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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.

### Author and article information

###### 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
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27 April 2018
10.14293/P2199-8442.1.SOP-MATH.YFVAQS.v1