On May 19th, 2016, Abdul-Lateef Haji-Ali successfully defended his PhD thesis entitled “Efficient Multilevel and Multi-index Sampling Methods for Stochastic Differential Equations”.
Committee Members:
Prof. Mike Giles (University of Oxford) External Examiner
Prof. Robert Sheichl (University of Bath)
Prof. Diogo Gomes (KAUST, CEMSE, AMCS)
Prof. Martin Mai (KAUST, PSE)
Prof. Raul Tempone (KAUST, CEMSE, AMCS) Thesis Advisor
Abstract:
Most problems in engineering
and natural sciences involve parametric equations in which the
parameters are
not known exactly due to measurement errors, lack of measurement data,
or even
intrinsic variability. In such problems, one objective is to compute
point or
aggregate values, called "quantities of interest". In such a setting,
the parametric equations must be accurately solved for
multiple values of the parameters to explore the dependence of the
quantities
of interest on these parameters, using various so-called "sampling
methods". In almost all cases, the parametric equations cannot be solved
exactly and suitable numerical discretization methods are required. The
high
computational complexity of these numerical methods coupled with the
fact that
the parametric equations must be solved for multiple values of the
parameters
make UQ problems computationally intensive, particularly when the
dimensionality of the underlying problem and/or the parameter space is
high. This thesis includes five published articles and one
soon-to-published one.
These articles are concerned with optimizing existing sampling methods,
namely
Multilevel Monte Carlo, and developing novel methods for high
dimensional
problems, namely Multi-index Monte Carlo and Multi-index Stochastic
Collocation. Assuming sufficient regularity of the underlying problem,
the
order of the computational complexity of these novel methods is, at
worst up to
a logarithmic factor, independent of the dimensionality of the problem.
The
articles also explore different applications, including an elliptic
partial
differential equation that models the flow of a fluid through a porous
medium
with random permeability and a stochastic particle system that models a
system
of coupled oscillators.
