Quadrature methods for elliptic PDEs with random diffusion

In this thesis, we consider elliptic boundary value problems with
random diffusion coefficients. Such equations arise in many
engineering applications, for example, in the modelling of
subsurface flows in porous media, such as rocks.
To describe the subsurface flow, it is convenient to use
Darcy's law. The key ingredient in this approach is the hydraulic
conductivity. In most cases, this hydraulic conductivity is approximated
from a discrete number of measurements and, hence, it is common to
endow it with uncertainty, i.e. model it as a random field.
This random field is usually characterized
by its mean field and its covariance function.
Naturally, this randomness propagates through the model which
yields that the solution is a random field as well.
The thesis on hand is concerned with the effective computation
of statistical quantities of this random solution, like the expectation,
the variance, and higher order moments.
In order to compute these quantities, a suitable representation of the
random field which describes the hydraulic conductivity needs to be
computed from the mean field and the covariance function.
This is realized by the Karhunen-Loeve expansion which
separates the spatial variable and the stochastic variable. In general, the
number of random variables and spatial functions used in this expansion
is infinite and needs to be truncated appropriately.
The number of random variables which are required depends on the
smoothness of the covariance function and grows with the desired accuracy.
Since the solution also depends on these random variables, each moment
of the solution appears as a high-dimensional Bochner integral over the
image space of the collection of random variables. This integral has to be
approximated by quadrature methods where each function evaluation
corresponds to a PDE solve.
In this thesis, the Monte Carlo, quasi-Monte Carlo, Gaussian tensor product, and
Gaussian sparse grid quadrature is analyzed to deal with this high-dimensional
integration problem.
In the first part, the necessary regularity requirements of the integrand and
its powers are provided in order to guarantee convergence of the different
methods.
It turns out that all the powers of the solution depend, like the solution itself,
anisotropic on the different random variables which means in this case that
there is a decaying dependence on the different random variables.
This dependence can be used to overcome, at least up to a certain extent, the
curse of dimensionality of the quadrature problem.
This is reflected in the proofs of the convergence rates of the different
quadrature methods which can be found in the second part of this thesis.
The last part is concerned with multilevel quadrature approaches to keep
the computational cost low. As mentioned earlier, we need to solve a partial
differential equation for each quadrature point.
The common approach is to apply a finite element approximation scheme on
a refinement level which corresponds to the desired accuracy.
Hence, the total computational cost is given by the product of the number
of quadrature points times the cost to compute one finite element solution
on a relatively high refinement level.
The multilevel idea is to use a telescoping sum decomposition of the quantity
of interest with respect to different spatial refinement levels and use
quadrature methods with different accuracies for each summand.
Roughly speaking, the multilevel approach spends a lot of quadrature points
on a low spatial refinement and only a few on the higher refinement levels.
This reduces the computational complexity but requires further regularity
on the integrand which is proven for the considered problems in this thesis.