Numerical methods for model reduction and the control
of partial differential equations
We discuss model reduction techniques for control problems arising
in the control of semi-discretized partial differential equations.
In particular we study the method of balanced truncation which
leads to a system of large scale Lyapunov equations. These equations
are solved via a low rank ADI method. We show explict decay rates
for the singular values of the Lyapunov solutions and demonstrate
the results for the control of the heat equation and the Stokes
John R. Gilbert
in Numerical Linear Algebra: Past, Present, and Future
Graph theory and graph algorithms appear in an amazing variety
of numerical computations: modeling path structure in sparse matrices,
modeling locality in parallel computing, providing data structures
and algorithms to represent, organize, and manipulate discretized
versions of continuous phenomena.
I will describe some of the ways graph algorithms and numerical
linear algebra have enriched each other in the past, and will
pose some challenges to be met by finding new ways for them to
interact in the future.
On the solution of indefinite linear systems
An important class of linear systems are ones whose associated
matrix is indefinite and can be presented in a 2x2 block form
with a zero block. Many applications lead to such a structure;
among them we mention constrained optimization, inverse problems,
and the linearized Navier-Stokes equations. In this talk we will
discuss some characteristics of such systems, and present recent
results related to solution techniques in cases where the (1,1)
block is singular or ill-conditioned. Joint work with Gene Golub.
Modelling tidal resonance and tidal power around Vancouver Island
As the tidal currents in some of the narrow passages around Vancouver
Island are among the largest in the world, their potential as
a source of renewable energy has received considerable interest.
Though there have been numerous models for Juan de Fuca Strait,
the Strait of Georgia, and the west coast of Vancouver Island,
only recently has a model been developed that provides adequate
resolution for the complicated network of channels north of Campbell
River where many of the strong flows are found. In this talk we
will briefly describe the finite element and control theory techniques
that are presently being used to simulate tidal flows and elevations
in this region. Resonance, tidal dissipation, and preliminary
results on the potential for power generation will all be discussed.
Numerical methods in climate research
Through numerical simulation we aim to better understand past,
present and future climate and climate variability. In this talk
I will briefly describe the climate system and global climate
models (GCMs), and then will highlight some numerical issues that
arise. Namely: 1) Gibbs oscillations in truncated spectral expansions;
2) "dynamical" convergence of GCM solutions and 3) artificial
boundary-induced wave breaking.