accuracy: Tools for Accurate and Reliable Statistical Computing

Most empirical social scientists are surprised that low-level numerical issues in software can have deleterious effects on the estimation process. Statistical analyses that appear to be perfectly successful can be invalidated by concealed numerical problems. We have developed a set of tools, contained in accuracy, a package for R and S-PLUS, to diagnose problems stemming from numerical and measurement error and to improve the accuracy of inferences. The tools included in accuracy include a framework for gauging the computational stability of model results, tools for comparing model results, optimization diagnostics, and tools for collecting entropy for true random numbers generation.

Fast and Accurate Coarsening Simulation with an Unconditionally Stable Time Step

We present Cahn-Hilliard and Allen-Cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy-controlled simulation. Our stability analysis is based on Eyre's theorem and unconditional von Neumann stability analysis, both of which we present. Numerical tests confirm the accuracy of the von Neumann approach, which is straightforward and should be widely applicable in phase-field modeling. We show that accuracy can be controlled with an unbounded time step Delta-t that grows with time t as Delta-t ~ t^alpha. We develop a classification scheme for the step exponent alpha and demonstrate that a class of simple linear algorithms gives alpha=1/3. For this class the speed up relative to a fixed time step grows with the linear size of the system as N/log N, and we estimate conservatively that an 8192^2 lattice can be integrated 300 times faster than with the Euler method.Comment: 14 pages, 6 figure

Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares

In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of H_infinity control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series

Numerical Awareness in Control

Algorithm development, sensitivity and accuracy issues, large-scale computations, and high-performance numerical softwar

Testing the Accuracy and Stability of Spectral Methods in Numerical Relativity

The accuracy and stability of the Caltech-Cornell pseudospectral code is evaluated using the KST representation of the Einstein evolution equations. The basic "Mexico City Tests" widely adopted by the numerical relativity community are adapted here for codes based on spectral methods. Exponential convergence of the spectral code is established, apparently limited only by numerical roundoff error. A general expression for the growth of errors due to finite machine precision is derived, and it is shown that this limit is achieved here for the linear plane-wave test. All of these tests are found to be stable, except for simulations of high amplitude gauge waves with nontrivial shift.Comment: Final version, as published in Phys. Rev. D; 13 pages, 16 figure

Stability and accuracy of numerical boundary conditions in aeroelastic analysis

This paper analyses the accuracy and numerical stability of coupling procedures in aeroelastic modelling. A two-dimensional model problem assuming unsteady inviscid flow past an oscillating wall leads to an even simpler one-dimensional model problem. Analysis of different numerical algorithms shows that in general the coupling procedures are numerically stable, but care is required to achieve accuracy when using very few timesteps per period of natural oscillation of the structure. The relevance of the analysis to fully three-dimensional applications is discussed