# Optimisation of the population Monte Carlo algorithm : application to cosmology.

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## Date

2015

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## Abstract

In this thesis we study the Population Monte Carlo (PMC) algorithm and utilise simulations to
improve the efficiency of the algorithm by optimising the algorithm parameters. We then ap
ply these optimisation results to a cosmological parameter estimation problem, specifically that
of determining the initial conditions for structure formation. We accomplish this by using cos
mic microwave background (CMB) data to constrain models with an admixture of adiabatic and
isocurvature modes.
We review the standard cosmological model and current cosmological probes used for cosmol
ogy and discuss the CMB anisotropy spectrum, which forms the basis for our cosmological
parameter estimation study. We briefly outline linear perturbation theory and initial conditions
that form the basis of the inflationary models considered in this thesis. We describe the adiabatic
and isocurvature perturbations and investigate their effect on the CMB anisotropy spectrum.
We outline the Bayesian parameter estimation methodology adopted in our study and review
Monte Carlo sampling, specifically the Markov Chain Monte Carlo (MCMC) and PMC algo
rithms explaining why these methods are used in Bayesian parameter estimation. We discuss
recent developments to the PMC and MCMC algorithms and discuss various applications of
these algorithms in cosmology.
We focus on optimising the performance of the PMC algorithm with respect to its algorithm pa
rameters that are specified initially. However, we first define a measure of efficiency, related to
the computational cost of the sampling algorithm and then use simulations to maximise this mea
sure with respect to the algorithm parameters. These algorithm parameters include the sample
size drawn at each iteration, the number of importance functions used, and the parameters that
characterise the importance functions. Before this though, we will first investigate the optimi
sation of the PMC algorithm for a multivariate Gaussian target distribution, and present results
for choosing the optimal algorithm parameters that maximise efficiency. We will also explore
the performance of PMC on more complex distributions such as the banana shaped, bimodal and
hypercube distribution, and discuss the advantages and shortfalls for these distributions.
We incorporate the results from the previous optimisation study by applying the PMC algorithm
to a cosmological parameter estimation problem. We constrain models with an admixture of
adiabatic and isocurvature perturbations using the nine-year data release from the Wilkinson Mi
crowave Anisotropy Probe (WMAP) experiment. We discuss challenges faced in sampling such
complex distributions, the modifications to the PMC sampler needed to achieve convergence,
and the efficiencies achieved in sampling these distributions. We present results on the marginal
and joint parameter distributions for all possible admixtures of adiabatic and isocurvature modes.
We then perform a principal component analysis to determine the degeneracies that arise from
the introduction of isocurvature modes. In comparison to similar studies undertaken with the
WMAP one-year and three-year datasets, we find that the allowed isocurvature fraction is more
tightly constrained than in previous studies.

## Description

Ph. D. University of KwaZulu-Natal, Durban 2015.

## Keywords

Monte Carlo method., Mathematical optimization., Cosmic background radiation., Cosmology., Theses -- Applied mathematics.