Automatic implementation of material laws: Jacobian calculation in a finite element code with TAPENADE

In an effort to increase the versatility of finite element codes, we explore the possibility of automatically creating the Jacobian matrix necessary for the gradient-based solution of nonlinear systems of equations. Particularly, we aim to assess the feasibility of employing the automatic differentiation tool TAPENADE for this purpose on a large Fortran codebase that is the result of many years of continuous development. As a starting point we will describe the special structure of finite element codes and the implications that this code design carries for an efficient calculation of the Jacobian matrix. We will also propose a first approach towards improving the efficiency of such a method. Finally, we will present a functioning method for the automatic implementation of the Jacobian calculation in a finite element software, but will also point out important shortcomings that will have to be addressed in the future.Comment: 17 pages, 9 figure

Elimination Techniques for Algorithmic Differentiation Revisited

All known elimination techniques for (first-order) algorithmic differentiation (AD) rely on Jacobians to be given for a set of relevant elemental functions. Realistically, elemental tangents and adjoints are given instead. They can be obtained by applying software tools for AD to the parts of a given modular numerical simulation. The novel generalized face elimination rule proposed in this article facilitates the rigorous exploitation of associativity of the chain rule of differentiation at arbitrary levels of granularity ranging from elemental scalar (state of the art) to multivariate vector functions with given elemental tangents and adjoints. The implied combinatorial Generalized Face Elimination problem asks for a face elimination sequence of minimal computational cost. Simple branch and bound and greedy heuristic methods are employed as a baseline for further research into more powerful algorithms motivated by promising first test results. The latter can be reproduced with the help of an open-source reference implementation

Growth and the pollution convergence hypothesis: A nonparametric approach

The pollution-convergence hypothesis is formalized in a neoclassical growth model with optimal emissions reduction: pollution growth rates are positively correlated with output growth (scale effect) but negatively correlated with emission levels (defensive effect). This dynamic law is empirically tested for two major and regulated air pollutants - nitrogen oxides (NOX) and sulfur oxides (SOX) - with a panel of 25 European countries spanning over years 1980-2005. Traditional parametric models are rejected by the data. However, more flexible regression techniques - semiparametric additive specifications and fully nonparametric regressions with discrete and continuous factors - confirm the existence of the predicted positive and defensive effects. By analyzing the spatial distributions of per capita emissions, we also show that cross-country pollution gaps have decreased over the period for both pollutants and within the Eastern as well as the Western European areas. A Markov modeling approach predicts further cross-country absolute convergence, in particular for SOX. The latter results hold in the presence of spatial non-convergence in per capita income levels within both regions.Air pollution, convergence, economic growth, mixed nonparametric regressions, distribution dynamics.

Growth and the pollution convergence hypothesis: a nonparametric approach

The pollution-convergence hypothesis is formalized in a neoclassical growth model with optimal emissions reduction: pollution growth rates are positively correlated with output growth (scale effect) but negatively correlated with emission levels (defensive effect). This dynamic law is empirically tested for two major and regulated air pollutants - nitrogen oxides (NOX) and sulfur oxides (SOX) - with a panel of 25 European countries spanning over years 1980-2005. Traditional parametric models are rejected by the data. However, more flexible regression techniques - semiparametric additive specifications and fully nonparametric regressions with discrete and continuous factors - confirm the existence of the predicted positive and defensive effects. By analyzing the spatial distributions of per capita emissions, we also show that cross-country pollution gaps have decreased over the period for both pollutants and within the Eastern as well as the Western European areas. A Markov modeling approach predicts further cross-country absolute convergence, in particular for SOX. The latter results hold in the presence of spatial non-convergence in per capita income levels within both regions.Air pollution, convergence, economic growth, mixed nonparametric regressions, distribution dynamics