Automated Discovery of Numerical Approximation Formulae Via Genetic Programming

This thesis describes the use of genetic programming to automate the discovery of numerical approximation formulae. Results are presented involving rediscovery of known approximations for Harmonic numbers and discovery of rational polynomial approximations for functions of one or more variables, the latter of which are compared to Padé approximations obtained through a symbolic mathematics package. For functions of a single variable, it is shown that evolved solutions can be considered superior to Padé approximations, which represent a powerful technique from numerical analysis, given certain tradeoffs between approximation cost and accuracy, while for functions of more than one variable, we are able to evolve rational polynomial approximations where no Padé approximation can be computed. Furthermore, it is shown that evolved approximations can be iteratively improved through the evolution of approximations to their error function. Based on these results, we consider genetic programming to be a powerful and effective technique for the automated discovery of numerical approximation formulae

On a class of rational matrices and interpolating polynomials related to the discrete Laplace operator

Let \dlap be the discrete Laplace operator acting on functions (or rational matrices) $f:\mathbf{Q}_L\to\mathbb{Q}$, where $\mathbf{Q}_L$ is the two dimensional lattice of size $L$ embedded in $\mathbb{Z}_2$. Consider a rational $L\times L$ matrix $\mathcal{H}$, whose inner entries $\mathcal{H}_{ij}$ satisfy \dlap\mathcal{H}_{ij}=0. The matrix $\mathcal{H}$ is thus the classical finite difference five-points approximation of the Laplace operator in two variables. We give a constructive proof that $\mathcal{H}$ is the restriction to $\mathbf{Q}_L$ of a discrete harmonic polynomial in two variables for any $L>2$. This result proves a conjecture formulated in the context of deterministic fixed-energy sandpile models in statistical mechanics.Comment: 18 pag, submitted to "Note di Matematica

On a class of rational matrices and interpolating polynomials related to the discrete Laplace operator

Let \dlap be the discrete Laplace operator acting on functions(or rational matrices) $f:\mathbf{Q}_L\rightarrow\mathbb{Q}$,where $\mathbf{Q}_L$ is the two dimensional lattice of size $L$embedded in $\mathbb{Z}_2$. Consider a rational $L\times L$ matrix $\mathcal{H}$, whose inner entries $\mathcal{H}_{ij}$ satisfy \dlap\mathcal{H}_{ij}=0. The matrix $\mathcal{H}$ is thus theclassical finite difference five-points approximation of theLaplace operator in two variables. We give a constructive proofthat $\mathcal{H}$ is the restriction to $\mathbf{Q}_L$ of adiscrete harmonic polynomial in two variables for any L>2. Thisresult proves a conjecture formulated in the context ofdeterministic fixed-energy sandpile models in statisticalmechanics

Representation of conformal maps by rational functions

The traditional view in numerical conformal mapping is that once the boundary correspondence function has been found, the map and its inverse can be evaluated by contour integrals. We propose that it is much simpler, and 10-1000 times faster, to represent the maps by rational functions computed by the AAA algorithm. To justify this claim, first we prove a theorem establishing root-exponential convergence of rational approximations near corners in a conformal map, generalizing a result of D. J. Newman in 1964. This leads to the new algorithm for approximating conformal maps of polygons. Then we turn to smooth domains and prove a sequence of four theorems establishing that in any conformal map of the unit circle onto a region with a long and slender part, there must be a singularity or loss of univalence exponentially close to the boundary, and polynomial approximations cannot be accurate unless of exponentially high degree. This motivates the application of the new algorithm to smooth domains, where it is again found to be highly effective

Platonic Gravitating Skyrmions

We construct globally regular gravitating Skyrmions, which possess only discrete symmetries. In particular, we present tetrahedral and cubic Skyrmions. The SU(2) Skyrme field is parametrized by an improved harmonic map ansatz. Consistency then requires also a restricted ansatz for the metric. The numerical solutions obtained within this approximation are compared to those obtained in dilaton gravity.Comment: 13 pages, 4 figure

Symmetric Contours and Convergent Interpolation

The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as applied to the multipoint Pad\'e approximants is the fact that given a germ of an algebraic function and a sequence of rational interpolants with free poles of the germ, if there exists a contour that is "symmetric" with respect to the interpolation scheme, does not separate the plane, and in the complement of which the germ has a single-valued continuation with non-identically zero jump across the contour, then the interpolants converge to that continuation in logarithmic capacity in the complement of the contour. The existence of such a contour is not guaranteed. In this work we do construct a class of pairs interpolation scheme/symmetric contour with the help of hyperelliptic Riemann surfaces (following the ideas of Nuttall \& Singh and Baratchart \& the author. We consider rational interpolants with free poles of Cauchy transforms of non-vanishing complex densities on such contours under mild smoothness assumptions on the density. We utilize $\bar\partial$-extension of the Riemann-Hilbert technique to obtain formulae of strong asymptotics for the error of interpolation