The mathematical formulation of many physical problems results in the task of inverting a compact operator. The only known sensible solution technique is regularization which poses a severe problem in itself. Classically one dealt with deterministic noise models and required the knowledge of smoothness of the solution or the overall error behaviour. We will show that we can guarantee an asymptotically almost optimal regularization for a physically motivated noise model under no assumptions for the smoothness and rather weak assumptions on the noise behaviour. An application to the determination of the gravitational field out of satellite data will be shown.

The paper presents the results of an asymptotic theory of axially symmetric cavity flows at small and zero cavitation number. The results have been obtained on the basis of the variational Riabouchinsky principle and the asymptotic theory of slender bodies. This variational-asymptotic approach has been applied to deduce asymptotic expansions for the shape of the cavity and the force exerted on the cavitator at small cavitation number. At the zero cavitation number an integro-differential equation for the shape of the free streamline has been obtained. An exact integral of the equation has been found and a one-parameter family of solutions has been constructed and which has refined earlier asymptotics of Levinson and Gurevich. The equation and asymptotic expansion are independent of the cavitator shape and in this sense are maximally accurate. Any further amendment of the equation of higher order of accuracy would be connected with the shape of the cavitator.

We consider the problem of modelling the flow of a slightly compressible fluid in a periodic fractured medium assuming that the fissures are thin with respect to the block size. As a starting point we used a formulation applied to a system comprising a fractured porous medium made of blocks and fractures separated by a thin layer which is considered as an interface. The inter-relationship between these three characteristics comprise the triple porosity model. The microscopic model consists of the usual equation describing Darcy flow with the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by $(\varepsilon \delta)^2$, where $\varepsilon$ is the size of a typical porous block, with $\delta$ representing the relative size of the fracture. We then consider a model with Robin type transmission conditions: a jump of the density across the interface block-fracture is taken into account and proportional to the flux by the mean of a function $(\varepsilon\delta)^{-\gamma}$, where $\gamma$ is a parameter. Using two-scale convergence, we get homogenized models which govern the global behaviour of the flow as $\varepsilon$ and $\delta$ tend to zero. The resulting homogenized problem is a dual-porosity type model that contains a term representing memory effects for $\gamma\le 1$, and it is a single porosity model with effective coefficients for $\gamma >1$.

In this paper a one-dimensional parabolic variational inequality which typically arises in option pricing of fixed rate mortgage loan is studied. The main goal is to study the properties of the free boundary. The monotonicity and $C^\infty$ smoothness of free boundary are proved and its behavior near expiry is considered as well.

We study the effect of stochastic volatility on option prices. In the fast mean-reversion model for stochastic volatility of [5], we show that there is a full asymptotic expansion for the option price, centered at the Black-Scholes price. We show how to callibrate the first two terms in the expansion with the implied volatility surface. We show, however, that this price does not converge in a strong sense to Black-Scholes as the mean-reversion rate increases.

We formulate a number of open problems for time-harmonic inverse electromagnetic scattering theory focusing on uniqueness theorems, the determination of the support of a scattering object and the determination of material parameters