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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Reduced Basis Solvers for Stochastic Galerkin Matr
ix Equations - Catherine Powell (University of Man
chester)
DTSTART;TZID=Europe/London:20180309T094500
DTEND;TZID=Europe/London:20180309T103000
UID:TALK102118AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/102118
DESCRIPTION:In the applied mathematics community\, reduced bas
is methods are typically used to reduce the comput
ational cost of applying sampling methods to param
eter-dependent partial differential equations (PDE
s). When dealing with PDE models in particular\,
repeatedly running computer models (eg finite elem
ent solvers) for many choices of the input paramet
ers\, is computationally infeasible. The cost of o
btaining each sample of the numerical solution is
instead sought by projecting the so-called high f
idelity problem into a reduced (lower-dimensional)
space. The choice of reduced space is crucial in
balancing cost and overall accuracy. In this tal
k\, we do not consider sampling methods. Rather\,
we consider stochastic Galerkin finite element met
hods (SGFEMs) for parameter-dependent PDEs. Here\,
the idea is to approximate the solution to the PD
E model as a function of the input parameters. We
combine finite element approximation in physical s
pace\, with global polynomial approximation on the
parameter domain. In the statistics community\, t
he term intrusive polynomial chaos approximation i
s often used. Unlike samping methods\, which requi
re the solution of many deterministic problems\, S
GFEMs yield a single very large linear system of e
quations with coefficient matrices that have a cha
racteristic Kronecker product structure. By refo
rmulating the systems as multiterm linear matrix e
quations\, we have developed [see: C.E. Powell\, D
. Silvester\, V.Simoncini\, An efficient reduced b
asis solver for stochastic Galerkin matrix equatio
ns\, SIAM J. Comp. Sci. 39(1)\, pp A141-A163 (2017
)] a memory-efficient solution algorithm which gen
eralizes ideas from rational Krylov subspace appro
ximation (which are known in the linear algebra co
mmunity). The new approach determines a low-rank a
pproximation to the solution matrix by performing
a projection onto a reduced space that is iterativ
ely augmented with problem-specific basis vectors.
Crucially\, it requires far less memory than stan
dard iterative methods applied to the Kronecker fo
rmulation of the linear systems. For test problems
consisting of elliptic PDEs\, and indefinite prob
lems with saddle point structure\, we are able to
solve systems of billions of equations on a standa
rd desktop computer quickly and efficiently.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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