High-Performance Computing and Four-Dimensional Data Assimilation: The Impact on Future and Current Problems

This is the final technical report for the project entitled: "High-Performance Computing and Four-Dimensional Data Assimilation: The Impact on Future and Current Problems", funded at NPAC by the DAO at NASA/GSFC. First, the motivation for the project is given in the introductory section, followed by the executive summary of major accomplishments and the list of project-related publications. Detailed analysis and description of research results is given in subsequent chapters and in the Appendix

Monte Carlo studies of two dimensional quantum spin systems

Spin-1/2 nearest neighbor Heisenberg antiferromagnet and XY model on a square lattice are studied via large scale quantum Monte Carlo simulations using a fast and efficient multispin coding algorithm on the Caltech/JPL MarkIIIfp parallel supercomputer, based on the Suzuki-Trotter transformation. We performed simulations with very good statistics on lattices as large as 128x128 spins, in the temperature range from 0.1 to 2.5 in units of the effective exchange coupling J. We calculated energy, specific heat, magnetic susceptibilities and also spin correlation functions from which we deduce the correlation lengths. For the Heisenberg model, at temperatures higher than J the results are in excellent agreement with high-temperature series expansion. At low temperatures the long wavelength behavior is essentially classical. Our data show that the correlation length and staggered susceptibility are quantitatively well described by the renormalized classical picture at the 2-loop level of approximation. From the divergence of the correlation length, we deduce the value of the quantum renormalized spin stiffness, [rho][subscript s]/J = 0.199(2). We give evidence that the correlation function is of the Ornstein-Zernicke type. By comparing the largest measured correlation lengths with neutron scattering experiments on La2CuO4, we deduce the value of effective exchange coupling J = 1450±30 K. By measuring the imaginary time-dependent correlation functions, we show that the dynamics of the model can be well understood within a Bose liquid-type picture. The spin waves are rather sharp throughout most of the Brillouin zone and the damping is weakly dependent on the wave vector. In the case of the XY model, convincing numerical evidence is obtained on square lattices as large as 96x96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless (KT) phase transition at kTc/J = 0.350(4). The correlation length and in-plane susceptibility diverge at Tc precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly on heating near Tc and exhibits a peak around kT/J = 0.45. We also measure the spin stiffness and the correlation function exponent below the transition temperature. Within the statistical accuracy of the measurements, the results are well described by the square root singularity (with a nonuniversal amplitude) below Tc, and they have the universal values in agreement with KT theory at Tc

Numerical Pricing of Derivative Claims: Path Integral Monte Carlo Approach

We propose a path integral Monte Carlo method for pricing of derivative securities. Metropolis algorithm is used to sample probability distribution of histories (paths) of the underlying security. The advantage of path integral approach is that complete information about the derivative security, including its parameter sensitivities is obtained in a single simulation. It is also possible to obtain results for multiple values of parameters in a single simulation. The algorithm is efficiently implemented on parallel machines using High-Performance Fortran. Keywords: Derivative securities, option pricing, Monte Carlo, path integrals, data parallel, HighPerformance Fortran Northeast Parallel Architectures Center (NPAC), Syracuse University, 111 College Place, Syracuse, NY 13244-4100 e-mail: [email protected] Mosaic http://www.npac.syr.edu/users/miloje/ The author acknowledges partial support from Center for Research in Parallel Computation. Copyright c fl1994 by Miloje S. Makiv..

Pricing of Options Using the Path Integral Monte Carlo Approach

A path integral Monte Carlo method is implemented in a parallel computing environment to price a family of derivative securities known as options. In the path integral method, we construct a probability distribution of asset price histories in a phase space where price is a function of time. An asset's price history is the path of asset price movements from the present time to the expiration time of the option. We sample these histories in their entirety using the Metropolis algorithm. This procedure allows us to obtain information about an asset's sensitivities to multiple parameter sets. Once we integrate the sampled histories, we can average over the terminal prices to estimate a single value of the asset's true price. Using standard financial models, we can then approximate the values for options of various durations. 1 Introduction An option is a derivative security of finite duration whose value depends on the price of an underlying asset(s), such as the equity share of a compan..