BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Greedy algorithms for optimal measurements selection in state esti mation using reduced models - Olga Mula (Université Paris-Dauphine) DTSTART:20180308T110000Z DTEND:20180308T114500Z UID:TALK102043@talks.cam.ac.uk CONTACT:INI IT DESCRIPTION:Co-authors: Peter BINEV (University of South Carolina)\, Albe rt COHEN (University Pierre et Marie Curie)\, James NICHOLS (University Pierre et Marie Curie)

Parametric PDEs of the general form
$$\\m athcal{P} (u\,a) = 0$$
are commonly used to describe many physical proc esses\, where $\\cal P$ is a differential operator\, $a$ is a high-dimensi onal vector of parameters and $u$ is the unknown solution belonging to som e Hilbert space $V$. A typical scenario in state estimation is the followi ng: for an unknown parameter $a$\, one observes $m$ independent linear mea surements 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 st ructure often allows to derive good approximations of it by linear spaces Vn of moderate dimension n. In this setting\, the observed measurements an d Vn can be combined to produce an approximation $u^*$ of $u$ up to accura cy
$$
\\Vert u -u^* \\Vert \\leq \\beta(V_n\, W_m) \\text{dist}(u\, V_n)
$$
where
$$
\\beta(V_n\, W_m) := \\inf_{v\\in V_n} \\frac {\\Vert P_{W_m} v \\Vert}{\\Vert v \\Vert}
$$
plays the role o f a stability constant. For a given $V_n$\, one relevant objective is to g uarantee that $\\beta(V_n\, W_m) \\geq \\gamma >0$ with a number of measur ements $m \\geq n$ as small as possible. We present results in this direct ion when the measurement functionals $\\ell_i$ belong to a complete dictio nary. If time permits\, we will also briefly explain ongoing research on h ow to adapt the reconstruction technique to noisy measurements.
Related Links LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR