The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we...

The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we...

The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we...

Let $A$ be a closed set of $M\simeq {\mathbb{R}}^n$, whose conormai cones $x+y_x^{*}\left(A\right)$, $x\in A$, have locally empty intersection. We first show in §1 that $\text{dist}\left(x,A\right)$, $x\in M\setminus A$ is a $C^1$ function. We then represent the n microfunctions of ${\mathcal{C}}_{A|X}$, $X\simeq {\mathbb{C}}^n$, using cohomology groups of ${\mathcal{O}}_X$ of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of ${\left.{\mathcal{C}}_{A|X}\right|}_{{\dot{\pi }}^{-1}\left(x_0\right)}$, $x_0\in \partial A$, satisfy the principle of the analytic continuation in the complex integral manifolds of ${\left\{H\left({\varphi }_i^C\right)\right\}}_{i=1,\mathrm{\dots },m}$, $\left\{{\varphi }_i\right\}$ being a base for the linear hull of ${\gamma }_{x_0}^{*}\left(A\right)$ in $T_{x_0}^{*}M$; in particular we get ${\left.{\mathrm{\Gamma }}_{A\times {}_{M}T{}^{*}{}_{M}X}\left({\mathcal{C}}_{A|X}\right)\right|}_{\partial A\times {}_M\dot{T}{}^{*}{}_{M}X}=0$. When $A$is a half space with $C^{\omega }$-boundary,...

Let $\Omega \subset {ℂ}^n$ be a bounded, simply connected $ℂ$-convex domain. Let $\alpha \in {\mathbb{Z}}_{+}^n$ and let $f$ be a function on $\Omega$ which is separately $C^{2{\alpha }_j-1}$-smooth with respect to $z_j$ (by which we mean jointly $C^{2{\alpha }_j-1}$-smooth with respect to $\mathrm{Re}{z}_j$, $\mathrm{Im}{z}_j$). If $f$ is $\alpha$-analytic on $\Omega \setminus f^{-1}\left(0\right)$, then $f$ is $\alpha$-analytic on $\Omega$. The result is well-known for the case ${\alpha }_i=1$, $1\le i\le n$, even when $f$ a priori is only known to be continuous.

Si prova resistenza locale della soluzione di una equazione di Riccati che si incontra in un problema di controllo ottimale. In ipotesi di regolarità per il costo si prova resistenza globale. Il problema astratto considerato è il modello di alcuni problemi di controllo ottimale governati da equazioni paraboliche con controllo sulla frontiera.