Investigating Uncertain Parameters in the Burgers Equation

The Burgers equation is a simple model of the differential
equations that are of interest in scientific computing.
The field of Uncertainty Quantification estimates the 
effect that errors or uncertainties in the input quantity
affect our computed output quantities.

In this talk, we look at two simple case studies involving
the Burgers equation.  For the steady Burgers equation,
we consider the effect of errors in the boundary conditions.
For the unsteady Burgers equation, we let the parameters be
the values of the initial condition at selected points,
and ask whether uncertainties in some parameters are more
important than others.

For the unsteady problem, we first try to identify the
parameters that are likely to have the most influence;
then, for the reduced system, we consider estimates using
both the Monte Carlo and the sparse grid approach.

Because the Burgers equation is relatively simple, it is
easy to program these examples, to analyze the results,
and to focus on some ideas of uncertainty quantification,
rather than on the details of solving the underlying 
differential equation.