Computing a Quantity of Interest from Observational
· Simon Foucart
· Przemyslaw Wojtaszczyk
Received: 13 June 2017 / Revised: 9 February 2018 / Accepted: 26 March 2018
© Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract Scientiﬁc problems often feature observational data received in the form
( f ),...,w
( f ) of known linear functionals applied to an unknown func-
tion f from some Banach space X , and it is required to either approximate f (the full
approximation problem) or to estimate a quantity of interest Q( f ). In typical exam-
ples, the quantities of interest can be the maximum/minimum of f or some averaged
quantity such as the integral of f , while the observational data consists of point eval-
uations. To obtain meaningful results about such problems, it is necessary to possess
additional information about f , usually as an assumption that f belongs to a certain
model class K contained in X . This is precisely the framework of optimal recovery,
Communicated by Wolfgang Dahmen.
This research was supported by the ONR Contracts N00014-15-1-2181, N00014-16-1-2706 the NSF
Grant DMS 1521067, DARPA through Oak Ridge National Laboratory; by the NSF Grant DMS 1622134;
and by National Science Centre, Poland Grant UMO-2016/21/B/ST1/00241.
Department of Mathematics, Texas A&M University, College Station, TX 77840, USA
Interdisciplinary Center for Mathematical and Computational Modelling, University of Warsaw,
ul. Tyniecka 15/17, 02-630 Warsaw, Poland
Institute of Mathematics, Polish Academy of Sciences, ul.
Sniadeckich 8, 00-656 Warsaw, Poland