The topic is differential equations (both partial and ordinary differential equations).
Unless otherwise noted, talks were Tuesdays, 12:15–14:00 in room 734, S-2.
Date | Speaker | Title |
---|---|---|
2008-01-08 12:15–13:00 |
Michael V. Klibanov | Globally convergent numerical method for a class of coefficient inverse problems
Abstract:
The development of globally convergent numerical methods
for coefficient inverse problems
has vital importance for the field of Inverse Problems.
The vast majority of current numerical methods converge locally.
In this talk a new globally convergent method will be presented
for a class of inverse problems for hyperbolic and parabolic equations.
This method has been developed during the entire year of 2007
by L. Beilina and M.V. Klibanov.
In addition, a new numerical method will be presented for
the popular problem of thermoacoustic tomography. |
2008-01-29 12:15–13:00 |
Peter Lindqvist | About positive eigenfunctions
Abstract:
If an eigenfunction of the Laplacian is positive in a bounded
domain and has boundary values zero
(the Dirichlet boundary value problem),
then it is unique, if normalized,
and the associated eigenvalue must be the principal frequency.
We present a simple proof of these facts
even for a non-linear eigenvalue problem.
The talk is rather elementary. |
2008-02-26 12:15–13:00 |
Espen R. Jakobsen | Error estimates for monotone approximation schemes of fully non-linear integro-PDEs
Abstract:
I will discuss some resent results on error bounds for monotone approximations
of convex fully non-linear 2nd order integro-PDEs
arising in optimal control theory.
These equations are degenerate and may contain
both Laplacian and fractional Laplacian type of terms,
and they appear e.g. in financial models.
Monotone schemes or schemes of positive type
are the only type of schemes that have been proved
to converge in this setting.
Nonmonotone schemes may not converge or converge to a wrong solution.
The task of proving error estimates is much more difficult
than that of proving convergence,
and I will do so using a sophisticated smoothing technique
developed by Krylov, Barles and myself for pure PDEs. The talk is based on several paper joint with Biswas, Camilli, Karlsen, and La Chioma. |
2008-03-04 12:15–13:15 |
Alex Hansen | Towards a Thermodynamic Description of Steady-State
Two-Phase Flow in Porous Media
Abstract:
Steady-state two-phase flow in porous media has received
very little attention compared to the instabilities that
occur in connection with flooding. At low flow rates,
steady state flow essentially consists of one fluid being
held in place by capillary forces whereas the other fluid flows.
However, at higher flow rates, both fluids will move and there will
be a incessant breakup and merging of fluid clusters. In this
regime, the flow settles to state which is independent of the initial
conditions. This opens for a statistical description of the flow
which is closely related to that of statistical mechanics and
thermodynamics.
We discuss this approach, based on a numerical model, both from a general point of view and using specific examples. In particular we will discuss the stability of the interface between two immiscible fluids flowing in parallel in a porous medium. |
2008-03-11 12:15–13:00 |
Bernhard Müller | Towards Higher Accuracy in Computational Fluid Dynamics
Abstract:
High order methods have increasingly been employed in computational
fluid dynamics, because they are more efficient than low order
methods for high accuracy requirements.
A brief review of recent developments in high order methods
is given with an emphasis on compressible flow computations.
Suitable forms of the compressible Navier-Stokes equations
and strictly stable high order finite difference methods
are presented. Their applications to the numerical simulation
of aeolian tones and of human phonation are discussed.
|
2008-03-11 13:15–14:00 |
Tuomo Kuusi | Lower semicontinuity of weak supersolutions to parabolic equations
Abstract:
We present a concise proof that gives the lower semicontinuity
of weak supersolutions. The result complements the theory of so-called
superparabolic functions, i.e. the parabolic counterpart for
superharmonic functions.
|
2008-04-15 12:15–13:00 |
Riikka Korte | The Obstacle Problem for a nonlinear parabolic equation
We construct a continuous solution
to the obstacle problem for evolutionary p-Laplace type equations.
Our method is similar to the classical Schwarz alternating method.
It demonstrates that the obstacle problem has a continuous solution
if and only if the corresponding Dirichlet boundary value problem
has a continuous solution. |
2008-04-15 13:15–14:00 |
Mats Ehrnström | Rotational water waves:
the nonlinear influence of vorticity on maximum principles
and related properties
The study of travelling water waves
classically concerns the existence and the properties
of a harmonic function satisfying a free-boundary problem.
When vorticity is present – so that the velocity field
has a nonvanishing curl – the main equation becomes nonlinear.
In this talk we show some of the differences and similarities
between the setting of irrotational and that of rotational flow.
In particular we show how many properties based on maximum principles
can be extended to water waves with vorticity,
and indicate in other circumstances what seems to be the problem.
Most properties discussed are geometrical in the sense that
they describe e.g. symmetry, the shape of the wave,
and the paths traversed by the fluid particles. |
2008-04-22 12:15–13:00 |
Hilde Sande | The solution of the Cauchy problem with large data for a model of a mixture of gases
Abstract:
We consider a 3×3 system of hyperbolic conservation laws
describing the one dimensional flow of a mixture of isentropic gases.
The system is strictly hyperbolic
and each family is either genuinely nonlinear or linear degenerate.
Thus, we have global existence of a weak solution of
the Cauchy problem with small initial data due to Glimm,
and the solution can be found by using either the Glimm scheme
or front tracking.
