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SUMMARY:High Dimensional Approximation via Sparse Occupancy Trees - Peter 
  Binev (University of South Carolina)
DTSTART:20190617T132000Z
DTEND:20190617T141000Z
UID:TALK126079@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Adaptive domain decomposition is often used in finite elements
  methods for solving partial differential equations in low space dimension
 s. The adaptive decisions are usually described by a tree. Assuming that c
 an find the (approximate) error for approximating a function on each eleme
 nt of the partition\, we have shown that a particular coarse-to-fine metho
 d provides a near-best approximation. This result can be extended to appro
 ximating point clouds any space dimension provided that we have relevant i
 nformation about the errors and can organize properly the data. Of course\
 , this is subject to the curse of dimensionality and nothing can be done i
 n the general case. In case the intrinsic dimensionality of the data is mu
 ch smaller than the space dimension\, one can define algorithms that defy 
 the curse. This is usually done by assuming that the data domain is close 
 to a low dimensional manifold and first approximating this manifold and th
 en the function defined by it. A few years ago\, together with Philipp Lam
 by\, Wolfgang Dahmen\, and Ron DeVore\, we proposed a direct method (witho
 ut specifically identifying any low dimensional set) that we called "spars
 e occupancy trees". The method defines a piecewise constant or linear appr
 oximation on general simplicial partitions. This talk considers an extensi
 on of this method to find a similar approximation on conforming simplicial
  partitions following an idea from a recent result together with Francesca
  Fierro and Andreas Veeser about near-best approximation on conforming tri
 angulations.
LOCATION:Seminar Room 1\, Newton Institute
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