RSS-Feed der Veranstaltungen

Seminar für Analysis und Numerik

Fri, 08 Oct 2010 00:00:00 +0200

Computational seismology from crust to core: Forward and inverse modeling element and adjoint techniques Abstract:
Waves emanating from earthquakes are most widely used and successful to image the interior structure of the Earth. Similar to medical imaging, seismic tomography poses an inverse problem to understand seismic ground motion in terms of 3D wavespeeds which in turn are the basis for many dynamic and compositional constraints on the Earth.
I will briefly touch upon this general framework at scales from oil exploration to deep earth structure, and focus on the tasks to be performed, most importantly the solution to the forward and inverse problems.
Specifically, I will present spectral-element methods which have gained special popularity in the field, in conjunction with adjoint-state techniques to compute misfit gradients. Highlighting some of the most important methodological trade-offs and computational issues, I will round up with a short view on potential alternatives. Freitag, 08. 10. 2010, 9:00 Uhr im grossen Hörsaal
Mathematisches Institut
Rheinsprung 21, Basel

Minimal model program and explicit birational geometry of 3-folds

Thu, 14 Oct 2010 00:00:00 +0200

In this series of lectures (approximately 10) basic concepts of the minimal model program (MMP) in dimensions 2 and 3 shall be reviewed. The preliminary request is only Hartshorne, Chapter 1. Dates: Thursday, 14h. The first talk is on 14.10.2010. Room: Kleiner Hörsaal, Mathematisches Institut, Rheinsprung 21

Seminar für Analysis und Numerik

Fri, 15 Oct 2010 00:00:00 +0200

Iterative linear solvers for PDE-constrained Optimization problems involving fluid flow
Abstract:
The numerical approximation of Partial Differential Equation (PDE) problems leads typically to large dimensional linear or linearised systems of equations. For problems where such PDEs provide only a constraint on an Optimization problem (so-called PDE-constrained Optimization problems), the systems are many times larger in dimension.
We will discuss the solution of such problems by preconditioned iterative techniques in particular where the PDEs in question are the steady Stokes equations describing incompressible fluid flow and some very recent work on the time-dependent diffusion equation. Freitag, 15.10.2010, 9.00 Uhr
im Grossen Hörsaal Mathematisches Institut
Rheinsprung 21, Basel

Seminar in Analysis und Numerik

Fri, 19 Nov 2010 00:00:00 +0100

abstract: In [1] a globally convergent numerical method for a Coefficient Inverse Problems (CIP) for a hyperbolic PDE was developed. Next, a two-stage numerical procedure was proposed in [2,3,4]. In this procedure the technique of [1] is used as the first stage. Next, the Adaptive Finite Element method is used as the second stage for the refinement. In [5] the first stage was verified on blind experimental data. The goal of the current publication is to demonstrate that the two-stage numerical procedure
applied to the same experimental data can significantly improve imaging results compared with the first stage only. Specifically, we now accurately reconstruct not only locations and refractive indices of dielectric abnormalities, as it was in [5], but their shapes as well. The analytical part of this talk will be focused on two recommendations for the mesh refinement in a posteriori error analysis for the adaptivity technique.
[1] Beilina, L.; Klibanov, M. V.; 2008: A globally convergent numerical method for a coefficient inverse problem; SIAM J. Sci. Comp., 31, 478-509
[2] Beilina, L.; Klibanov, M. V.; 2010: Synthesis of global convergence and adaptivity for a hyperbolic coefficient inverse problem in 3D; J. Inverse and Ill-posed Problems, 18, 85-132
[3] Beilina, L.; Klibanov, M. V.; 2010: A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem; Inverse Problems, 26, 045012
[4] Beilina, L.; Klibanov, M. V.; Kokurin, M. Yu; 2010: Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem; Journal of Mathematical Sciences, Springer, 167, 279-325
[5] Klibanov, M. V.; Fiddy, M. A.; Beilina, L.; Pantong, N.; Schenk, J.; 2010: Picosecond scale experimental verification of a globally convergent numerical method for a coefficient inverse problem, Inverse Problems, 26, 045003, 2010. Freitag, 19. 11. 2010, 9.00 Uhr im Grossen Hörsaal
Mathematisches Institut
Rheinsprung 21, Basel

Jérôme Tambour: "LVMB manifolds and triangulations of spheres"

Fri, 25 Feb 2011 00:00:00 +0100

It is not easy to construct examples of non k ̈ahler compact complex manifold. For instance, every algebraic variety, and every Riemann surface, is ka ̈hler. The classical examples of such varieties are Hopf manifolds (1948) and Calabi-Eckmann manifolds (1953) which are complex structures on product of spheres Sp × Sq (with p and q odd). Santiago Lopez de Medrano, Alberto Verjovsky and Laurent Meersseman gave a generalization of this construction. The interest of theirs manifolds, known as LVM manifolds, stand in the fact that it is practical to compute some of their topological invariants (homology and cohomology for example). They are also endowed with a very nice action of a compact torus.

