Event Detail

Event Type: 
Thursday, September 6, 2012 - 04:00 to 06:00
GLK 113

Speaker Info

Mathematics Department, Oregon State University

Random fields are frequently used in computational simulations of real-life
processes. In particular, they are used in modeling of flow and transport in
porous media. Porous media as they arise in geological formations are
intrinsically deterministic but there is significant uncertainty involved in
determination of their properties. Physical properties such as permeability
and porosity are often represented as random fields and consequently the differential equations describing flow and transport in porous media are stochastic in nature. In many situations description of properties of the porous media is aided by a limited number of observations at fixed points. These observations constrain the randomness of the field and lead to conditional simulations.

In this work we propose a method of modeling random fields for conditional
simulations which respect the observed data. An advantage of our method is
that in the case that additional data becomes available it can be easily
incorporated into subsequent representations. The proposed method is based
on infinite series representations of random fields. We provide truncation
error estimates which bound the discrepancy between the truncated series and
the random field. We additionally provide the expansions for some processes
that have not yet appeared in the literature.

There are several approaches to efficient numerical computations for partial
differential equations with random data. In this work we compare the
solutions of flow and transport equations obtained by conditional
simulations with Monte Carlo (MC) and stochastic collocation (SC) methods.
Due to its simplicity MC method is one of the most popular methods allowing
treatment of uncertainty in models despite being quite computationally
expensive. The SC method is very similar to the MC method in that it can be
implemented via repetitive execution of existing deterministic solvers. Its
potential has not yet been completely explored. The computational power of
the SC method depends on the choice of collocation points at which the flow
and transport equations are solved. We show that for both methods the
conditioning on measurements helps to reduce the uncertainty of the
solutions of the flow and transport equations. This especially holds in the
neighborhood of the conditioning points. We show that conditioning reduces
the variances of solutions helping to quantify the uncertainty in the output
of the flow and transport equations.

Advisers: Mina Ossiander and Malgorzata Peszynska.