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.

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.

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.

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.

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.

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.

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.

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.

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.

Abstract: We review basic properties of the degenerate and singular evolution equation

u_t=(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.

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.

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).

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.

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.

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.