In optical tomography, conventionally the diffusion approximation (DA) to the radiative transport equation (RTE) with a constant refractive index is used. The existence, uniqueness, and non-uniqueness of the forward and the inverse problem using conventional DA has been studied in the past. In this report, we investigate the existence, uniqueness, and non-uniqueness of optical tomography forwar...

In this paper we consider the two-dimensional linear problem of wave–body interaction with surface tension effects being taken into account. We suggest a criterion for unique solvability of the problem based on symmetrization of boundary integral equations. The criterion allows us to develop an algorithm for detecting non-uniqueness (finding trapped modes) for given geometries of bodies; exampl...

We construct solutions to vector valued Burgers type equations perturbed by a multiplicative space-time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to...

Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the wea...

This master’s thesis presents a proven-correct implementation of a distributed topology construction algorithm based upon the Thallner topology construction method for creating a minimal ∆-node connected fault-tolerant overlay graph. The algorithm works in asynchronous fault-tolerant distributed systems augmented with failure detectors. A detailed proof shows that given a perfect propose module...

The input–output block decoupling problem by state feedback is studied for linear time-varying singular systems. First, an algorithm, the regularization algorithm, is developed such that the system can be made by state feedback to have a unique impulse-free solution. Second, another algorithm, the block decoupling algorithm, is proposed, which provides sufficient conditions for the solvability ...

We consider residual-based stabilised finite element methods for the generalised Oseen problem. The unique solvability based on a modified stability condition and an error estimate are proved for inf-sup stable discretisations of velocity and pressure. The analysis highlights the role of an additional stabilisation of the incompressibility constraint. It turns out that the stabilisation terms o...

We prove some Liouville properties for suband supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in OrnsteinUhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully n...

We consider boundary value problems and transmission problems for strongly elliptic second-order systems with boundary conditions on a compact nonclosed Lipschitz surface S with Lipschitz boundary. The main goal is to find conditions for the unique solvability of these problems in the spaces H , the simplest L2-spaces of the Sobolev type, with the use of potential type operators on S . We also ...