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SUMMARY:Computing the spectrum of differential operators: a resolvent-base
 d approach.  - Andrew Horning (Cornell University)
DTSTART:20190717T150000Z
DTEND:20190717T160000Z
UID:TALK127720@talks.cam.ac.uk
CONTACT:59181
DESCRIPTION:During the past 50 years\, numerical methods for differential 
 eigenvalue problems have developed primarily within the `discretize-then-s
 olve' framework. These methods compute eigenvalues and eigenfunctions of a
  differential operator L by discretizing L to obtain a matrix eigenvalue p
 roblem. Subsequently\, the matrix eigenvalue problem can be solved using s
 tandard techniques from numerical linear algebra. Motivated by recent deve
 lopments in mathematical software for highly adaptive computations with fu
 nctions\, we invert this popular paradigm by designing an eigensolver that
  manipulates L\, rather than intermediate discretizations\, and approximat
 es only the functions that L acts on. This `solve-then-discretize' strateg
 y allows us to leverage spectrally accurate approximation schemes for func
 tions while overcoming fundamental difficulties that have undermined their
  application to differential eigenvalue problems in the past. The resultin
 g eigensolver is efficient\, scalable\, and well-conditioned\, and is capa
 ble of resolving eigenfunctions that exhibit rapid oscillations\, layers\,
  and other challenging features. Moreover\, the solve-then-discretize appr
 oach reveals two key ideas that open the door to computing exotic (yet phy
 sically relevant) spectral properties of differential operators that have 
 no finite-dimensional analogues\, such as the continuous spectrum and asso
 ciated spectral measure.
LOCATION:MR21\, Centre for Mathematical Sciences
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