Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds

In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a $C^0$ function (derivative-discontinuity at a point), a rate of convergence of $n+1$ is obtained in $R^n$. Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function---a strongly localized function that is used for modeling sharp gradients. Then, the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in $n$-dimensional parallelepiped elements.Comment: 22 page

Monte Carlo Simulation of Quantum Computation

The many-body dynamics of a quantum computer can be reduced to the time evolution of non-interacting quantum bits in auxiliary fields by use of the Hubbard-Stratonovich representation of two-bit quantum gates in terms of one-bit gates. This makes it possible to perform the stochastic simulation of a quantum algorithm, based on the Monte Carlo evaluation of an integral of dimension polynomial in the number of quantum bits. As an example, the simulation of the quantum circuit for the Fast Fourier Transform is discussed.Comment: 12 pages Latex, 2 Postscript figures, to appear in Proceedings of the IMACS (International Association for Mathematics and Computers in Simulation) Conference on Monte Carlo Methods, Brussels, April 9

Speeding-up the execution of credit risk simulations using desktop grid computing: A case study

This paper describes a case study that was undertaken at a leading European Investment bank in which desktop grid computing was used to speed-up the execution of Monte Carlo credit risk simulations. The credit risk simulations were modelled using commercial-off-the-shelf simulation packages (CSPs). The CSPs did not incorporate built-in support for desktop grids, and therefore the authors implemented a middleware for desktop grid computing, called WinGrid, and interfaced it with the CSP. The performance results show that WinGrid can speed-up the execution of CSP-based Monte Carlo simulations. However, since WinGrid was installed on non-dedicated PCs, the speed-up achieved varied according to users’ PC usage. Finally, the paper presents some lessons learnt from this case study. It is expected that this paper will encourage simulation practitioners and CSP vendors to experiment with desktop grid computing technologies with the objective of speeding-up simulation experimentation

Using a desktop grid to support simulation modelling

Simulation is characterized by the need to run multiple sets of computationally intensive experiments. We argue that Grid computing can reduce the overall execution time of such experiments by tapping into the typically underutilized network of departmental desktop PCs, collectively known as desktop grids. Commercial-off-the-shelf simulation packages (CSPs) are used in industry to simulate models. To investigate if Grid computing can benefit simulation, this paper introduces our desktop grid, WinGrid, and discusses how this can be used to support the processing needs of CSPs. Results indicate a linear speed up and that Grid computing does indeed hold promise for simulation

Leveraging simulation practice in industry through use of desktop grid middleware

This chapter focuses on the collaborative use of computing resources to support decision making in industry. Through the use of middleware for desktop grid computing, the idle CPU cycles available on existing computing resources can be harvested and used for speeding-up the execution of applications that have “non-trivial” processing requirements. This chapter focuses on the desktop grid middleware BOINC and Condor, and discusses the integration of commercial simulation software together with free-to-download grid middleware so as to offer competitive advantage to organizations that opt for this technology. It is expected that the low-intervention integration approach presented in this chapter (meaning no changes to source code required) will appeal to both simulation practitioners (as simulations can be executed faster, which in turn would mean that more replications and optimization is possible in the same amount of time) and the management (as it can potentially increase the return on investment on existing resources)

Theory of Feshbach molecule formation in a dilute gas during a magnetic field ramp

Starting with coupled atom-molecule Boltzmann equations, we develop a simplified model to understand molecule formation observed in recent experiments. Our theory predicts several key features: (1) the effective adiabatic rate constant is proportional to density; (2) in an adiabatic ramp, the dependence of molecular fraction on magnetic field resembles an error function whose width and centroid are related to the temperature; (3) the molecular production efficiency is a universal function of the initial phase space density, the specific form of which we derive for a classical gas. Our predictions show qualitative agreement with the data from [Hodby et al, Phys. Rev. Lett. {\bf{94}}, 120402 (2005)] without the use of adjustable parameters

Identification of nanoindentation-induced phase changes in silicon by in situ electrical characterization

In situ electrical measurements during nanoindentation of Czochralski grown p-type crystalline silicon (100) have been performed using a conducting diamond Berkovich indenter tip. Through-tip current monitoring with a sensitivity of ∼10pA and extraction of current-voltage curves at various points on the complete load-unload cycle have been used to track the phase transformations of silicon during the loading and unloading cycle. Postindent current-voltage curves prove to be extremely sensitive to phase changes during indentation, as well as to the final phase composition within the indented volume. For example, differences in the final structure are detected by current-voltage measurements even in an unloading regime in which only amorphous silicon is expected to form. The electrical measurements are interpreted with the aid of previously reported transmission electron microscopy and Raman microspectroscopy measurements.This work was funded by the Australian Research Council and WRiota Pty Ltd

Complex Energy Spectrum and Time Evolution of QBIC States in a Two-Channel Quantum wire with an Adatom Impurity

We provide detailed analysis of the complex energy eigenvalue spectrum for a two-channel quantum wire with an attached adatom impurity. The study is based on our previous work [Phys. Rev. Lett. 99, 210404 (2007)], in which we presented the quasi-bound states in continuum (or QBIC states). These are resonant states with very long lifetimes that form as a result of two overlapping continuous energy bands one of which, at least, has a divergent van Hove singularity at the band edge. We provide analysis of the full energy spectrum for all solutions, including the QBIC states, and obtain an expansion for the complex eigenvalue of the QBIC state. We show that it has a small decay rate of the order $g^6$, where $g$ is the coupling constant. As a result of this expansion, we find that this state is a non-analytic effect resulting from the van Hove singularity; it cannot be predicted from the ordinary perturbation analysis that relies on Fermi's golden rule. We will also numerically demonstrate the time evolution of the QBIC state using the effective potential method in order to show the stability of the QBIC wave function in comparison with that of the other eigenstates.Comment: Around 20 pages, 50 total figure

Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models

We study the probability density function $P(h_m,L)$ of the maximum relative height $h_m$ in a wide class of one-dimensional solid-on-solid models of finite size $L$. For all these lattice models, in the large $L$ limit, a central limit argument shows that, for periodic boundary conditions, $P(h_m,L)$ takes a universal scaling form $P(h_m,L) \sim (\sqrt{12}w_L)^{-1}f(h_m/(\sqrt{12} w_L))$, with $w_L$ the width of the fluctuating interface and $f(x)$ the Airy distribution function. For one instance of these models, corresponding to the extremely anisotropic Ising model in two dimensions, this result is obtained by an exact computation using transfer matrix technique, valid for any $L>0$. These arguments and exact analytical calculations are supported by numerical simulations, which show in addition that the subleading scaling function is also universal, up to a non universal amplitude, and simply given by the derivative of the Airy distribution function $f'(x)$.Comment: 13 pages, 4 figure