20 Dicembre, 2016 15:00 oclock

Sezione di Geometria, Algebra e loro applicazioni

Open problems on 2-factorizations and new powerful methods to attack them

Tommaso Traetta, Department of Mathematics, Ryerson University

Aula Seminari Terzo Piano

abstract

abstract

A 2-factorization of a graph G is a set F of spanning 2-regular subgraphs (i.e., 2-factors) whose edge-sets partition the edge-set of G. It is well known that G has a 2-factorization if and only if it is regular of even degree. However, if we specify t 2-factors, say F1, F2, ..., Ft, and ask for the factorization F to contain ?_i factors isomorphic to Fi, then the problem becomes much harder. When t = 1
we have the well-known Oberwolfach problem, attributed to G. Ringel who posed it in 1956. The case t = 2 is known as the Hamilton-Waterloo problem, whereas for greater values of t we speak of the generalized Oberwolfach problem. The case where t ? {1, 2} and G is the complete (equipartite) graph is the most studied one. Nonetheless, these problems are still open although they have received much attention lately.

In this talk I will emphasize the connection between 2-factorizations and sharply transitive sets of permutations. Then, I will focus on some of the most recent results and their powerful algebraic methods.

This seminar is organized within the PRIN 2012 Research project «Geometric Structures, Combinatorics and their Applications» Grant Registration number 2012XZE22K, funded by MIUR - Project coordinator Prof.ssa Norma Zagaglia

14 Dicembre, 2016 11:00

Sezione di Finanza Quantitativa

Shall I sell or shall I wait? Optimal liquidation under partial information with price impact.

Katia Colaneri, Università di Perugia

Aula Seminari Terzo Piano

abstract

abstract

We study the problem of a trader who wants to maximize the expected reward from liquidating a given stock position. We model the stock price dynamics as a geometric pure jump process with local characteristics driven by an unobservable finite-state Markov chain and the liquidation rate. This refl ects uncertainty about the state of the market and feedback effects from trading. We use stochastic fi ltering to reduce the optimization
problem under partial information to an equivalent one under complete information. This leads to a control problem for piecewise deterministic Markov processes (in short PDMP).
We apply control theory for PDMPs to our problem. In particular, we derive the optimality equation for the value function and characterize the value function as unique viscosity solution of the associated dynamic programming equation. Finally we provide adetailed analysis on some specific examples.

This is a joint work with Zehra Eksi, Rüdiger Frey and Michaela Szölgyenyi.

14 Dicembre, 2016 13:15 oclock

Sezione di Analisi

On entire solutions of an elliptic system modelling phase separation

Nicola Soave, Politecnico di Milano

Aula seminari 3° piano

abstract

abstract

We consider a system of elliptic equations defined in the whole space, arising in the analysis of phase separation phenomena in singularly perturbed problems driven by strong competition. We present several results concerning existence and qualitative properties of the solutions, with particular emphasis to their symmetry and rigidity.

12 Dicembre, 2016 14:30 oclock

Sezione di Geometria, Algebra e loro applicazioni

Using spacial graphs to study reoccurring patterns of scrambled genes

Natasha Jonoska, University of South Florida

Aula seminari III piano

abstract

abstract

Nucleotide rearrangements occur at both evolutionary and developmental levels, and are often studied through model organisms such as ciliate species Oxytricha and Stylonychia. These processes can be modeled by 4-regular rigid vertex graphs, called assembly graphs. They are closely related to double occurrence words, chord diagrams, and circle graphs. Edges of these graphs represent double-stranded DNA molecules, while vertices correspond to DNA recombination sites. We present graph invariants of these assembly graphs and investigate genome-wide the range of scrambled gene architectures that describe the precursor-product relationships. We find that there are two general patterns, reoccurring genome wide, that describe over 90% of the Oxytricha’s scrambled genes. We further investigate the patterns of interleaving genes and find that there are specific star-like graph structures that describe most complex interleaving patterns.

