Title: Computation and Application of Gramian Based Model Reduction

Speaker:  Danny C. Sorensen
          Noah Harding Professor
          Computational and Applied Mathematics
          Rice University

 
                 Abstract

Model reduction seeks to replace a large-scale system of differential
or difference equations by a system of substantially lower dimension that 
has nearly the same response characteristics.  Balanced truncation is a 
dimension reduction technique which provides a means to approximate a linear 
time invariant dynamical system with a reduced order system that has excellent 
approximation properties. These include a global a priori error bound and 
preservation of asymptotic stability.   Balanced reduction is within
the broader class of Gramian based methods which include Proper Orthogonal
Decomposition (POD).  This talk shall review this relationship and 
then discuss several applications and extensions of Balanced Reduction. 

To construct the transformations 
required to obtain balanced reduction, large scale Lyapunov equations must 
be approximately solved in low rank factored form.   This intensive 
computational task has recently been made tractable through algorithmic 
advances which include new methods for solving Lyapunov equations that 
are capable of solving problems on the order of 1 million state variables on a 
high end workstation.  


These techniques have been extended to a class of descriptor systems  which 
includes  the semidiscrete Oseen equations with time independent advection.  
This is accomplished by eliminating the algebraic equation using a projection.
However, we have been able to develop a solution scheme that does not
require explicit construction and application of this projector.

We have also applied balanced truncation to a compartmental
neuron model to construct  a low dimensional, morphologically accurate,
quasi-active integrate and fire model.   The complex model of dimension 6000
is faithfully approximated with a low order model with 10 variables.
More precisely, 10 variables suffice to reliably capture the soma voltage 
for a number of true morphologies over a broad (in space and time) class of 
synaptic input patterns. This savings will permit, for the first time, one to 
simulate large networks of biophysically accurate cells over realistic time 
spans.  This application will eventually require nonlinear model
reduction and new techniques for this aspect shall also be presented.