Parametric PDEs of the general for m

$$\\mathcal{P} (u\,a) = 0$$

are commonly u sed to describe many physical processes\, where $\ \cal P$ is a differential operator\, $a$ is a high -dimensional vector of parameters and $u$ is the u nknown solution belonging to some Hilbert space $V $. A typical scenario in state estimation is the f ollowing: for an unknown parameter $a$\, one obser ves $m$ 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 $W_m = \\tex t{span}\\{w_1\,...\,w_m\\}$. The goal is to recove r an approximation $u^*$ of $u$ from the measureme nts. Due to the dependence on a the solutions of t he PDE lie in a manifold and the particular PDE st ructure 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) := \\in f_{v\\in V_n} \\frac{\\Vert P_{W_m} v \\Vert}{\\Ve rt v \\Vert}

$$

plays the role of a st ability 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 measuremen ts $m \\geq n$ as small as possible. We present re sults in this direction when the measurement funct ionals $\\ell_i$ belong to a complete dictionary. If time permits\, we will also briefly explain ong oing research on how to adapt the reconstruction t echnique to noisy measurements.

Rela ted Links LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR