Quaderni MOX
Pubblicazioni
del Laboratorio di Modellistica e Calcolo Scientifico MOX. I lavori riguardano prevalentemente il campo dell'analisi numerica, della statistica e della modellistica matematica applicata a problemi di interesse ingegneristico. Il sito del Laboratorio MOX è raggiungibile
all'indirizzo mox.polimi.it
Trovati 1349 prodotti
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01/2009 - 15/01/2009
Miglio, Edie; Sgarra, Carlo;
A Finite Element Framework for Option Pricing with the Bates Model | Abstract | | In the present paper we present a finite element approach for option pricing in the framework of a well-known stochastic volatility model with
jumps, the Bates model. In this model the asset log-returns are assumed to follow a jump-diffusion model where the jump component consists of a Lévy process of compound Poisson type, while the volatility behavior is described by a stochastic differential equation of CIR type, with a meanreverting drift term and a diffusion component correlated with that of the
log-returns. Like in all the Lévy models, the option pricing problem can be formulated in terms of an integro-differential equation: for the Bates model the unknown F(S, V, t) (the option price) of the pricing equation depends on three independent variables and the differential operator part turns out to be of parabolic kind, while the nonlocal integral operator is calculated with respect to the Lévy measure of the jumps. In this paper we will present a variational formulation of the problem suitable for a finite element approach. The numerical results obtained for european options will be compared with those obtained with different methods. |
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28/2008 - 10/12/2008
D'Angelo, Carlo; Quarteroni, Alfio
On the coupling of 1D and 3D diffusion-reaction equations. Applications to tissue perfusion problems | Abstract | | In this paper we consider the coupling between two
diffusion-reaction problems, one taking place in a three-dimensional
domain, the other in a one-dimensional subdomain.
This coupled problem is the simplest model of fluid flow in a
three-dimensional porous medium featuring fractures that can be
described by one-dimensional manifolds. In particular this model
can provide the basis for a multiscale analysis of blood flow
through tissues, in which the capillary network is represented as a
porous matrix, while the major blood vessels are described by thin
tubular structures embedded into it: in this case, the model allows
the computation of the 3D and 1D blood pressures respectively in the
tissue and in the vessels.
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27/2008 - 25/11/2008
Quarteroni, Alfio
Mathematical Models in Science and Engineering
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26/2008 - 24/11/2008
Aletti, Giacomo; May, Caterina; Secchi, Piercesare
A Central Limit Theorem, and related results, for two-color randomly reinforced urn | Abstract | | We prove a Central Limit Theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are
able to show that the distribution of the urn limit composition has no point
masses. |
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25/2008 - 20/11/2008
Detomi, Davide; Parolini, Nicola; Quarteroni, Alfio
Mathematics in the wind
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24/2008 - 19/11/2008
Bacchelli, Valeria; Veneziani, Alessandro; Vessella, Sergio
Corrosion detection in a 2D domain with a polygonal boundary | Abstract | | We consider the problem of quantitative non-destructive evaluation
of corrosion in a 2D domain representing a thin metallic plate. Corro-
sion damage is assumed to occur in an inaccessible part of the domain.
Reconstruction of the damaged profile is possible by measuring an electro-
static current properly induced by a potential in an accessible part of the
boundary (electrical impedance tomography). We present here numerical
methods and results based on a formulation of the problem introduced
and analyzed in Bacchelli-Vessella, Inverse Problems, 22 (2006), where
the corroded profile is represented by a polygonal boundary. We resort
in particular to the Landweber method and the Brakhage semi-iterative
scheme. Numerical results show the reliability of this approach in general
situations, including nongraph corroded boundaries |
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23/2008 - 28/10/2008
Hysing, Shu-Ren; Turek, Stefan; Kuzmin, Dmitri; Parolini, Nicola; Burman, Erik; Ganesan, Sashikumaar; Tobiska, Lutz
Quantitative benchmark computations of two-dimensional bubble dynamics | Abstract | | Benchmark configurations for quantitative validation and comparison of incompressible interfacial flow
codes, which model two-dimensional bubbles rising in liquid columns, are proposed. The benchmark
quantities: circularity, center of mass, and mean rise velocity are defined and measured to monitor
convergence towards a reference solution. Comprehensive studies are undertaken by three independent
research groups, two representing Eulerian level set finite element codes, and one representing an ALE
moving grid approach.
The first benchmark test case considers a bubble with small density and viscosity ratios which
undergoes moderate shape deformation. The results from all codes agree very well allowing for target
reference values to be established. For the second test case, a bubble with a very low density compared
to that of the surrounding fluid, the results for all groups are in good agreement up to the point of
break up, after which all three codes predict different bubble shapes. This highlights the need for the
research community to invest more effort in obtaining reference solutions to problems involving break
up and coalescence.
Other research groups are encouraged to participate in these benchmarks by contacting the authors
and submitting their own data. The reference data for the computed benchmark quantities can also
be supplied for validation purposes |
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22/2008 - 23/10/2008
Nobile, Fabio; Tempone, Raul
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients | Abstract | | We consider the problem of numerically approximating statistical moments
of the solution of a time dependent linear parabolic partial differential
equation (PDE), whose coefficients and/or forcing terms are spatially
correlated random fields. The stochastic coefficients of the PDE are approximated
by truncated Karhunen-Lo eve expansions driven by a finite number
of uncorrelated random variables. After approximating the stochastic coefficients
the original stochastic PDE turns into a new deterministic parametric
PDE of the same type, the dimension of the parameter set being
equal to the number of random variables introduced.
After proving that the solution of the parametric PDE problem is analytic
with respect to the parameters, we consider global polynomial approximations
based on tensor product, total degree or sparse polynomial spaces
and constructed by either a Stochastic Galerkin or a Stochastic Collocation
approach. We derive convergence rates for the different cases and present
numerical results that show how these approaches are a valid alternative to
the more traditional Monte Carlo Method for this class of problems |
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