2,060 research outputs found
Fully faithful Fourier-Mukai functors and generic vanishing
The aim of this mainly expository note is to point out that, given an
Fourier-Mukai functor, the condition making it fully faithful is an instance of
\emph{generic vanishing}. We test this point of view on some fairly classical
examples, including the strong simplicity criterion of Bondal and Orlov, the
standard flip and the Mukai flop.Comment: In memory of Alexandru T. Lasc
Exponential Runge-Kutta methods for stiff kinetic equations
We introduce a class of exponential Runge-Kutta integration methods for
kinetic equations. The methods are based on a decomposition of the collision
operator into an equilibrium and a non equilibrium part and are exact for
relaxation operators of BGK type. For Boltzmann type kinetic equations they
work uniformly for a wide range of relaxation times and avoid the solution of
nonlinear systems of equations even in stiff regimes. We give sufficient
conditions in order that such methods are unconditionally asymptotically stable
and asymptotic preserving. Such stability properties are essential to guarantee
the correct asymptotic behavior for small relaxation times. The methods also
offer favorable properties such as nonnegativity of the solution and entropy
inequality. For this reason, as we will show, the methods are suitable both for
deterministic as well as probabilistic numerical techniques
Multi-scale control variate methods for uncertainty quantification in kinetic equations
Kinetic equations play a major rule in modeling large systems of interacting
particles. Uncertainties may be due to various reasons, like lack of knowledge
on the microscopic interaction details or incomplete informations at the
boundaries. These uncertainties, however, contribute to the curse of
dimensionality and the development of efficient numerical methods is a
challenge. In this paper we consider the construction of novel multi-scale
methods for such problems which, thanks to a control variate approach, are
capable to reduce the variance of standard Monte Carlo techniques
Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality
We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis
inequality for surfaces. The proof is based on the theory of Generic Vanishing
and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative
algebra. Along the way we give a positive answer, in the setting of K\"ahler
manifolds, to a question of Green-Lazarsfeld on the vanishing of higher direct
images of Poincar\'e bundles. We indicate generalizations to arbitrary
Fourier-Mukai transforms.Comment: 12 pages; some improvements according to suggestions from the
referees, to appear in Duke Math.
Asymptotic preserving Implicit-Explicit Runge-Kutta methods for non linear kinetic equations
We discuss Implicit-Explicit (IMEX) Runge Kutta methods which are
particularly adapted to stiff kinetic equations of Boltzmann type. We consider
both the case of easy invertible collision operators and the challenging case
of Boltzmann collision operators. We give sufficient conditions in order that
such methods are asymptotic preserving and asymptotically accurate. Their
monotonicity properties are also studied. In the case of the Boltzmann
operator, the methods are based on the introduction of a penalization technique
for the collision integral. This reformulation of the collision operator
permits to construct penalized IMEX schemes which work uniformly for a wide
range of relaxation times avoiding the expensive implicit resolution of the
collision operator. Finally we show some numerical results which confirm the
theoretical analysis
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