The talk will mainly deal with a generalization due to Bosio of the LVM manifolds, emphazing the combinatorial aspect of the LVM manifolds. These new manifolds are known as LVMB manifolds. In particular, our aim will be to show the very strong connection between LVMB manifolds, toric varieties and triangulations of spheres.

abstract

Perlen-Kolloquium: Michel Waldschmidt

Thu, 03 Mar 2011 00:00:00 +0100

On the so-called Fermat-Pell Equation

Nathan Ilten: "Upgrades and Downgrades of p-divisors"

Fri, 04 Mar 2011 00:00:00 +0100

A p-divisor on a normal variety Y is a divisor satisfying some positivity properties, where the usual integral or rational coefficients have been replaced by polyhedral coeffi- cients. K. Altmann and J. Hausen have shown that there is a correspondence between p-divisors and affine T-varieties, i.e. normal varieties together with some effective torus action. Given some T-variety X, one can try to ”upgrade” the torus action by considering some larger torus acting on X, or ”downgrade” the torus action by considering the action of some subtorus. I will discuss how these upgrading and downgrading procedures change the corresponding p-divisors. Time permitting, I will present an application of the upgrade procedure dealing with the p-divisors of Cox rings of certain T-varieties. This is joint work with R. Vollmert.

Marc Chardin: "Torsion in the symmetric algebra and images of rational maps"

Fri, 11 Mar 2011 00:00:00 +0100

In this lecture, I will present a way of computing the image of a rational map using a free resolution of the symmetric algebra. In this setting, the method was initiated by Jean-Pierre Jouanolou, and further developed by him, by Laurent Busé and by myself. The origin of this method is the work of people in Geometric Modeling, motivated by a simple question: how to represent the intersection of two surfaces parametrized by rational functions? Their approach was first put on firm mathematical bases by David Cox and several collaborators. The key point in this approach is to control the torsion in the symmetric algebra. This is performed using a construction of Herzog, Simis and Vasconcelos, that gives information on the equations of Rees algebras.

Seminar Algebra & Geometry: Lucas Fresse

Fri, 01 Apr 2011 00:00:00 +0200

On some geometric properties of orbital varieties To a nilpotent element x in a reductive Lie algebra, one can attach several algebraic varieties which play roles in representation theory: its nilpotent orbit; the intersection of its nilpotent orbit with a Borel subalgebra (the irreducible components of this intersection are called orbital varieties); the fiber over x of the Springer resolution. There is a close relation between the Springer fiber over x and the orbital varieties attached to x. In this talk, we rely on this relation in order to study two properties of orbital varieties: the smoothness, and the property to contain a dense B-orbit. We concentrate on type A. We provide several criteria which suggest that the two mentioned properties are related. This is a joint work with Anna Melnikov.

Seminar Algebra & Geometry: Enrico Schlesinger

Fri, 08 Apr 2011 00:00:00 +0200

On the problem of connectedness for the Hilbert schemes of space curves
Grothendieck has constructed the Hilbert scheme that parametrizes all subschemes of P^N with a given Hilbert polynomial, and Hartshorne in his thesis has shown that it is connected. For a curve in P³ the Hilbert polynomial is given by the degree d and by the arithmetic genus g. I will explain why, even if one is primarily interested in smooth curves, the correct class of curve to look at is that of locally Cohen Macaulay curves. These are parametrized by an open subscheme H_d,g of the full Hilbert scheme.

It is an open question whether H_d,g is connected whenever nonempty. This question was motivated by a result by
Martin-Deschamps and Perrin: they showed that H_d,g always has an irreducible component made up by "extremal curves"; these curves have the largest cohomology among curves in H_d,g. So there is no obstruction from semicontinuity that prevents the possibility that any smooth curve be specialized to an extremal curve.

I will discuss the state of affairs about this question, and briefly describe work in progress (with the help of Macaulay 2) showing that curves of type (a,a+4) on a smooth quadric surface are in the connected component of extremal curves; this problem was raised in Hartshorne's papers "On the connectedness of the Hilbert scheme of curves in P³" Comm. Alg. 28, 2000 and "Questions of connectedness of the Hilbert scheme of curves in P³" in the volume for Abhyankar's 70th (2004), and was still on the open problems list of the Workshop "Components of the Hilbert Schemes" (AIM Palo Alto 2010).