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CATEGORIES:Numerical Analysis
SUMMARY:A general framework for numerically stable reconst
ructions in Hilbert spaces - Ben Adcock (Simon Fra
ser University)
DTSTART;TZID=Europe/London:20110317T150000
DTEND;TZID=Europe/London:20110317T160000
UID:TALK29937AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/29937
DESCRIPTION:Many computational problems require the reconstruc
tion of a function-a signal\nor image\, for exampl
e-from a collection of its samples. In abstract t
erms\, we\nseek to recover an element of a Hilbert
space in a particular basis (or frame)\,\ngiven i
ts measurements (inner products) with respect to a
nother\, fixed basis.\nWhilst this problem has bee
n studied extensively in the last several decades\
,\nexisting methods are prone to be numerically un
stable\, and may require rather\nrestrictive condi
tions on the type of element to be reconstructed i
n order to\nensure convergence.\n\nThe purpose of
this talk is to describe a new approach for this p
roblem. It\ntranspires that the abstract reconstr
uction problem has a straightforward and\nwell-pos
ed infinite-dimensional formulation. Using certai
n operator-theoretic\nconsiderations\, we can deri
ve a finite-dimensional analogue\, suitable for\nc
omputations\, that retains such structure. This y
ields a convergent numerical\nmethod that is both
numerically stable and\, as it turns out\, near-op
timal.\nMoreover\, techniques from compressed sens
ing can also be incorporated to\naccurately recove
r sparse signals whilst significantly undersamplin
g.\n\nOne example of this framework\, with applica
tion to spectral methods for\nhyperbolic PDEs\, is
the accurate recovery of a piecewise smooth funct
ion from\nits (discrete or continuous) Fourier or
orthogonal polynomial coefficients. We\nshall des
cribe this example in detail\, and discuss a numbe
r of advantages over\nmore common techniques.\n\nT
his is joint work with Anders Hansen (Cambridge)\n
LOCATION:CMS\, MR14
CONTACT:Dr Hansen
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