Mario Annunziato is researcher in Mathematics, in the field of Numerical Analysis,
at the "Università degli Studi di Salerno" since the year 2004. He has been also member of the "Gruppo
Nazionale di Calcolo Scientifico, Istituto Nazionale di Alta Matematica" since 2005 till 2017.
His interests focuse to the numerical solution of Partial Differential Equations (PDE)
and Integral Equations, related to stochastic processes and stochastic optimal control, and to modeling and applications of stochastic processes.
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.
iii) to provide equation modeling for random phenomena.
From 2004 to 2013 he has taught to courses and exercises of Numerical Analysis at the Science Faculty
of the University.
Address: Dr. Mario Annunziato, Dipartimento di Fisica "E. Caianiello", Università degli
Studi di Salerno, Via Giovanni Paolo II, 132 - 84084 Fisciano (SA), ITALY
Office n. 8 | e-mail: mannunzi [at] am-research.it | Tel: +39 089 96 3372
Official Projects, financial support and partecipations
-
The "Università degli Studi di Salerno" provides "Fondi di Ateneo per la
ricerca di base (FARB ex 60%)" for the financial support of fundamental research.
FARB financial support:
- Metodi numerici innovativi per equazioni dinamiche o evolutive, associate a
processi stocastici, anche per problemi di controllo ottimale, years 2011 - 2020.
-
The "Gruppo Nazionale per il Calcolo Scientifico of the Istituto Nazionale di Alta Matematica
(Indam-GNCS)" provides some financial support since the year 2007.
-
The European Science Foundation provided the exchange grants OPTPDE
n. 3875 and n. 4099 for the research project Optimal control with Fokker-Planck equation for stochastic
systems for the years 2012-13.
-
Partecipation to the
Multi-ITN STRIKE - Novel Methods in Computational Finance
(Marie Curie International Training Network) in the research unit of
Prof. Alfio Borzì
at the Würzburg University.
-
Governative financial support from "Fondo per il finanziamento delle attività base di ricerca"
year 2017 (L. 232/2016 Art. 1, comma 295).
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 [9,11,12,27].
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. 27, 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 (maximizing) an objective function of the state. Mathematically it is a constrained
minimization problem.
For stochastic models, in the current scientific literature, 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. 34
32, 31,
30, 29, 28, 24, 23, 22,
21, 20,
19, 18, 17,
16, 15, 14, 13,
10, 8.
Publications
-
M. Annunziato, A. Borzì,
Fokker-Planck Analysis of Superresolution Microscopy Images.
Math. Comput. Appl., Vol. 28 (2023) 113.
DOI: 10.3390/mca28060113.
-
M. Annunziato,
Book Reviews: Stochastic Modelling of Reaction-Diffusion Processes.
By Radek Erban and S. Jonathan Chapman.
SIAM REVIEW. Vol. 64. Pag.1093-1094.
DOI: 10.1137/22N975597.
-
M. Annunziato, A. Borzì,
A sequential quadratic Hamiltonian scheme to compute optimal relaxed controls.
ESAIM: Control, Optimisation and Calculus of Variations, 27 (2021) 49.
DOI: 10.1051/cocv/2021041.
-
M. Annunziato, A. Borzì,
A Fokker-Planck approach to the reconstruction of a cell membrane potential.
SIAM J. Sci. Comp. Vol. 43-3 (2021) B623-B649.
DOI:10.1137/20M131504X.
-
M. Annunziato, A. Borzì,
A Fokker-Planck control framework for stochastic systems.
EMS Surveys in Mathematical Sciences 2018. DOI:10.4171/EMSS/27.
-
M. Annunziato, H. Gottschalk,
Calibration of Lévy processes using optimal control of Kolmogorov equations
with periodic boundary conditions,
Math. Modelling and Analysis,
Vol. 23 (2018) 390-413. DOI:10.3846/mma.2018.024.
S. Roy, M. Annunziato, A. Borzì, C. Klingenberg,
A Fokker-Planck approach to control collective motion,
Comp. Optim. Appl. 69 (2018) 423-459
M. Annunziato, E. Messina,
A positive and monotone numerical scheme for Volterra-Renewal equations with space fluxes,
Journal of Comp. Math. 37 (2019) 33-47
B. Gaviraghi, M. Annunziato and A. Borzì,
A Fokker-Planck based approach to control jump processes.
Chapter 26 in: M. Ehrhardt, M. Günther and
J. ter Maten (eds) “Novel methods in Computational finance”,
Vol. 25 Mathematics in Industry 2017.
