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SUMMARY:A general framework for numerically stable reconstructions in Hilb
 ert spaces - Ben Adcock (Simon Fraser University)
DTSTART:20110317T150000Z
DTEND:20110317T160000Z
UID:TALK29937@talks.cam.ac.uk
CONTACT:Dr Hansen
DESCRIPTION:Many computational problems require the reconstruction of a fu
 nction-a signal\nor image\, for example-from a collection of its samples. 
  In abstract terms\, we\nseek to recover an element of a Hilbert space in 
 a particular basis (or frame)\,\ngiven its measurements (inner products) w
 ith respect to another\, fixed basis.\nWhilst this problem has been studie
 d extensively in the last several decades\,\nexisting methods are prone to
  be numerically unstable\, and may require rather\nrestrictive conditions 
 on the type of element to be reconstructed in order to\nensure convergence
 .\n\nThe purpose of this talk is to describe a new approach for this probl
 em.  It\ntranspires that the abstract reconstruction problem has a straigh
 tforward and\nwell-posed infinite-dimensional formulation.  Using certain 
 operator-theoretic\nconsiderations\, we can derive a finite-dimensional an
 alogue\, suitable for\ncomputations\, that retains such structure.  This y
 ields a convergent numerical\nmethod that is both numerically stable and\,
  as it turns out\, near-optimal.\nMoreover\, techniques from compressed se
 nsing can also be incorporated to\naccurately recover sparse signals whils
 t significantly undersampling.\n\nOne example of this framework\, with app
 lication to spectral methods for\nhyperbolic PDEs\, is the accurate recove
 ry of a piecewise smooth function from\nits (discrete or continuous) Fouri
 er or orthogonal polynomial coefficients.  We\nshall describe this example
  in detail\, and discuss a number of advantages over\nmore common techniqu
 es.\n\nThis is joint work with Anders Hansen (Cambridge)\n
LOCATION:CMS\, MR14
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