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SUMMARY:Reduced Basis Solvers for Stochastic Galerkin Matrix Equations - C
 atherine Powell (University of Manchester)
DTSTART:20180309T094500Z
DTEND:20180309T103000Z
UID:TALK102253@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:In the applied mathematics community\, reduced basis methods a
 re typically used to reduce the computational cost of applying sampling me
 thods to parameter-dependent partial differential equations (PDEs).  When 
 dealing with PDE models in particular\, repeatedly running computer models
  (eg finite element solvers) for many choices of the input parameters\, is
  computationally infeasible. The cost of obtaining  each sample of the num
 erical solution is instead sought by projecting the so-called high fidelit
 y problem into a reduced (lower-dimensional) space. The choice of reduced 
 space is crucial in balancing cost and overall accuracy.   In this talk\, 
 we do not consider sampling methods. Rather\, we consider stochastic Galer
 kin finite element methods (SGFEMs) for parameter-dependent PDEs. Here\, t
 he idea is to approximate the solution to the PDE model as a function of t
 he input parameters. We combine finite element approximation in physical s
 pace\, with global polynomial approximation on the parameter domain. In th
 e statistics community\, the term intrusive polynomial chaos approximation
  is often used. Unlike samping methods\, which require the solution of man
 y deterministic problems\, SGFEMs yield a single very large linear system 
 of equations with coefficient matrices that have a characteristic Kronecke
 r product structure.   By reformulating the systems as multiterm linear ma
 trix equations\, we have developed [see: C.E. Powell\, D. Silvester\, V.Si
 moncini\, An efficient reduced basis solver for stochastic Galerkin matrix
  equations\, SIAM J. Comp. Sci. 39(1)\, pp A141-A163 (2017)] a memory-effi
 cient solution algorithm which generalizes ideas from rational Krylov subs
 pace approximation (which are known in the linear algebra community). The 
 new approach determines a low-rank approximation to the solution matrix by
  performing a projection onto a reduced space that is iteratively augmente
 d with problem-specific basis vectors. Crucially\, it requires far less me
 mory than standard iterative methods applied to the Kronecker formulation 
 of the linear systems. For test problems consisting of elliptic PDEs\, and
  indefinite problems with saddle point structure\, we are able to solve sy
 stems of billions of equations on a standard desktop computer quickly and 
 efficiently.
LOCATION:Seminar Room 1\, Newton Institute
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