B. Gaviraghi, M. Annunziato and A. Borzì,
Splitting methods for Fokker-Planck equations
related to jump-diffusion processes.
Chapter 25 in: M. Ehrhardt, M. Günther and
J. ter Maten (eds) “Novel methods in Computational finance”,
Vol 25 Mathematics in Industry 2017.
T. H. Breitenbach, M. Annunziato, A. Borzì,
On the optimal control of random walks with jumps and barriers.
Methodol. Comput. Appl. Probab. Vol. 20 (2018) pp. 435-462
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)
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
T. H. Breitenbach, M. Annunziato, A. Borzì,
On the optimal control of random walks.
IFAC papers online 49 (2016),
pp. 248-253
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
V. Thalhofer, M. Annunziato, A. Borzì,
Stochastic modelling and control of antibiotic subtilin production,
J. Math. Biol., Vol. 73 (2016), pp. 727-749
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
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
M. Annunziato, A. Borzì,
Optimal control of a class of piecewise deterministic processes,
European Journal of Applied Mathematics
Vol. 25 (2014), pp. 1-25
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
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
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
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
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
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
M. Annunziato, E. Messina,
Numerical treatment of a Volterra Integral Equation with Space Maps,
APNUM 60 (2010) pp. 809-815
M. Annunziato, A. Borzì,
Fast solvers of Fredholm optimal control problems,
Numer. Math. Theor. Meth. Appl. 3 (2010), pp. 431-448.
M. Annunziato,
A finite difference method for piecewise deterministic processes with memory II,
Mathematical Modelling and Analysis, 14 (2009) pp. 139-158
M. Annunziato, Analysis of upwind method for piecewise deterministic Markov processes,
Comp. Meth. Appl. Math. Vol. 8 (2008) No. 1, pp. 3-20
M. Annunziato, A finite difference method for piecewise deterministic processes with memory,
Mathematical Modelling and Analysis, 12 (2007) 157-178
M. Annunziato, Non-Gaussian equilibrium distributions arising from the Langevin equation,
Physical Review E 65 (2002) 21113
M. Annunziato, P. Grigolini, B.J. West,
Canonical and Non-Canonical Equilibrium Distribution,
Physical Review E 64 (2001) 11107.
M. Annunziato, P. Grigolini,
Stochastic versus dynamic approach to Lévy statistics in the presence of an external
perturbation,
Physics Letters A 269 (2000) 31.
M. Annunziato, P. Grigolini, J. Riccardi,
A Fluctuation-Dissipation Process without Time Scale,
Physical Review E 61 (2000) 4801.
Lectures
- ICTT28 - Roma, Sept. 23th 2024,
28TH INTERNATIONAL CONFERENCE ON TRANSPORT THEORY.
On the reconstruction of potential fields by the observation of the stochastic motion,
via Fokker-Planck optimal control.
- 109° Congresso SIF - Salerno Sept. 11th-18th 2023,
109° Congresso Nazionale della Società italiana di Fisica.
On a method for reconstructing potential fields with an application
to superresolution microscopy.
- QuantMOCOTE - Würzburg, March 8th 2023,
Mini Workshop: Modelling, Optimization and Control of
Quantum Systems in Technology and Education.
Statistical outcomes of point processes to the Langevin equation.
- ControlPV2018 - Pavia, Sept 19th 2018,
Optimal Control and Mean Field Games.
Optimal control of multi particle stochastic system with mean field approximation.
- FGI 2017 - Paderborn, Sept 25th 2017,
18th French - German - Italian Conference on Optimization.
A numerical solver for the Fokker-Planck optimal control of stochastic jump-diffusion processes.
Abstract.
- CPDE 2016 - Bertinoro, June 14th 2016,
2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations.
On the optimal control of random walks.
Abstract.
- WIAS - Berlin, Nov 12th 2015,
Direct and Inverse Problems for PDEs with Random Coefficients,
An application of the Fokker-Planck-Kolmogorov optimal control framework to the calibration of
Lévy processes (invited lecture).
Abstract.
- INRIA - Sophia Antipolis, Jul 1st 2015,
27 th IFIP TC7 Conference 2015 on System Modelling and Optimization,
Optimal control of subdiffusion processes with the fractional Fokker-Planck control framework.
Abstract.
- CWI - Amsterdam, Feb. 2nd 2015, Fokker-Planck optimal control of anomalous diffusion
processes (invited lecture).
- Universität Würzburg, Oct. 2nd 2014,
Multi-ITN STRIKE Mini-Workshop in Stochastic Computing and Optimization,
A new trend in optimal control of stochastic processes (invited lecture)
Abstract
- Würzburg July 2014.