30 Novembre, 2016 11:30

MOX Seminar

Modeling Inelastic Effects in Reconstituted Crosslinked F-Actin Networks

J. F. Rodriguez, Chemistry, Materials, and Chemical Engineering Department “Giulio Natta”, Politecnico di Milano

Aula Seminari "Saleri" VI Piano MOX-Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

The interplay between the conventional cross-linked actin filaments by means of actin-binding proteins, and physical bundling, by means of spatial organization of nucleating factors can be an alternative to understand several inelastic effects that take place in the cytoskeleton. In this regard, in vitro actin networks without active molecular motors have been considered to be in thermodynamic equilibrium[1]. However, recent experiments performed by Schmoller et
al. [5] in artificially reconstructed crosslinked actin networks have shown this networks to be non-equilibrium networks. In their experiments they have observed that, the internal stress trapped during network formation gives this material a unique behavior. Differently to most soft materials, such as rubber and living soft tissues, where nonlinear deformations irreversibly alter the mechanical properties of the material by causing a pronounced softening when cyclically deformed, reconstituted crosslinked actin networks show softening and hardening effects
when the network is subject to cyclic shear strain. As a continuation of a previous work [2], here we propose a constitutive model within the framework of continuum mechanics for the inelastic stress-strain behavior of reconstituted crosslinked actin networks. The network will be described using an homogenized framework based on the eight chain model [3]. A dynamic model for the crosslinks is introduced in order to account for the inelastic response
of the network. In this regard, we assume that crosslinks can be disrupted in either a reversible or irreversible manner. Reversible disruption of a crosslink is modeled as a two state process [4] in which the transition rates are modulated by the out of equilibrium forces. On the contrary, irreversible crosslink disruption will be modeled as a Bell-like bond rupture. These effects are introduced in the model by considering the bundle contour length, Lc as an
stochastic variable dependent on the reversible and irreversible dynamics of the crosslink.

References:
[1] O. Lieleg, J. Kayser, G. Brambilla, L. Cipelletti, A.R. Bausch, Slow dynamics and internal
stress relaxation in bundled cytoskeletal networks. Nat Mater, 10(3): 236–242,2011.
[2] H. Lopez, J.F. Rodriguez, Microstructural model for cyclic hardening in F-actin networks
crosslinked by -actinin. J. Mech. Phys. Solids, 91: 28–39, 2016.
[3] J.S. Palmer, M.C. Boyce, Constitutive modeling of the stress-strain behavior of F-actin
filament networks. Acta biomaterialia, 4(3): 597–612, 2008.
[4] R. Phillips, J. Kondev, J. Theriot, Physical Biology of the cell. Garland Science, 2009.
[5] K.M. Schmoller, P. Fernandez, R.C. Arevalo, D.L. Blair, A.R. Bausch, Cyclic hardening in bundled actin networks. Nat Commun 1, 2010.

contact: christian.vergara@polimi.it

30 Novembre, 2016 13:15 oclock

Sezione di Analisi

The Fisher-KPP problem with doubly nonlinear diffusion

Alessandro Audrito, Università degli Studi di Torino, Universidad Autonoma de Madrid

Aula seminari 3° piano

abstract

abstract

The famous Fisher-KPP reaction-diffusion model combines linear diffusion with the typical KPP reaction term, and appears in a number of relevant applications in biology and chemistry. It is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 <= u(x, t) <= 1 of the problem posed in the real line. The existence of propagation waves with finite speed has been confirmed in some related models and disproved in others. We investigate here the corresponding theory when the linear diffusion is replaced by the “slow” and “fast” doubly nonlinear diffusion. In the first case, we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions (we show a TW asymptotic behaviour also in the critical case that we call “pseudo-linear”, i.e., when the operator is still nonlinear but has homogeneity one. In the second one, we prove the non-existence of travelling waves while the propagation is exponentially fast in space for large times. Moreover, in the “fast” diffusion case, we show precise bounds for the level sets of general solutions.

