The equation arises not by analyzing the individual positions and velocity of each particle in the fluid but by considering a probability distribution f(x,v,t) for the position and velocity of a typical particle—that is, the probability that the particle occupies a given very small region of space centered at the position x, and has velocity nearly equal to v (thus occupying a very small region of velocity space), at an instant of time t.
Ludwig Boltzmann (18441906)
Recently, kinetic equations have found novel applications in the study of emergent behaviors in complex systems characterized by the spontaneous formation of spatiotemporal structures as a result of simple local interactions between agents. These include the study of birds’ flocks, insects’ swarming, crowd dynamics, opinions’ formation, stock markets and wealth distributions. Clearly, the basic entities in these fields differ from physical particles in that they already have an intermediate complexity themselves and are commonly denoted as agents (see [6]).
Milling behavior in fishes (left) and flocking of starlings (right)
One of the major difficulties in applying the classical toolbox of kinetic theory to these new fields is the lack of fundamental principles which define the microscopic dynamic. A degree of uncertainty is therefore implicitly embedded in such models, since most modeling parameters can be assigned only as statistical information from experimental results (see [4] for recent surveys).
In spite of the vast amount of existing research, both theoretically and numerically (see [3,8]), the study of kinetic equations has mostly remained deterministic and ignored uncertainty. In reality, there are many sources of uncertainties that can arise in these equations:
Incomplete knowledge of the interaction mechanism between particles/agents.
Imprecise measurements of the initial and boundary data.
Other sources of uncertainty like forcing and geometry, etc.
Understanding the impact of these uncertainties is critical to the simulations of the complex kinetic systems to validate the kinetic models, and will allow scientists and engineers to obtain more reliable predictions and perform better risk assessment.
Simulation of a flock attacked by a predator using the method developed in [1]
The development of numerical methods for kinetic equations presents several difficulties due to the high dimensionality and the intrinsic structural properties of the solution. Nonnegativity of the distribution function, conservation of invariant quantities, entropy dissipation and steady states are essential in order to compute qualitatively correct solutions. Preservation of these structural properties is even more challenging in presence of uncertainties which contribute to curse of dimensionality.
Uncertainty quantification (UQ) in kinetic equations represents a computational challenge for many reasons. Simple UQ tasks such as the estimation of statistical properties of the solution typically require multiple calls to a deterministic solver. A single solver call is already very expensive for such complex mathematical models.
Convergence acceleration of MicroMacro Monte Carlo versus standard Monte Carlo
Researchers at DMCS have a recognized international experience in the numerical analysis of kinetic equations with fundamental contributions in the development of fast spectral methods, Monte Carlo methods and asymptotic preserving schemes (see the recent review in [3] for example). Recently they developed novel approaches to UQ of kinetic equations based on generalized Polynomial Chaos expansions at a particle level in order to reduce the problem dimension and maintain the main physical properties of the solution (see [2]) and on micromacro Monte Carlo techniques which using control variate estimators based on the local equilibrium are capable to accelerate the slow statistical convergence of Monte Carlo methods (see [4, Chapter 5]).
References
Albi G., Pareschi L. (2013), Binary interaction algorithms for the simulation of flocking and swarming dynamics. Multiscale Modeling & Simulation 11, 129.
Carrillo J.A., Pareschi L., Zanella M. (2017), Particle gPC methods for mean field models of swarming with uncertainties, arXiv:1712.01677.
Dimarco G., Pareschi L. (2015), Numerical methods for kinetic equations. Acta Numerica.
Jin S., Pareschi L., eds (2018), Uncertainty quantification in hyperbolic and kinetic equations. SEMASIMAI Series in Applied Mathematics, Springer.
Mouhot C., Pareschi L. (2006), Fast algorithms for computing the Boltzmann collision operator. Mathematics of Computation 75, 18331852.
Pareschi L., Toscani G. (2013), Interacting multiagents systems: kinetic equations and Monte Carlo methods. Oxford University Press, (2013)
Pareschi L., Toscani G., Villani C. (2003), Spectral methods for the non cutoff Boltzmann equation and numerical grazing collision limit. Numerische Mathematik 93, 527548.
Villani C. (2002), A survey of mathematical topics in kinetic theory. Handbook of fluid mechanics, S. Friedlander and D. Serre, Eds. Elsevier Publ. NorthHolland vol. I, 71305.
Contact
Prof. Lorenzo Pareschi
Department of Mathematics and Computer Science
University of Ferrara
Via Machiavelli 30, 44121 Ferrara
email: lorenzo.pareschi@unife.it
]]>The article is avaible in full version at this link: https://www.sciencedirect.com/science/article/pii/S0004370218305964#br0050
Keywords
Authors
Received 16 April 2017, Revised 7 September 2018, Accepted 20 September 2018, Available online 17 October 2018 on
However, describing the real world is difficult because it usually includes many entities connected by a wide variety of relationships and the information we collect is often uncertain and incomplete.
To deal with the complexity of real world domains, AI researchers have developed powerful methods based on logic, while to deal with uncertainty they have devised clever techniques based on probability theory and statistics.
Until recently these two lines of research advanced independently because of the difficulties in combining them. The last twenty years have seen the fast growth of the field of Statistical Relational Artificial Intelligence (StarAI) which aims at overcoming these difficulties.
Researchers of DMCS are actively involved in StarAI by developing methods that combine logic programming with probability theory. They developed algorithm for drawing conclusions from probabilistic logic programs (inference) and algorithms that are able to autonomously build descriptions of the world from data (learning).
They developed the web application
where all the algorithms developed by the group can be tried online, even on datasets supplied by the users.
Probabilistic Logic Programming (PLP) is a powerful tool that has a large number of applications, from medicine to marketing, social networks, natural language processing, games, biology and genetics.
In medicine, it can be used to perform causal reasoning and rigorously identify the effects of treatments, avoiding pitfalls such as the Simpson Paradox, see http://cplint.ml.unife.it/example/inference/simpson.swinb
In marketing and social networks, we can predict with PLP whether a marketing action such as sending promotional material regarding a product to a number of clients can have a viral effect on the social network of all clients, see
http://cplint.ml.unife.it/example/inference/viral.swinb
In social networks, as well as in other kind of graphs such as biological or telecommunication networks, we can compute the probability of a connection between two nodes, a problem knwon as “path reliability”, see http://cplint.ml.unife.it/example/inference/path.swinb Or, we can compute the probability of the existence of a link between two individuals