We extend the existence result for this system to larger initial data. In particular, we can by reducing the total variation and the supremum of one variable, allow large total variation for the other two variables. We show this both using the Glimm scheme and front tracking. For front tracking we are able to show that the number of fronts is finite in finite time, and therefore we do not need to remove any fronts or introduce non-physical fronts. The key point in both methods is choosing a suitable Glimm functional and show that it is decreasing in time. This requires detailed analysis of all possible interactions. We present a technique of splitting an interaction into simpler interactions in order to obtain the needed estimates. We give numerical examples both using the Glimm scheme and front tracking. |
2008-04-22 13:15–14:00 |
Petri Juutinen | On the evolution governed by the infinity Laplacian
Abstract:
We review basic properties
of the degenerate and singular evolution equation
ut=(D2u(Du/|Du|))·(Du/|Du|), which is a parabolic version of the increasingly popular infinity Laplace equation. We discuss existence and uniqueness of solutions, interior and boundary Lipschitz estimates, and Harnack's inequality. A characterization involving “fundamental solutions” is also given. |
2008-04-29 12:15–13:00 |
Marte Hatlo | Adaptive FEM for Inverse Electromagnetic Scattering
Abstract:
We apply a mesh-adaptive method to an inverse electromagnetic scattering
problem. The method is based on an a posteriori error estimate which
couples residuals of the computed solution to weights in the
reconstruction. The new element in the present work is the introduction
of absorbing and mirror boundary conditions in the formulation of the
forward problem, and thus a new a posteriori error estimate.
The inverse problem consists of reconstructing the dielectric permittivity, ε(x), from data measured on parts of the surface of the given domain, given the wave input on other parts. By solving the wave equation with the same input, the material variables are in principle obtained by fitting the computed solution to the measured data. The inverse problem is formulated as an optimal control problem, where we solve equations expressing stationarity of an associated Lagrangian. An a posteriori error estimate for the Lagrangian couples residuals of the computed solution to interpolation errors and steers the adaptivity of the finite element mesh. The method is hybrid in the sense that different numerical methods, finite elements and finite differences, are used in different parts of the computational domain. Lagrangian couples residuals of the computed solution to interpolation errors and steers the adaptivity of the finite element mesh. The method is hybrid in the sense that different numerical methods, finite elements and finite differences, are used in different parts of the computational domain. The usefulness of the adaptive error control is shown in numerical examples where a two dimensional structure is recovered using data measured at the boundary. |
2008-04-29 13:15–14:00 |
Harald Hanche-Olsen | Differential inclusions
Abstract:
The purpose of this talk is not to discuss current research, but to
explain some basic, well established, but little known facts.
Differential inclusions are a way to make sense of differential
equations of the form
ẋ=f(x,t)
where f may be discontinuous in x.
We shall cover the basic definitions and the existence theorem, and
give suitable conditions for uniqueness. If time permits we shall
discuss applications to generalized characteristics for solutions of
conservation laws. Sources: A. F. Filippov (Differential equations
with discontinuous righthand sides) and C. M. Dafermos (Hyperbolic
conservation laws in continuum physics).
|
2008-05-06 12:15–13:00 |
Teemu Lukkari | Removability results for solutions of nonlinear elliptic equations |
2008-05-13 12:15–13:00 |
Marte Godvik | Solutions for the Aw–Rascle model with vacuum
Abstract:
We will consider the Aw–Rascle traffic flow model with vacuum.
The model is a 2×2 system of hyperbolic conservations laws.
The main difficulty is that the presence of vacuum
makes us unable to control the total variation
of the conservative variables.
By considering the model in Eulerian form we show existence of weak entropy solutions of the Cauchy problem. Our strategy is to consider slightly modified systems for which we can control the total variation and thus get existence results by the Glimm scheme. A uniqueness result is established from a Kružkov-type entropy condition by considering the model in Lagrangian form. |
2008-05-27 12:15–13:00 |
Giuseppe Coclite | Conservation Laws with Singular Nonlocal Sources
Abstract:
In this lecture we consider a one-dimensional
hyperbolic conservation law with integral source
that contains a singular nonlinear term in the origin.
In several gas-dynamics and traffic models
we have fluxes or sources depending on the
reciprocal of the density: our equation is an
integral regularization of such models.
The sharp assumptions on the integral kernel are satisfied
by several Green's functions of elliptic problems,
in these cases our equation is equivalent to an
hyperbolic-elliptic system similar to the ones
associated to the Camassa–Holm, Degasperis–Procesi
and radiating gases models.
We work on the initial-boundary value problem with homogenous
Dirichlet boundary conditions
and prove the existence of weak solutions
that are almost everywhere positive.
The results were obtained in collaboration with
Professor Mario M. Coclite.
|
2008-06-24 13:15–14:00 |
Fritz Gesztesy | Some Remarks on Generalized Polar Decompositions of Closed Hilbert Space Operators
Abstract:
We study (vast) generalizations of the polar decomposition
of densely defined, closed linear operators in Hilbert spaces.
Some applications to relatively (form) bounded
and relatively (form) compact perturbations
of self-adjoint (resp., normal and sectorial)
operators will be given.
This is based on recent joint wok with Mark Malamud, Marius Mitrea, and Sergey Naboko. |