ItemAccurate Solution of Weighted Least Squares by Iterative MethodsBobrovnikova, Elena Y.; Vavasis, Stephen A. (Cornell University, 1997-02-06)We consider the weighted least-squares (WLS) problem with a very ill-conditioned weight matrix. Weighted least-squares problems arise in many applications including linear programming, electrical networks, boundary value problems, and structures. Because of roundoff errors, standard iterative methods for solving a WLS problem with ill-conditioned weights may not give the correct answer. Indeed, the difference between the true and computed solution (forward error) may be large. We propose an iterative algorithm, called MINRES-L, for solving WLS problems. The MINRES-L method is the application of MINRES, a Krylov-space method due to Paige and Saunders, to a certain layered linear system. Using a simplified model of the effects of round off error, we prove that MINRES-L gives answers with small forward error. We present computational experiments for some applications. ItemLocal correlation energies of two-electron atoms and model systemsHuang, Chien-Jung; Umrigar, C.J. (Cornell University, 1997-01)We present nearly-local definitions of the correlation energy density, and its potential and kinetic components, and evaluate them for several two-electron systems. This information should provide valuable guidance in constructing better correlation functionals than those in common use. In addition, we demonstrate that the quantum chemistry and the density functional definitions of the correlation energy rapidly approach one another with increasing atomic number. ItemQuality Mesh Generation in Higher DimensionsMitchell, Scott A.; Vavasis, Stephen A. (Cornell University, 1996-12)We consider the problem of triangulating a d-dimensional region. Our mesh generation algorithm, called QMG, is a qradtree-based algorithm that can triangulate any polyhedral region including nonconvexregions with holes. Furthermore, our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles. Finally, our algorithm is guaranteed never to over refine the domain in the sense that the number of simplices produced by QMG is bounded above by a factor times the number produced by any competing algorithm, where the factor depends on the aspect ratio bound satisfied by the competing algorithm. The QMG algorithm has been implemented in C++ and is used as a mesh generator for the finite element method. ItemModel fermion Monte Carlo with correlated pairs IIKalos, M.H.; Schmidt, K.E. (Cornell University, 1996-12)Correlated dynamics can produce stable algorithms for excited states of quantum many-body problems. We study a variety of harmonic oscillator problems to demonstrate the kinds of correlations needed. We show that marginally correct dynamics that produce a stable overlap with an antisymmetrictrial function give the correct fermion ground state. ItemSymmetry, Nonlinear Bifurcation Analysis, and Parallel ComputationWohlever, J.C. (Cornell University, 1996-10)In the natural and engineering sciences the equations which model physical systems with symmetry often exhibit an invariance with respect to a particular group "G" of linear transformations. "G" is typically a linear representation of a symmetry group "g" which characterizes the symmetry of the physical system. In this work, we will discuss the natural parallelism which arises while seeking families of solutions to a specific class of nonlinear vector equations which display a special type of group invariance, referred to as equivariance. The inherent parallelism stems for a global de-coupling, due to symmetry, of the full nonlinear equations which effectively splits the original problem into a set of smaller problems. Numerical results from asymmetry-adapted numerical procedure, (MMcontcm.m), written in MultiMATLAB are discussed. ItemMonte Carlo Optimization of Trial Wave Functions in Quantum Mechanics and Statistical MechanicsNightingale, M.P.; Umrigar, C.J. (Cornell University, 1996-10)This review covers applications of quantum Monte Carlo methods to quantum mechanical problems in the study of electronic and atomic structure, as well as applications to statistical mechanical problems both of static and dynamic nature. The common thread in all these applications is optimization of many-parameter trial states, which is done by minimization of the variance of the local energy or, more generally for arbitrary eigenvalue problems, minimization of the variance of the configurational eigenvalue. ItemA critical assessment of the Self-Interaction Corrected Local Density Functional method and its algorithmic implementationGoedecker, S.; Umrigar, C.J. (Cornell University, 1996-10)We calculate the electronic structure of several atoms and small molecules by direct minimization of the Self-Interaction Corrected Local Density