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DTSTART:19810329T020000
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UID:news-1464@dmi.unibas.ch
DTSTAMP:20230509T105935
DTSTART;TZID=Europe/Zurich:20230512T110000
DTEND;TZID=Europe/Zurich:20230512T120000
SUMMARY:Seminar in Numerical Analysis: Martin Eigel (WIAS Berlin)
LOCATION:
DESCRIPTION:Weighted least squares methods have been examined thouroughly t
o obtain quasi-optimal convergence results for a chosen (polyno
mial) basis of a linear space. A focus in the analysis lies on
the construction of optimal sampling measures and the derivatio
n of a sufficient sample complexity for stable reconstructions.
When considering holomorphic functions such as solutions of co
mmon parametric PDEs, the anisotropic sparsity they exhibit can
be exploited to achieve improved results adapted to the consid
ered problem. In particular, the sparsity of the data transfers
to the solution sparsity in terms of polynomial chaos coeffici
ents. When using nonlinear model classes, it turns out that the
known results cannot be used directly. To obtain comparable a
priori rates, we introduce a new weighted version of Stechkin's
lemma. This enables to obtain optimal complexity results for a
model class of low-rank tensor trains. We also show that the s
olution sparsity results in sparse component tensors and sketch
how this can be realised in practical algorithms. A nice appli
cation is the reconstruction of Galerkin solutions for parametr
ic PDEs. With this, a provably converging a posteriori adaptive
algorithm can be derived for linear model PDEs with non-affine
coefficients.\r\n \r\nFor further information about the semin
ar, please visit this webpage [https://dmi.unibas.ch/de/forschu
ng/mathematik/seminar-in-numerical-analysis/].
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