem. This is a new and unexplored framework in the field of optimization.
The solution can be found by formulating the minimization problem as an optimality system of PDEs, in order to find the reduced gradient of the objective and its vanishing point.
The aim of this research is to develop numerical methods to solve the optimality system and special minimization techniques for this non-linear optimization problem.
Refs. 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 10, 8.

Publications

  1. NEW! B. Gaviraghi, M. Annunziato and A. Borzì, Splitting methods for Fokker-Planck equations related to jump-diffusion processes & A Fokker-Planck based approach to control jump processes. Chapters 25 & 26 in: M. Ehrhardt, M. Günther and J. ter Maten (eds) “Novel methods in Computational finance”, Mathematics in Industry, to appear 2017.
  2. NEW! T. H. Breitenbach, M. Annunziato, A. Borzì, On the optimal control of random walks with jumps and barriers. Methodol. Comput. Appl. Probab.(2017) - in press
  3. B. Gaviraghi, A. Schindele, M. Annunziato, A. Borzì, On Optimal Sparse-Control Problems Governed by Jump-Diffusion Processes. Applied Mathematics, Vol. 7 (2016) pp. 1978-2004 (DOI: 10.4236/am.2016.716162)
  4. B. Gaviraghi, M. Annunziato, A. Borzì, Analysis of splitting methods for solving a partial integro-differential Fokker-Planck equation. Applied Mathematics and Computation, Vol. 294 (2017) pp. 1-17
  5. T. H. Breitenbach, M. Annunziato, A. Borzì, On the optimal control of random walks. IFAC papers online 49 (2016), pp. 248-253
  6. S. Roy, M. Annunziato, A. Borzì, A Fokker-Planck Feedback Control-Constrained Approach for Modeling Crowd Motion, J. Comp. and Theor. Transport 45 (2016) pp. 442-458
  7. V. Thalhofer, M. Annunziato, A. Borzì, Stochastic modelling and control of antibiotic subtilin production, J. Math. Biol., Vol. 73 (2016), pp. 727-749
  8. M. Annunziato, A. Borzì, M. Magdziarz, A. Weron, A fractional Fokker-Planck control framework for subdiffusion processes, Optimal Contr. Appl. and Meth. Vol. 37 (2016), pp. 290-304
  9. M. Annunziato, A. Borzì, F. Nobile, R. F. Tempone, On the connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck control frameworks, Applied Mathematics Vol. 5 (2014), pp. 2476-2484
  10. M. Annunziato, A. Borzì, Optimal control of a class of piecewise deterministic processes, European Journal of Applied Mathematics Vol. 25 (2014), pp. 1-25
  11. M. Annunziato, A. Borzì, Fokker-Planck-based control of a two level open quantum system, Math. Models and Meth. in Appl. Sci. Vol. 23 (2013) No. 11, pp. 2039-2064
  12. M. Annunziato, A. Borzì, A Fokker-Planck control framework for multidimensional stochastic processes, Journal of Comp. and Appl. Math., Vol. 237 (2013) No. 1, pp. 487-507
  13. M. Annunziato, A. Borzì, On a Fokker-Planck approach to control open quantum systems, IEEE Xplore: Nonlinear Dynamics of Electronic Systems, Proceedings of NDES 2012
  14. M. Annunziato, On the Action of a Semi-Markov Process on a System of Differential Equations, Math. Mod. Analysis Vol. 17 (2012) No. 5, pp. 650-672
  15. M. Annunziato, H. Brunner, E. Messina, Asymptotic stability of solutions to Volterra-renewal Equation with Space Maps, Journal of Mathematical Analysis and Applications, Vol. 395 (2012) No. 2, pp. 766-775
  16. M. Annunziato, A. Borzì, Optimal control of probability density functions of stochastic processes, Mathematical Modelling and Analysis, Vol. 15 (2010) No. 4, pp. 393-407
  17. M. Annunziato, E. Messina, Numerical treatment of a Volterra Integral Equation with Space Maps, APNUM 60 (2010) pp. 809-815
  18. M. Annunziato, A. Borzì, Fast solvers of Fredholm optimal control problems, Numer. Math. Theor. Meth. Appl. 3 (2010), pp. 431-448.
  19. M. Annunziato, A finite difference method for piecewise deterministic processes with memory II, Mathematical Modelling and Analysis, 14 (2009) pp. 139-158
  20. M. Annunziato, Analysis of upwind method for piecewise deterministic Markov processes, Comp. Meth. Appl. Math. Vol. 8 (2008) No. 1, pp. 3-20
  21. M. Annunziato, A finite difference method for piecewise deterministic processes with memory, Mathematical Modelling and Analysis, 12 (2007) 157-178
  22. M. Annunziato, Non-Gaussian equilibrium distributions arising from the Langevin equation, Physical Review E 65 (2002) 21113
  23. M. Annunziato, P. Grigolini, B.J. West, Canonical and Non-Canonical Equilibrium Distribution, Physical Review E 64 (2001) 11107.
  24. M. Annunziato, P. Grigolini, Stochastic versus dynamic approach to Lévy statistics in the presence of an external perturbation, Physics Letters A 269 (2000) 31.
  25. M. Annunziato, P. Grigolini, J. Riccardi, A Fluctuation-Dissipation Process without Time Scale, Physical Review E 61 (2000) 4801.

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