High frequency surface estimation

Mountain generation can be considered a part of the general theory of surface estimation; In this thesis, two methods have been presented to generate fractals--fast Fourier method and a new generalized stochastic subdivision method. Also, a new surface estimation method has been introduced that deals with points of unequal powers. The uniqueness of this method is the usage of splines to calculate the arc lengths between the points, as opposed to Euclidean distances used in Kriging. The fast Fourier technique has been used to generate mountains in particular; also, some extensions have been suggested, whereby different sets of mountains can be obtained by modifying some parameters. This method is global and has the advantages of simplicity and efficiency; it also provides exact spectral control. The search for a more localized method resulted in the new generalized stochastic subdivision technique. The choice of an autocorrelation function is pivotal here. The only significant differences between the fractal subdivision method and this new technique are the increased neighborhood size, boundary conditions and the need to solve a system of equations for each subdivision level; The source code for these techniques was implemented on SGI machines, using C with GL as a graphics standard

Entropy Production of Doubly Stochastic Quantum Channels

We study the entropy increase of quantum systems evolving under primitive, doubly stochastic Markovian noise and thus converging to the maximally mixed state. This entropy increase can be quantified by a logarithmic-Sobolev constant of the Liouvillian generating the noise. We prove a universal lower bound on this constant that stays invariant under taking tensor-powers. Our methods involve a new comparison method to relate logarithmic-Sobolev constants of different Liouvillians and a technique to compute logarithmic-Sobolev inequalities of Liouvillians with eigenvectors forming a projective representation of a finite abelian group. Our bounds improve upon similar results established before and as an application we prove an upper bound on continuous-time quantum capacities. In the last part of this work we study entropy production estimates of discrete-time doubly-stochastic quantum channels by extending the framework of discrete-time logarithmic-Sobolev inequalities to the quantum case.Comment: 24 page

On the discretization of backward doubly stochastic differential equations

In this paper, we are dealing with the approximation of the process (Y,Z) solution to the backward doubly stochastic differential equation with the forward process X . After proving the L2-regularity of Z, we use the Euler scheme to discretize X and the Zhang approach in order to give a discretization scheme of the process (Y,Z)