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Order reduction methods for solving large-scale differential matrix Riccati equations
We consider the numerical solution of large-scale symmetric differential
matrix Riccati equations. Under certain hypotheses on the data, reduced order
methods have recently arisen as a promising class of solution strategies, by
forming low-rank approximations to the sought after solution at selected
timesteps. We show that great computational and memory savings are obtained by
a reduction process onto rational Krylov subspaces, as opposed to current
approaches. By specifically addressing the solution of the reduced differential
equation and reliable stopping criteria, we are able to obtain accurate final
approximations at low memory and computational requirements. This is obtained
by employing a two-phase strategy that separately enhances the accuracy of the
algebraic approximation and the time integration. The new method allows us to
numerically solve much larger problems than in the current literature.
Numerical experiments on benchmark problems illustrate the effectiveness of the
procedure with respect to existing solvers
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
The symplectic and twistor geometry of the general isomonodromic deformation problem
Hitchin's twistor treatment of Schlesinger's equations is extended to the
general isomonodromic deformation problem. It is shown that a generic linear
system of ordinary differential equations with gauge group SL(n,C) on a Riemann
surface X can be obtained by embedding X in a twistor space Z on which sl(n,C)
acts. When a certain obstruction vanishes, the isomonodromic deformations are
given by deforming X in Z. This is related to a description of the deformations
in terms of Hamiltonian flows on a symplectic manifold constructed from affine
orbits in the dual Lie algebra of a loop group.Comment: 35 pages, LATE
Symmetries and pre-metric electromagnetism
The equations of pre-metric electromagnetism are formulated as an exterior
differential system on the bundle of exterior differential 2-forms over the
spacetime manifold. The general form for the symmetry equations of the system
is computed and then specialized to various possible forms for an
electromagnetic constitutive law, namely, uniform linear, non-uniform linear,
and uniform nonlinear. It is shown that in the uniform linear case, one has
four possible ways of prolonging the symmetry Lie algebra, including
prolongation to a Lie algebra of infinitesimal projective transformations of a
real four-dimensional projective space. In the most general non-uniform linear
case, th effect of non-uniformity on symmetry seems inconclusive in the absence
of further specifics, and in the uniform nonlinear case, the overall difference
from the uniform linear case amounts to a deformation of the electromagnetic
constitutive tensor by the electromagnetic fields strengths, which induces a
corresponding deformation of the symmetry Lie algebra that was obtained in the
uniform linear case.Comment: 53 pages. Annalen der Physik (Leipzig) (2005), to be publishe
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