Parallel algorithm with spectral convergence for nonlinear integro-differential equations

We discuss a numerical algorithm for solving nonlinear integro-differential equations, and illustrate our findings for the particular case of Volterra type equations. The algorithm combines a perturbation approach meant to render a linearized version of the problem and a spectral method where unknown functions are expanded in terms of Chebyshev polynomials (El-gendi's method). This approach is shown to be suitable for the calculation of two-point Green functions required in next to leading order studies of time-dependent quantum field theory.Comment: 15 pages, 9 figure

Scalable Inference for Markov Processes with Intractable Likelihoods

Bayesian inference for Markov processes has become increasingly relevant in recent years. Problems of this type often have intractable likelihoods and prior knowledge about model rate parameters is often poor. Markov Chain Monte Carlo (MCMC) techniques can lead to exact inference in such models but in practice can suffer performance issues including long burn-in periods and poor mixing. On the other hand approximate Bayesian computation techniques can allow rapid exploration of a large parameter space but yield only approximate posterior distributions. Here we consider the combined use of approximate Bayesian computation (ABC) and MCMC techniques for improved computational efficiency while retaining exact inference on parallel hardware

FAST IMPLEMENTATION TECHNIQUES OF MULTICHANNEL DIGITAL FILTERS FOR COLOR IMAGE PROCESSING USING MATRIX DECOMPOSITIONS

For the processing of color images, multivariable 3-input, 3-output 2-D digital filters are used, considering decomposition in the R, G and B components. Assuming that the three image components are decorrelated, three independent single-input, single-output (SISO) two-dimensional (2-D) digital filters are needed for the processing of each monochromatic image. Additional processing is needed for the correlated noise components in each chan- nel. The requirement of very fast processing dictates the use of special purpose hardware implementations. The VLSI array processors, which are special purpose, locally intercon- nected computing networks, are ideally suited for the fast implementation of digital filters, since they maximize concurrency by exploiting both parallelism and pipelining. In this paper fast implementation architectures of 3-input, 3-output 2-D multi-input digital filters for color image processing that are based on matrix decompositions are presented. The resulting structures are modular, regular, have high inherent parallelism and are easily pipelined, so that they may be implemented via VLSI array processors

Large scale ab-initio simulations of dislocations

We present a novel methodology to compute relaxed dislocations core configurations, and their energies in crystalline metallic materials using large-scale ab-intio simulations. The approach is based on MacroDFT, a coarse-grained density functional theory method that accurately computes the electronic structure with sub-linear scaling resulting in a tremendous reduction in cost. Due to its implementation in real-space, MacroDFT has the ability to harness petascale resources to study materials and alloys through accurate ab-initio calculations. Thus, the proposed methodology can be used to investigate dislocation cores and other defects where long range elastic effects play an important role, such as in dislocation cores, grain boundaries and near precipitates in crystalline materials. We demonstrate the method by computing the relaxed dislocation cores in prismatic dislocation loops and dislocation segments in magnesium (Mg). We also study the interaction energy with a line of Aluminum (Al) solutes. Our simulations elucidate the essential coupling between the quantum mechanical aspects of the dislocation core and the long range elastic fields that they generate. In particular, our quantum mechanical simulations are able to describe the logarithmic divergence of the energy in the far field as is known from classical elastic theory. In order to reach such scaling, the number of atoms in the simulation cell has to be exceedingly large, and cannot be achieved with the state-of-the-art density functional theory implementations

Wealth, income, earnings and the statistical mechanics of flow systems

This paper looks at empirical data from economics regarding wealth, earnings and income, alongside a flow model for an economy based on the general Lotka-Volterra models of Levy & Solomon. The data and modelling suggest that a simple economic system might provide a tractable model for giving an exact statistical mechanical solution for an 'out of equilibrium' flow model. This might also include an exact mathematical definition of a 'dissipative structure' derived from maximum entropy considerations. This paper is primarily a qualitative discussion of how such a mathematical proof might be achieved

Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations

A new algorithm is presented to find exact traveling wave solutions of differential-difference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the equations might admit polynomial solutions in tanh. Examples illustrate the key steps of the algorithm. Parallels are drawn through discussion and example to the tanh-method for partial differential equations. The new algorithm is implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute traveling wave solutions of nonlinear polynomial differential-difference equations. Use of the package, implementation issues, scope, and limitations of the software are addressed.Comment: 19 pages submitted to Computer Physics Communications. The software can be downloaded at http://www.mines.edu/fs_home/wherema