In the following there is a list of projects for interested students. If you would like to work on one of those projects please get in contact with me. The projects can be scaled to fit for a PhD, honours or master's thesis.
Pseudo marginal Metropolis Hastings
This project is in computational statistics and deals with Markov chain Monte Carlo algorithms. Particle methods, and in particular the pseudo-marginal Metropolis Hastings (PMMH) approach, are used in Bayesian inference where it is computationally difficult or too expensive to compute the likelihood. This situation arises in settings such as nonlinear time series in Finance. Quasi-Monte Carlo methods (QMC) are techniques from computational mathematics which have been employed in a range of applications, for instance, in computational finance and uncertainty quantification. This project aims to develop a combination of particle methods and QMC, for big data settings.
- D. Gunawan, M.-N. Tran, K. Suzuki, J. Dick, R. Kohn, Computationally Efficient Bayesian Estimation of High Dimensional Copulas with Discrete and Mixed Margins. arXiv:1608.06174 [stat.ME]
PDE's with random coefficients
Partial differential equations (PDE) are a staple tool in the modeling world. Some input data is required to set up the PDE. In contemporary research one studies the case where this input data is only known to have a certain distribution. The aim is to study numerical solutions and simulations of such PDEs.
- Kuo, F. Y.; Schwab, Ch.; Sloan, I. H. Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond. ANZIAM J. 53 (2011), no. 1, 1-37.
- Kuo, Frances Y.; Schwab, Christoph; Sloan, Ian H. Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50 (2012), no. 6, 3351-3374.
This project has its roots in number theory. The classic paper in this area is
- H. Weyl, Ueber die Gleichverteilung von Zahlen mod. Eins. Math. Ann., 77, 313-352, 1916.
Another classic paper is the lower bound on the so-called L_2 discrepancy by Klaus Roth
- K. Roth, On irregularities of distribution. Mathematika, 1, 73-79, 1954.
The aim in this project is to find constructions of point sets and sequences in the unit cube with small discrepancy.
Fast QMC matrix-vector products
A fast matrix-vector multiplication for lattice rules and polynomial lattice rules was introduced in
This method can significantly reduce the computation time in linear applications. The aim of this project is to extend this idea to other methods and apply it in computational finance and uncertainty quantification.
Higher order quasi-Monte Carlo
This project is in Numerical Analysis and Computational Mathematics, but also uses some wavelet theory and harmonic analysis. Quasi-Monte Carlo rules are equal weight quadrature rules over the s-dimensional unit cube to approximate integrals. The emphasis here is on the case where the integrands are smooth in which case fast rates of convergence of the integration error can be achieved.
Point distributions on the sphere
This project deals with point sets on the sphere which have small spherical cap discrepancy. Numerical experiments indicate that certain constructions of points achieve the optimal rate of convergence, but a proof of the corresponding statement is missing.
Markov chain quasi-Monte Carlo
This project is in the areas of Statistics, Numerical Analysis and Computational Mathematics. It deals with a combination of Markov chain Monte Carlo (MCMC) methods with quasi-Monte Carlo (QMC). The idea is to replace the pseudo-random numbers with quasi-Monte Carlo type point sets to achieve an improvement in the convergence of the MCMC method.
- Su Chen, Consistency and convergence rate of Markov chain quasi Monte Carlo with examples. Stanford University, 2011.
- S. Chen, J. Dick and A.B. Owen, Consistency of Markov Chain Quasi-Monte Carlo on continuous state spaces. Ann. Stat., 39, 679--701, 2011. doi: 10.1214/10-AOS831 For a blog entry and preprint version of this paper see here.
- J. Dick, D. Rudolf and H. Zhu, Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo. Submitted, 2013. For an arXiv version see here.
- Seth Tribble, Markov chain Monte Carlo Algorithms using completely uniformly distributed driving sequences. PhD Thesis, Stanford University, 2007.
- H. Zhu and J. Dick, Discrepancy bounds for deterministic acceptance-rejection samplers. Electron. J. Stat., 8, 678--707, 2014. DOI:10.1214/14-EJS898 For the arXiv version see here.