Narrow Results

This paper introduces a novel approach to approximate a broad range of reaction–convection–diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves an accuracy of $O(h^k)$ in the energy norm, where $k$ represents the underlying polynomial degree. To validate the approach, a series of numerical experiments had been conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favorable performance of the current approach.

The ancestral sequence reconstruction problem asks to predict the DNA or protein sequence of an ancestral species, given the sequences of extant species. Such reconstructions are fundamental to comparative genomics, as they provide information about extant genomes and the process of evolution that gave rise to them. Arguably the best method for ancestral reconstruction is maximum likelihood estimation. Many effective algorithms for accurately computing the most likely ancestral sequence have been proposed. We consider the less-studied problem of computing the expected reconstruction error of a maximum likelihood reconstruction, given the phylogenetic tree and model of evolution, but not the extant sequences. This situation can arise, for example, when deciding which genomes to sequence for a reconstruction project given a gene-tree phylogeny (The Taxon Selection Problem). In most applications, the reconstruction error is necessarily very small, making Monte Carlo simulations very inefficient for accurate estimation. We present the first practical algorithm for this problem and demonstrate how it can be used to quickly and accurately estimate the reconstruction accuracy. We then use our method as a kernel in a heuristic algorithm for the taxon selection problem. The implementation is available at http://www.mcb.mcgill.ca/ blanchem/mlerror