Numerical Complete Solution for Random Genetic Drift by Energetic Variational Approach

In this paper, we focus on numerical solutions for random genetic drift problem, which is governed by a degenerated convection-dominated parabolic equation. Due to the fixation phenomenon of genes, Dirac delta singularities will develop at boundary points as time evolves. Based on an energetic variational approach (EnVarA), a balance between the maximal dissipation principle (MDP) and least action principle (LAP), we obtain the trajectory equation. In turn, a numerical scheme is proposed using a convex splitting technique, with the unique solvability (on a convex set) and the energy decay property (in time) justified at a theoretical level. Numerical examples are presented for cases of pure drift and drift with semi-selection. The remarkable advantage of this method is its ability to catch the Dirac delta singularity close to machine precision over any equidistant grid.Comment: 22 pages, 11 figures, 2 table

Speciation dynamics of an agent-based evolution model in phenotype space

This dissertation is an exploration of phase transition behavior and clustering of populations of organisms in an agent-based model of evolutionary dynamics. The agents in the model are organisms, described as branching-coalescing random walkers, which are characterized by their coordinates in a two-dimensional phenotype space. Neutral evolutionary conditions are assumed, such that no organism has a fitness advantage regardless of its phenotype location. Lineages of organisms evolve by limiting the maximum possible offspring distance from their parent(s) (mutability, which is the only heritable trait) along each coordinate in phenotype space. As mutability is varied, a non-equilibrium phase transition is shown to occur for populations reproducing by assortative mating and asexual fission. Furthermore, mutability is also shown to change the clustering behavior of populations. Random mating is shown to destroy both phase transition behavior and clustering. The phase transition behavior is characterized in the asexual fission case. By demonstrating that the populations near criticality collapse to universal scaling functions with appropriate critical exponents, this case is shown to belong to the directed percolation universality class. Finally, lineage behavior is explored for both organisms and clusters. The lineage lifetimes of the initial population of organisms are found to have a power-law probability density which scales with the correlation length exponent near critical mutability. The cluster centroid step-sizes obey a probability density function that is bimodal for all mutability values, and the average displays a linear dependence upon mutability in the supercritical range. Cluster lineage tree structures are shown to have Kingman\u27s coalescent universal tree structure at the directed percolation phase transition despite more complicated lineage structures. --Abstract, page iii

Reaction-diffusion processes and their interdisciplinary applications

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 16-12-201

Approximating Solutions to Convection-Diffusion Equations by Tensor Train Decompositions

A finite volume method for solving general time-homogeneous convection-diffusion equations with zero source term is presented. Computational efficiency of the method is improved by performing linear algebra in the tensor train format. To our knowledge this is the first time that the tensor train format and the finite volume method have been combined for this purpose. Finite volume methods, tensors and tensor decompositions are reviewed by summarising prominent texts on each respective topic. We extend a finite volume method for convection equations to include diffusion terms and show that the method preserves integrals and positivity. The time discretisation uses an explicit Euler step that leads to a sequence of linear systems of equations defining a discrete approximation of the solution at some final time. The recurrence is stepped forwards in time by performing algebraic operations in the tensor train format. In some cases, this leads to a significant increase in computational efficiency. We use our tensor train implementation of the finite volume method to approximate the allele frequency spectrum in three populations by solving the Wright-Fisher diffusion equation. Our method did not appear to outperform current methods for approximating the allele frequency spectrum. However, we develop some interesting and efficient tools for approximating the allele frequency spectrum if the solution to the Wright-Fisher diffusion equation is known in tensor train format

Geometrical Methods for the Analysis of Simulation Bundles

Efficiently analyzing large amounts of high dimensional data derived from the simulation of industrial products is a challenge that is confronted in this thesis. For this purpose, simulations are considered as abstract objects and assumed to be living in lower dimensional space. The aim of this thesis is to characterize and analyze these simulations, this is done by examining two different approaches. Firstly, from the perspective of manifold learning using diffusion maps and demonstrating its application and merits; the inherent assumption of manifold learning is that high dimensional data can be considered to be located on a low dimensional abstract manifold. Unfortunately, this can not be verified in practical applications as it would require the existence of several thousand datasets, where in reality only a few hundred are available due to computational costs. To overcome these restrictions, a new way of characterizing the set of simulations is proposed where it is assumed that transformations send simulations to other simulations. Under this assumption, the theoretical framework of shape spaces can be applied wherein a quotient space of a pre-shape space (the space of simulations shapes) modulo a transformation group is used. It is propound to add into this setting, the construction of positive definite operators that are assumed invariant to specific transformations. They are built using only one simulation and as a consequence all other simulations can be projected to the eigen-basis of these operators. A new representation of all simulations is thus obtained based on the projection coefficients in a very much analogous way to the use of the Fourier transformation. The new representation is shown to be significantly reduced, depending on the smoothness of the data. Several industrial applications for time dependent datasets from engineering simulations are provided to demonstrate the usefulness of the method and put forward several research directions and possible new applications