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BEGIN:VEVENT
DTSTAMP:20200313T134710Z
DTSTART:20200327T140000Z
DTEND:20200327T150000Z
SUMMARY:[Cancelled] Heike Fassbender - A new framework for solving Lyapun
ov (and other matrix) equations
UID:{http://www.columbasystems.com/customers/uom/gpp/eventid/}cw1-k206a61
c-ufeuuc
DESCRIPTION:Prof. Heike Fassbender from University of Oxford will be spea
king at the seminar.\nAbstract: We will consider model order reduction f
or stable linear time-invariant (LTI) systems \\[\\dot{x}=Ax+Bu\,\\quad
y=Cx\\] with real\, large and sparse system matrices. In particular\, $A
$ is a square $n \\times n$ matrix\, $B$ is rectangular $n \\times m\,$
and $C$ is $p \\times n.$ Among the many existing model order reduction
methods our focus will be on (approximate) balanced truncation. The meth
od makes use of the two Lyapunov equations \\[A\\mathfrak{P}+\\mathfrak{
P}A^T=-BB^T\,\\] and \\[A^T \\mathfrak{Q}+\\mathfrak{Q}A=-C^TC.\\] The s
olutions $\\mathfrak{P}$ and $\\mathfrak{Q}$ of these equations are call
ed the controllability and observability Gramians\, respectively. The ba
lanced truncation method transforms the LTI system into a balanced form
whose controllability and observability Gramians become diagonal and equ
al\, together with a truncation of those states that are both difficult
to reach and to observe. One way to solve these large-scale Lyapunov equ
ations is via the Cholesky factorâ€“alternating direction implicit (CFâ€“ADI
) method which provides a low rank approximation to the exact solution m
atrix $\\mathfrak{P}$\, $\\mathfrak{Q}$ resp.. After reviewing existing
solution techniques\, in particular the CF-ADI method\, we will present
and analyze a system of ODEs\, whose solution for $t \\rightarrow \\inft
y$ is the Gramian $\\mathfrak{P}.$ We will observe that the solution evo
lves on a manifold and will characterize numerical methods whose approxi
mate low-rank solution evolves on this manifold as well. This will allow
us to give a new interpretation of the ADI method.
STATUS:TENTATIVE
TRANSP:TRANSPARENT
CLASS:PUBLIC
LOCATION:Frank Adams 1\, Alan Turing Building\, Manchester
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