Abstract

Co-authors: Peter BINEV (University of South Carolina), Albert COHEN (University Pierre et Marie Curie), James NICHOLS (University Pierre et Marie Curie)

Parametric PDEs of the general form
$$\mathcal{P} (u,a) = 0$$
are commonly used to describe many physical processes, where $\cal P$ is a differential operator, $a$ is a high-dimensional vector of parameters and $u$ is the unknown solution belonging to some Hilbert space $V$. A typical scenario in state estimation is the following: for an unknown parameter $a$, one observes $m$ independent linear measurements of $u(a)$ of the form $\ell_i(u) = (w_i, u), i = 1, ..., m$, where $\ell_i \in V'$ and $w_i$ are the Riesz representers, and we write $W_m = \text{span}\{w_1,...,w_m\}$. The goal is to recover an approximation $u^{$ of $u$ from the measurements. Due to the dependence on a the solutions of the PDE lie in a manifold and the particular PDE structure often allows to derive good approximations of it by linear spaces Vn of moderate dimension n. In this setting, the observed measurements and Vn can be combined to produce an approximation $u}$ of $u$ up to accuracy
$$
\Vert u -u^* \Vert \leq \beta(V_n, W_m) \text{dist}(u, V_n)
$$
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. If time permits, we will also briefly explain ongoing research on how to adapt the reconstruction technique to noisy measurements.

Related Links

• https://hal.archives-ouvertes.fr/hal-01638177/document – Preprint