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SUMMARY:Linear Algebra Methods for Parameter-Dependent Partial Differentia
 l Equations - Howard  Elman (University of Maryland\, College Park)
DTSTART:20180110T113000Z
DTEND:20180110T123000Z
UID:TALK97504@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:We discuss some recent developments in solution algorithms for
  the linear algebra problems that arise from parameter-dependent partial d
 ifferential equations (PDEs). In this setting\, there is a need to solve l
 arge coupled algebraic systems (which come from stochastic Galerkin method
 s)\, or large numbers of standard spatially discrete systems (from Monte C
 arlo or stochastic collocation methods).  The ultimate goal is solutions t
 hat represent surrogate approximations that can be evaluated cheaply for m
 ultiple values of the parameters\, which can be used effectively for simul
 ation or uncertainty quantification. <br><span><br>Our focus is on represe
 nting parameterized solutions in reduced-basis or low-rank matrix formats.
   We show that efficient solution algorithms can be built from multigrid m
 ethods designed for the underlying discrete PDE\, in combination with meth
 ods for truncating the ranks of iterates\, which reduce both cost and stor
 age requirements of solution algorithms. These ideas can be applied to the
  systems arising from many ways of treating the parameter  spaces\, includ
 ing stochastic Galerkin and collocation.  In addition\, we present new app
 roaches for solving the dense systems that arise from reduced-order models
  by preconditioned iterative methods and we show that such approaches can 
 also be combined with empirical interpolation methods to solve the algebra
 ic systems that arise from nonlinear PDEs.&nbsp\;</span>
LOCATION:Seminar Room 1\, Newton Institute
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