Numerical solution of conservative finite-dimensional stochastic Schrodinger equations

The paper deals with the numerical solution of the nonlinear Ito stochastic differential equations (SDEs) appearing in the unravelling of quantum master equations. We first develop an exponential scheme of weak order 1 for general globally Lipschitz SDEs governed by Brownian motions. Then, we proceed to study the numerical integration of a class of locally Lipschitz SDEs. More precisely, we adapt the exponential scheme obtained in the first part of the work to the characteristics of certain finite-dimensional nonlinear stochastic Schrodinger equations. This yields a numerical method for the simulation of the mean value of quantum observables. We address the rate of convergence arising in this computation. Finally, an experiment with a representative quantum master equation illustrates the good performance of the new scheme.Comment: Published at http://dx.doi.org/10.1214/105051605000000403 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Geography of Fields in Extra Dimensions: String Theory Lessons for Particle Physics

String theoretical ideas might be relevant for particle physics model building. Ideally one would hope to find a unified theory of all fundamental interactions. There are only few consistent string theories in D=10 or 11 space-time dimensions, but a huge landscape in D=4. We have to explore this landscape to identify models that describe the known phenomena of particle physics. Properties of compactified six spatial dimensions are crucial in that respect. We postulate some useful rules to investigate this landscape and construct realistic models. We identify common properties of the successful models and formulate lessons for further model building.Comment: To be published in "Perspectives on String Phenomenology" (World Scientific

Numerical evaluation of two and three parameter Mittag-Leffler functions

The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method for the efficient computation of the ML function based on the numerical inversion of its Laplace transform (LT): an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the LT, with the aim of minimizing the computational effort and reduce the propagation of errors. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. The application to the three parameter ML (also known as Prabhakar) function is also presented.Comment: Accepted for publication in SIAM Journal on Numerical Analysi

Two-points problem for an evolutional first order equation in Banach space

Two-point nonlocal problem for the first order differential evolution equation with an operator co- efficient in a Banach space X is considered. An exponentially convergent algorithm is proposed and justified in assumption that the operator coefficient is strongly positive and some existence and unique- ness conditions are fulfilled. This algorithm leads to a system of linear equations that can be solved by fixed-point iteration. The algorithm provides exponentially convergence in time that in combination with fast algorithms on spatial variables can be efficient treating such problems. The efficiency of the proposed algorithms is demonstrated by numerical examples