Approximation (SIC-LDA) functional. To do this we first derive an expression for the gradient of this functional under the constraint that the orbitals be orthogonal and show that previously given expressions do not correctly incorporate this constraint. In our atomic calculations the SIC-LDA yields total energies, ionization energies and charge densities that are superior to results obtained with the Local Density Approximation (LDA). However, for molecules SIC-LDA gives bond lengths and reaction energies that are inferior to those obtained from LDA. The nonlocal BLYP functional, which we include as a representative GGA functional, out performs both LDA and SIC-LDA forall ground state properties we considered. ItemEuropean Option Pricing with Fixed Transaction CostsAiyer, Ajay Subramanian (Cornell University, 1996-09)In this paper, we study the problem of European option pricing in the presence of fixed transaction costs. The problems of optimal portfolio selection and option pricing in the presence of proportional transaction costs has been extensively studied in the mathematical finance literature. However, much less is known when we have fixed transaction costs. In this paper, we show that calculating the price of an European optioninvolves calculating the value functions of two stochastic impulse control problems and we obtain the explicit expressions for the resultant quasi-variational ine qualities satisfied by the value functions and then carry out a numerical calculation of the option price. ItemOptimal Portfolio Selection with Fixed Transactions Costs in the presence of Jumps and Random DriftAiyer, Ajay Subramanian (Cornell University, 1996-09)In this paper, we study the general problem of optimal portfolio selection with fixed transactions costs in the presence of jumps. We extend the analysis of Morton and Pliska to this setting by modeling the return processes of the risky assets in the investor's portfolio as jump-diffusion processes and derive the expression for the related optimal stopping time problem of a Markov process with jumps and explicitly solve it in the situation when the portfolio consists only of one risky asset. We also provide an asymptotic analysis of our model with one risky asset following the ideas of Wilmott and Atkinson. In the process, we also obtain a solution for the "Merton problem" generalized to the situation when there is credit risk. Finally, we consider the case where the drift of the stockprice process is random and unobservable and obtain expressions for the optimal trading policies. ItemNon-normal Dynamics and Hydrodynamic StabilityBaggett, Jeffrey Scott (Cornell University, 1996-08)This thesis explores the interaction of non-normality and nonlinearity incontinuous dynamical systems. A solution beginning near a linearly stable fixed point may grow large by a linear mechanism, if the linearization is non-normal, until it is swept away by nonlinearities resulting in a much smaller basin of attraction than could possibly be predicted by the spectrum of the linearization. Exactly this situation occurs in certain linearly stable shear flows, where the linearization about the laminar flow may be highly non-normal leading to the transient growth of certain small disturbances by factors which scale with the Reynolds number. These issues are brought into focus in Chapter 1 through the study of atwo-dimensional model system of ordinary differential equations proposed by Trefethen, et al. [Science, 261, 1993]. In Chapter 2, two theorems are proved which show that the basin of attraction of a stable fixed point, in systems of differential equations combining a non-normal linear term with quadratic nonlinearities, can decrease rapidly as the degree of non-normality is increased, often faster than inverse linearly. Several different low-dimensional models of transition to turbulence are examined in Chapter 3. These models were proposed by more than a dozen authors for a wide variety of reasons, but they all incorporate non-normal linear terms and quadratic nonlinearities. Surprisingly, in most cases, the basin of attraction of the "laminar flow" shrinks much faster than the inverse Reynolds number. Transition to turbulence from optimally growing linear disturbances, streamwise vortices, is investigated in plane Poiseuille and plane Couette flows in Chapter4. An explanation is given for why smaller streamwise vortices can lead to turbulence in plane Poiseuille flow. In plane Poiseuille flow, the transient linear growth of streamwise streaks caused by non-normality leads directly to a secondary instability. Certain unbounded operators are so non-normal that the evolution of infinitesimal perturbations to the fixed point is entirely unrelated to the spectrum, even as i to infinity. Two examples of this phenomenonare presented in Chapter 5.