Greedy measurement selection for state estimation

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.