Internationale Tage der Nachhaltigkeit (International sustainability Days) "Euro-Ibsa".
Best Control of Uncertainty (invited lecture).
- NetCO2014 Tours. New Trends in Optimal Control 2014,
Optimal control of stochastic processes via probability density distribution function control.
- Italien-German training for stochastic modeling of financial
crisis at the University of Wuppertal (Invited lecture, Dec 9th - 16th 2013).
- InterDyn2013 Paris.
Workshop on Modeling and Control of Large Interacting Dynamical Systems 10-12 Sept. 2013,
Dauphine University. An innovative framework for the optimal
control of stochastic processes (invited lecture).
- International School of Mathematics "G. Stampacchia"
59th Workshop Nonlinear Optimization: a Bridge from Theory to Applications 11-6-2013. An optimal control framework for piecewise deterministic
processes.
- Oberwolfach 31-1-2013. Probability density function optimal control of piecewise determistic processes.
- MMA2012 Tallinn. On the action of semi-Markov processes on differential equations.
Abstract
- 7th European Conference on Elliptic and Parabolic Problems, Gaeta 2012.
Fokker Planck-based stochastic optimal control.
- Universität Würzburg, June 30th 2011, Seminar: Numerical solution to the Liouville Master
Equation for piecewise deterministic processes.
Abstract
- MMA 2011 Sigulda, An Optimal Control of Probability Density Function of
One-Dimensional Stochastic Processes
Conference Abstracts
- SIAM OP 2011 Darmstadt, An Optimal Control of Probability Density Functions of Stochastic Processes with
the Fokker-Planck Formulation
Conference Abstracts
- EMG 2010 Ischia, Fast numerical schemes for optimal control problems with Fredholm constraints
- SIMAI 2010 Cagliari, Robust and fast method for Fredholm optimal control problem
- MMA 2010 Druskininkai, Numerical schemes for the solution to differential equations driven by semi-Markov processes
Conference Abstract
- Convegno Parma 2009 - Equazioni Integrali:
recenti sviluppi numerici e nuove applicazioni, Fast solvers
of Fredholm optimal control problems (with A. Borzì).
- Lecture at Trier Universität 2009, Numerical solution of the Liouville Master Equation for Piecewise Deterministic Processes.
- ENUMATH 2009 Uppsala, High order numerical method for piecewise deterministic processes
with ENO scheme.
- GNCS 2009 Montecatini Terme, Trattamento numerico di un'equazione integrale di Volterra con
mappa sullo spazio (with E. Messina).
- ICCAM 2008 Ghent, Numerical treatment of a Volterra Integral Equation
with Space Maps (with E. Messina).
Conference Abstract
- MMA-AMOE 2008 Tartu, A finite difference method for piecewise deterministic processes
with memory: monotonicity and conservativity.
- XVIII Congresso UMI Bari, Alcuni risultati sul trattamento numerico della
``Liouville Master Equation'' per Processi Deterministici a Tratti.
- SciCADE'07 Saint Malo, Multi-dimensional piecewise deterministic Markov
processes: a first order numerical treatment.
Conference abstract
- MMA2007 Trakai, A numerical treatment of the Liouville-Master Equation for piecewise
deterministic processes with memory: convergence and monotonicity
(invited lecture).Conference abstract
- NTSEE 2006 Bielefeld, On a finite difference scheme for
piecewise-deterministic processes with memory and its parallel
algorithm implementation.
- Innovative Methods for Solving Evolutionary Problems with Memory
CAPRI, A finite difference scheme for
piecewise deterministic processes with memory.
Preprints
- M. Annunziato, H. Gottschalk, Calibration of Lévy Processes
using Optimal Control of Kolmogorov Equations with Periodic Boundary Conditions,
e-print:arXiv:1506.08439
- M. Annunziato, E. Messina, Numerical treatment of a Volterra Integral Equation
with Space Maps, Preprint n. 20-2008 DMI - Università di Salerno (outdated).
- M. Annunziato, A Legendre polynomials series for the distribution functions of
a piecewise deterministic relaxation Markov process,
Preprint n. 14-2008 DMI - Università di Salerno.
- M. Annunziato, On a finite difference method for
piecewise deterministic processes with memory and its parallel algorithm
implementation,
Preprint n. 12-2006 DMI - Università di Salerno (outdated).
- M. Annunziato, A finite difference method for
piecewise deterministic Markov processes,
Preprint n.10-2006 DMI - Università di Salerno,
e-print: arxiv.org/abs/math.NA/0606588 (outdated).
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