A numerical method for Mean Field Games on networks

We propose a numerical method for stationary Mean Field Games defined on a network. In this framework a correct approximation of the transition conditions at the vertices plays a crucial role. We prove existence, uniqueness and convergence of the scheme and we also propose a least squares method for the solution of the discrete system. Numerical experiments are carried out

A rescaled method for RBF approximation

A new method to compute stable kernel-based interpolants has been presented by the second and third authors. This rescaled interpolation method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allow us to consider its error and stability properties. First, we prove that the method is an instance of the Shepard\u2019s method, when certain weight functions are used. In particular, the method can reproduce constant functions. Second, it is possible to define a modified set of cardinal functions strictly related to the ones of the not-rescaled kernel. Through these functions, we define a Lebesgue function for the rescaled interpolation process, and study its maximum - the Lebesgue constant - in different settings. Also, a preliminary theoretical result on the estimation of the interpolation error is presented. As an application, we couple our method with a partition of unity algorithm. This setting seems to be the most promising, and we illustrate its behavior with some experiments

An ergodic problem for Mean Field Games: qualitative properties and numerical simulations

This paper is devoted to some qualitative descriptions and some numerical results for ergodic Mean Field Games systems which arise, e.g., in the homogenization with a small noise limit. We shall consider either power type potentials or logarithmic type ones. In both cases, we shall establish some qualitative properties of the effective Hamiltonian $\bar H$ and of the effective drift $\bar b$. In particular we shall provide two cases where the effective system keeps/looses the Mean Field Games structure, namely where $\nabla_P \bar H(P,\alpha)$ coincides or not with $\bar b(P, \alpha)$. On the other hand, we shall provide some numerical tests validating the aforementioned qualitative properties of $\bar H$ and $\bar b$. In particular, we provide a numerical estimate of the discrepancy $\nabla_P \bar H(P,\alpha)-\bar b(P, \alpha)$

A rescaled method for RBF approximation

In the recent paper [8], a new method to compute stable kernel-based interpolants has been presented. This \textit{rescaled interpolation} method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allow us to consider its error and stability properties

Mercury's geochronology revised by applying Model Production Functions to Mariner 10 data: geological implications

Model Production Function chronology uses dynamic models of the Main Belt Asteroids (MBAs) and Near Earth Objects (NEOs) to derive the impactor flux to a target body. This is converted into the crater size-frequency-distribution for a specific planetary surface, and calibrated using the radiometric ages of different regions of the Moon's surface. This new approach has been applied to the crater counts on Mariner 10 images of the highlands and of several large impact basins on Mercury. MPF estimates for the plains show younger ages than those of previous chronologies. Assuming a variable uppermost layering of the Hermean crust, the age of the Caloris interior plains may be as young as 3.59 Ga, in agreement with MESSENGER results that imply that long-term volcanism overcame contractional tectonics. The MPF chronology also suggests a variable projectile flux through time, coherent with the MBAs for ancient periods and then gradually comparable also to the NEOs.Comment: Accepted by Journal Geophysical Research Letter

Ontology Learning and Semantic Annotation: a Necessary Symbiosis

Semantic annotation of text requires the dynamic merging of linguistically structured information and a ?world model?, usually represented as a domain-specific ontology. On the other hand, the process of engineering a domain-ontology through semi-automatic ontology learning system requires the availability of a considerable amount of semantically annotated documents. Facing this bootstrapping paradox requires an incremental process of annotation-acquisition-annotation, whereby domain-specific knowledge is acquired from linguistically-annotated texts and then projected back onto texts for extra linguistic information to be annotated and further knowledge layers to be extracted. The presented methodology is a first step in the direction of a full ?virtuous? circle where the semantic annotation platform and the evolving ontology interact in symbiosis. As a case study we have chosen the semantic annotation of product catalogues. We propose a hybrid approach, combining pattern matching techniques to exploit the regular structure of product descriptions in catalogues, and Natural Language Processing techniques which are resorted to analyze natural language descriptions. The semantic annotation involves the access to the ontology, semi-automatically bootstrapped with an ontology learning tool from annotated collections of catalogues

A new chronology for the Moon and Mercury

In this paper we present a new method for dating the surface of the Moon, obtained by modeling the incoming flux of impactors and converting it into a size distribution of resulting craters. We compare the results from this model with the standard chronology for the Moon showing their similarities and discrepancies. In particular, we find indications of a non-constant impactor flux in the last 500 Myr and also discuss the implications of our findings for the Late Heavy Bombardment hypothesis. We also show the potential of our model for accurate dating of other inner Solar System bodies, by applying it to Mercury.Comment: 27 pages, 13 figures, 1 table; accepted by A