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Greedy measurement selection for state estimation

1 , * , 2 , 3 , 4

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      Abstract

      Parametric PDEs of the general form

      \[ \mathcal{P}(u,a)=0 \]
      are commonly used to describe many physical processes, where $\mathcal{P}$ is a differential operator, a is a high-dimensional vector of parameters and u is the unknown solution belonging to some Hilbert space V. Typically one observes m linear measurements of u(a) of the form $\ell_i(u)=\langle w_i,u \rangle$, $i=1,\dots,m$, where $\ell_i\in V'$ and $w_i$ are the Riesz representers, and we write $W_m = \text{span}\{w_1,\ldots,w_m\}$. The goal is to recover an approximation $u^*$ of u from the measurements. The solutions u(a) lie in a manifold within V which we can approximate by a linear space $V_n$, where n is of moderate dimension. The structure of the PDE ensure that for any a the solution is never too far away from $V_n$, that is, $\text{dist}(u(a),V_n)\le \varepsilon$. In this setting, the observed measurements and $V_n$ can be combined to produce an approximation $u^*$ of u up to accuracy
      \[ \Vert u -u^*\Vert \leq \beta^{-1}(V_n,W_m) \, \varepsilon \]
      where
      \[ \beta(V_n,W_m) := \inf_{v\in V_n} \frac{\Vert P_{W_m}v\Vert}{\Vert v \Vert} \]
      plays the role of a stability constant. For a given $V_n$, one relevant objective is to guarantee that $\beta(V_n,W_m)\geq \gamma >0$ with a number of measurements $m\geq n$ as small as possible. We present results in this direction when the measurement functionals $\ell_i$ belong to a complete dictionary.

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      Affiliations
      [1 ]Université Pierre et Marie Curie
      [2 ]Sorbonne Université
      [3 ]University of South Carolina
      [4 ]Paris Dauphine
      [* ]Correspondence: james.ashton.nichols@ 123456gmail.com
      Journal
      ScienceOpen Posters
      ScienceOpen
      27 April 2018
      10.14293/P2199-8442.1.SOP-MATH.YFVAQS.v1
      Copyright © 2018

      This work has been published open access under Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com.

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