Todor V. Gurov

A new iterative Monte Carlo approach for inverse matrix problem

A new approach of iterative Monte Carlo algorithms for the well-known inverse matrix problem is presented and studied. The algorithms are based on a special techniques of iteration parameter choice, which allows to control the convergence of the algorithm for any column (row) of the matrix using different relaxation parameters. The choice of these parameters is controlled by a posteriori criteria for every Monte Carlo iteration. The presented Monte Carlo algorithms are implemented on a SUN Sparkstation. Numerical tests are performed for matrices of moderate in order to show how work the algorithms. The algorithms under consideration are well parallelized.

Femtosecond relaxation of hot electrons by phonon emission in presence of electric field

Abstract The femtosecond relaxation of an initial distribution of electrons which interact with phonons in presence of applied electric field is studied numerically. The evolution at such a time scale cannot be described in terms of Boltzmann transport. Here, the Barker–Ferry equation is utilized as a quantum-kinetic model of the process. The numerical treatment of the original formulation of the Barker–Ferry equation becomes difficult since coordinates and time variables are coupled by the field. A transformation which decouples coordinates and time variables in the equation is proposed. A randomized iterative Monte Carlo algorithm is developed to solve the transformed equation. The quantum character of the equation is investigated. An instantaneously created initial condition is favored above the physically more adequate generation term in order to point out the quantum effects. Simulation results are obtained for GaAs material at different evolution times. Effects of collisional broadening and retardation are observed already in the fieldless case. The intracollisional field effect is clearly demonstrated as an effective change of the phonon energy, which depends on the field direction and the evolution time. Moreover, the collisional broadening and retardation are affected by the applied field. The observed phenomena are understood from the structure and the properties of the model equation.

Quasi-Monte Carlo integration on the grid for sensitivity studies

In this paper we present error and performance analysis of quasi-Monte Carlo algorithms for solving multidimensional integrals (up to 100 dimensions) on the grid using MPI. We take into account the fact that the Grid is a potentially heterogeneous computing environment, where the user does not know the specifics of the target architecture. Therefore parallel algorithms should be able to adapt to this heterogeneity, providing automated load-balancing. Monte Carlo algorithms can be tailored to such environments, provided parallel pseudorandom number generators are available. The use of quasi-Monte Carlo algorithms poses more difficulties. In both cases the efficient implementation of the algorithms depends on the functionality of the corresponding packages for generating pseudorandom or quasirandom numbers. We propose efficient parallel implementation of the Sobol sequence for a grid environment and we demonstrate numerical experiments on a heterogeneous grid. To achieve high parallel efficiency we use a newly developed special grid service called Job Track Service which provides efficient management of available computing resources through reservations.

Estimates of the computational complexity of iterative Monte Carlo algorithm based on Green's function approach

In this work an iterative Monte Carlo algorithm for solving elliptic boundary value problems is studied. The algorithm uses the local integral presentation by Greens function. The integral transformation kernel is obtained applying the adjoint operator on Levys function. Such a kernel can be used as a transition density function of a Markov process for estimating the solution. The studied approach leads to a random process, which is called a ball process. The corresponding Monte Carlo algorithm is presented. This algorithm is similar to the well-known grid-free spherical process used for solving simple elliptic problems, however instead of moving to a random point on the sphere, a move is made to a point into a maximal ball, which is located “not far from the boundary of the ball”. The selection Monte Carlo algorithm for solving the above mentioned problem is described. An estimation for the efficiency of the selection Monte Carlo algorithm is obtained. The estimate of the averaged number of moves for reaching the ϵ-strip of the boundary of the domain for the studied random process is obtained. It is proved that the algorithm efficiency depends on the radius of the maximal ball, lying inside the domain Ω in which the problem is defined and on the parameters of the operator under consideration. Some numerical examples are performed. The results show that the obtained theoretical estimates can be used for a wide class of elliptic boundary value problems.

Ultra-fast Semiconductor Carrier Transport Simulation on the Grid

We consider the problem of computer simulation of ultra-fast semiconductor carrier transport. The mathematical description of this problem includes quantum kinetic equations whose approximate solving is a computationally very intensive problem. In order to reduce the computational cost we use recently developed Monte Carlo methods as a numerical approach. We study intra-collision field effect, i.e. effective change of phonon energy, which depends on the field direction and the evolution time. In order to obtain results for different evolution times in a reasonable time-frame, we implement simulation on the computational grid. We split the task into thousands of subtasks (jobs) which are sent to different grid sites to be executed. In this paper we present new results for inhomogeneous case in the presence of electric field, and we describe our grid implementation scheme.

Femtosecond evolution of spatially inhomogeneous carrier excitations part II: stochastic approach and grid implementation

We present a stochastic approach for solving the quantum-kinetic equation introduced in Part I. A Monte Carlo method based on backward time evolution of the numerical trajectories is developed. The computational complexity and the stochastic error are investigated numerically. Variance reduction techniques are applied, which demonstrate a clear advantage with respect to the approaches based on symmetry transformation. Parallel implementation is realized on a GRID infrastructure.

Femtosecond evolution of spatially inhomogeneous carrier excitations part i: kinetic approach

The ultrafast evolution of optically excited carriers which propagate in a quantum wire and interact with three dimensional phonons is investigated. The equation, relevant to this physical problem, is derived by a first principle approach. The electron-phonon interaction is described on a quantum-kinetic level by the Levinson equation, but the evolution problem becomes inhomogeneous due to the spatial dependence of the initial condition. The initial carrier distribution is assumed Gaussian both in energy and space coordinates, an electric field can be applied along the wire. A stochastic method, described in Part II of the work, is used for solving the equation. The obtained simulation results characterize the space and energy dependence of the evolution in the zero field case. Quantum effects introduced by the early time electron-phonon interaction are analyzed.

A Parallel Monte Carlo Method for Electron Quantum Kinetic Equation

We study a parallel Monte Carlo (MC) method for investigation of a quantum kinetic equation which accounts for the action of the electric field during the process of electron-phonon interaction. Optimization of the presented parallel algorithm is done using variance reduction techniques and parallel random sequences from the Scalable Parallel Random Number Generator (SPRNG) library. The developed code written in C is parallelized with MPI and OpenMP codes.

Statistical Algorithms for Simulation of Electron Quantum Kinetics in Semiconductors - Part II

In this work we solve the Barker-Ferry equation which accounts for the quantum character of the electron-phonon interaction in semiconductors in the framework of the Monte Carlo (MC) method. The first part of the work considers the zero electric field formulation of the equation in spherical coordinates. Different MC algorithms for solving the equation are suggested and investigated.In the second part of the work we consider the case of an applied electric field. It is shown that the second algorithm from the first part can be successfully modified to account for the cylindrical symmetry of the task.

Monte carlo grid application for electron transport

In this paper we present a Grid application developed for electron transport problems called SALUTE (Stochastic ALgorithms for Ultra-fast Transport in sEmiconductors). We consider a physical model of a femtosecond relaxation of optically excited electrons which interact with phonons in an one-band semicondoctor. The electron-phonon interaction is switched on after a laser pulse creates an initial electron distribution. The Barker-Ferry equation is utilized as a quantum-kinetic model of the process under consideration. Two cases of this process are investigated – with and without an applied electric field. The electric field causes shift in the replicas, population of the semiclassically forbidden regions and influences the broadening and retardation of the electron distribution. The paper describes Grid implementation of these CPU-intensive algorithms. Using this application innovative results for different materials can be obtained. Here we present the first version of SALUTE which is used to obtain innovative results for GaAs materials. The results from a number of tests on MPI-enabled Grid are shown and disscussed.