In games, we can exploit PLP for endowing games with intelligence and variation. For example, we can use it to randomly generate 2D maps as in http://cplint.ml.unife.it/example/inference/tile_map.swinb 

In biology, it can be used to predict whether a molecule can cause cancer by looking at the chemical structure of the formula. The predictive model is automatically constructed from the descriptions of molecules for which the cancerogenic or mutagenic effect has been determined in the laboratory, see http://cplint.ml.unife.it/example/learning/muta.pl 

In genetics, it is possible to predict the genotype of an individual knowing the genotype of (some of) its ancestors using Mendel’s laws of inheritance, see http://cplint.ml.unife.it/example/inference/mendel.pl and http://cplint.ml.unife.it/example/inference/bloodtype.pl CC BYSA 3.0, Madeleine Price Ball 

The power of PLP can also be illustrated by its ability of solving puzzles, such as the Monty Hall puzzle (http://cplint.ml.unife.it/example/inference/monty.swinb), the truel, or duel among three opponents (http://cplint.ml.unife.it/example/inference/truel.swinb), the coupon collector problem (http://cplint.ml.unife.it/example/inference/coupon.swinb) or the threeprisoners puzzle (http://cplint.ml.unife.it/example/inference/jail.swinb). Humans have difficulty in providing the correct answer for these problems while PLP can compute the correct solution very quickly. 

PLP can also be used for dealing with continuous random variables. The picture on the left shows the probability distribution of the prediction of the position of an object moving in 2 dimensions made using a Kalman filter. 
Publications:
Fabrizio Riguzzi, Elena Bellodi, Evelina Lamma, Riccardo Zese, and Giuseppe Cota. Probabilistic logic programming on the web. Software: Practice and Experience, 46(10):13811396, © Wiley, October 2016. [ bib  DOI  .pdf ]
Fabrizio Riguzzi. The distribution semantics for normal programs with function symbols. International Journal of Approximate Reasoning, 77:1  19, © Elsevier, October 2016. [ bib  DOI  .pdf  http ]
Nicola Di Mauro, Elena Bellodi, and Fabrizio Riguzzi. Banditbased MonteCarlo structure learning of probabilistic logic programs. Machine Learning, 100(1):127156, © Springer International Publishing, July 2015. The original publication is available at http://link.springer.com. [ bib  DOI  .pdf ]
Elena Bellodi and Fabrizio Riguzzi. Structure learning of probabilistic logic programs by searching the clause space. Theory and Practice of Logic Programming, 15(2):169212, © Cambridge University Press, 2015. [ DOI  http ]
Fabrizio Riguzzi and Terrance Swift. Welldefinedness and efficient inference for probabilistic logic programming under the distribution semantics. Theory and Practice of Logic Programming, 13(Special Issue 02  25th Annual GULP Conference):279302, © Cambridge University Press, March 2013. [ bib  DOI  http ]
Fabrizio Riguzzi and Terrance Swift. The PITA system: Tabling and answer subsumption for reasoning under uncertainty. Theory and Practice of Logic Programming, 27th International Conference on Logic Programming (ICLP'11) Special Issue, Lexington, Kentucky 610 July 2011, 11(45):433449, © Cambridge University Press, 2011. [ DOI http ]
Contact:
Prof. Dr. Fabrizio Riguzzi
Dipartimento di Matematica e Informatica, Università di Ferrara
Blocco B, Polo Scientifico Tecnologico
Via Saragat 1, 44122, Ferrara, Italy
]]>The first Mathematical Teachings
(italian)
1802  1940Mathematics in Ferrara in the nineteenth and early twentieth centuries
(italian)
1945  Modern TimesMathematics in the modern university
(italian)
1391  Modern TimesHistorical Buildings of the Mathematical Teachings
(italian)
1936  1959Mathematics in Annali dell'Università di Ferrara
(italian)
Literature
Alessandra Fiocca, Luigi Pepe, La lettura di matematica nell'Università di Ferrara, dal 1602 al 1771. Annali Un. Ferrara Sez. VII , 31 (1985), pp. 125167.
Alessandra Fiocca, Luigi Pepe, L'Università e le Scuole per gli Ingegneri a Ferrara. Annali Univ. Ferrara Sez. VII, 32 (1986), pp. 125166.
Alessandra Fiocca, Luigi Pepe L'insegnamento della matematica nell'Università di Ferrara dal 1771 al 1942, in Università e cultura a Ferrara e Bologna, Firenze, Olschki, 1989, pp. 179.
Maria Teresa Borgato, Luigi Pepe, La matematica nella prima serie degli Annali dell’Università di Ferrara: due lavori di M. Beloch e W. Gröbner, Ferrara, Tip. Litografia Artigiana, 1997.
A special thank you goes to Prof. Luigi Pepe for his support in compiling the history of the Department.
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