English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Reduced model spaces, such as reduced basis and polynomial chaos, are linearspaces $V_n$ of finite dimension $n$ which are designed for the efficientapproximation of families parametrized PDEs in a Hilbert space $V$. Themanifold $\mathcal{M}$ that gathers the solutions of the PDE for all admissibleparameter values is globally approximated by the space $V_n$ with somecontrolled accuracy $\epsilon_n$, which is typically much smaller than whenusing standard approximation spaces of the same dimension such as finiteelements. Reduced model spaces have also been proposed in [13] as a vehicle todesign a simple linear recovery algorithm of the state $u\in\mathcal{M}$corresponding to a particular solution when the values of parameters areunknown but a set of data is given by $m$ linear measurements of the state. Themeasurements are of the form $\ell_j(u)$, $j=1,\dots,m$, where the $\ell_j$ arelinear functionals on $V$. The analysis of this approach in [2] shows that therecovery error is bounded by $\mu_n\epsilon_n$, where $\mu_n=\mu(V_n,W)$ is theinverse of an inf-sup constant that describe the angle between $V_n$ and thespace $W$ spanned by the Riesz representers of $(\ell_1,\dots,\ell_m)$. Areduced model space which is efficient for approximation might thus beineffective for recovery if $\mu_n$ is large or infinite. In this paper, wediscuss the existence and construction of an optimal reduced model space forthis recovery method, and we extend our search to affine spaces. Our basicobservation is that this problem is equivalent to the search of an optimalaffine algorithm for the recovery of $\mathcal{M}$ in the worst case errorsense. This allows us to perform our search by a convex optimization procedure.Numerical tests illustrate that the reduced model spaces constructed from ourapproach perform better than the classical reduced basis spaces.

Optimal reduced model algorithms for data-based state estimation