23 Novembre, 2016 12:00 oclock

Sezione di Finanza Quantitativa

Advancing the universality of pricing methods for Asian options to any underlying process – applications in financial markets

Ioannis Kyriakou, Cass Business School

Aula Seminari III piano

abstract

abstract

We propose an accurate method for pricing arithmetic Asian options in a general model setting by means of a lower bound approximation. In particular, we derive analytical expressions for the lower bound in the Fourier domain. This is then recovered by a single univariate inversion and sharpened using an optimization technique.
Our proposed method can be applied to computing the prices and price sensitivities of Asian options with fixed or floating strike price, discrete or continuous averaging, under a wide range of stochastic dynamic models, including exponential Lévy, mean-reverting, affine stochastic volatility models, and the constant elasticity of variance diffusion. We present evidence of notable performance and robustness of our optimized lower bound for different test cases.
We then focus on applications in financial markets where the use of Asian options is prevalent, such as energy commodity and freight markets, exploiting the analytical tractability, computational competence and precision of our valuation framework. Asian options are favoured by risk managers being cost-saving hedging instruments and due to their ability to mitigate problems relating to market manipulation of the underlying in thinly traded markets.

22 Novembre, 2016 15:30

MOX Seminar

A coupling strategy for nonlocal and local diffusion models with applications to nonlocal elasticity

Marta D' Elia, Sandia National Laboratories, Dept 1441 Optimization and Uncertainty Quantification

Aula Seminari "Saleri" VI Piano MOX Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

The use of nonlocal models in science and engineering applications has been steadily increasing over the past decade. The ability of nonlocal theories to accurately capture effects that are difficult or impossible to represent by local Partial Differential Equation (PDE) models motivates and drives the interest in this type of simulations. However, the improved accuracy of nonlocal models comes at the price of a significant increase in computational costs compared to, e.g., traditional PDEs. In particular, a complete nonlocal simulation remains computationally untenable for many science and engineering applications. As a result, it is important to develop local-to-nonlocal coupling strategies, which aim to combine the accuracy of nonlocal models with the computational efficiency of PDEs. The basic idea is to use the more efficient PDE model everywhere except in those parts of the domain that require the improved accuracy of the nonlocal model. We develop and analyze an optimization-based method for the coupling of nonlocal and local problems with mixed volume constraints and boundary conditions. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. We prove that the resulting optimization problem is well-posed and discuss its implementation using Sandia's agile software components toolkit. Numerical results for nonlocal diffusion in one- and three-dimensions illustrate key properties of the optimization-based coupling method; these numerical tests provide the groundwork for the development of efficient and effective engineering analysis tools. As an application, we also present results for a static peridynamic model.

contact: simona.perotto@polimi.it

22 Novembre, 2016 11:00 oclock

Sezione di Finanza Quantitativa

Survival in Speculative Markets

Pietro Dindo,  Ca Foscari University of Venice

Aula Seminari III piano

The use of nonlocal models in science and engineering applications has been steadily increasing over the past decade. The ability of nonlocal theories to accurately capture effects that are difficult or impossible to represent by local Partial Differential Equation (PDE) models motivates and drives the interest in this type of simulations. However, the improved accuracy of nonlocal models comes at the price of a significant increase in computational costs compared to, e.g., traditional PDEs. In particular, a complete nonlocal simulation remains computationally untenable for many science and engineering applications. As a result, it is important to develop local-to-nonlocal coupling strategies, which aim to combine the accuracy of nonlocal models with the computational efficiency of PDEs. The basic idea is to use the more efficient PDE model everywhere except in those parts of the domain that require the improved accuracy of the nonlocal model. We develop and analyze an optimization-based method for the coupling of nonlocal and local problems with mixed volume constraints and boundary conditions. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. We prove that the resulting optimization problem is well-posed and discuss its implementation using Sandia's agile software components toolkit. Numerical results for nonlocal diffusion in one- and three-dimensions illustrate key properties of the optimization-based coupling method; these numerical tests provide the groundwork for the development of efficient and effective engineering analysis tools. As an application, we also present results for a static peridynamic model.

contact: simona.perotto@polimi.it

16 Novembre, 2016 13:15 oclock

Sezione di Analisi

Convergence of proximal gradient methods

Silvia Villa, Politecnico di Milano

Sala del consiglio 7° piano

abstract

abstract

First order methods have recently been widely applied to solve convex optimization problems in a
variety of areas including machine learning and signal processing.
In particular, proximal gradient algorithms (a.k.a. forward-backward splitting algorithms) and their
accelerated variants have received considerable attention. These algorithms are easy to implement
and suitable for solving high dimensional problems thanks to the low memory requirement of each iteration.
In this talk I will present some recent convergence results for this class of methods.