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# Quasi-optimality of approximate solutions in normed vector spaces

Roman Andreev

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## To cite this version:

Roman Andreev. Quasi-optimality of approximate solutions in normed vector spaces. 2016. hal-01338040

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Preprint submitted on 27 Jun 2016

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Distributed under a Creative Commons Attribution 4.0 International License
QUASI-OPTIMALITY OF APPROXIMATE SOLUTIONS IN NORMED VECTOR SPACES

ROMAN ANDREEV†

ABSTRACT. We discuss quasi-optimality of approximate solutions to operator equations in normed vector spaces, defined either by Petrov-Galerkin projection or by residual minimization. Examples demonstrate the sharpness of the estimates.

Let $X$ and $Y$ be real normed vector spaces. Let $B: X \to Y'$ be a linear operator. Fix $u \in X$ – the “unknown”. Let $X_h \times Y_h \subset X \times Y$ be nontrivial finite-dimensional subspaces. Abbreviate

$$ (1) \qquad \gamma_h := \inf_{w \in X_h \setminus \{0\}} \|Bw\|_{Y'_h} / \|w\|_X \quad \text{and} \quad \|B\| := \sup_{w \in (u+X_h) \setminus \{0\}} \|Bw\|_{Y'_h} / \|w\|_X. $$

Throughout, we assume the “discrete inf-sup condition”: $\gamma_h > 0$. We define $B_h: X_h \to Y'_h$ by $w \mapsto (Bw)|_{Y'_h}$. In the first proposition we require $\dim X_h = \dim Y_h$. In the second we admit $\dim Y_h \ge \dim X_h$.

**Proposition 1.** Suppose $\dim X_h = \dim Y_h$. Then there exists a unique $u_h \in X_h$ such that

$$ (2) \qquad \langle Bu_h, v \rangle = \langle Bu, v \rangle \quad \forall v \in Y_h. $$

The mapping $u \mapsto u_h$ is linear with $\|u_h\|_X \le \gamma_h^{-1} \|Bu\|_{Y'_h}$ and satisfies the quasi-optimality estimate:

$$ (3) \qquad \|u - u_h\|_X \le (1 + \gamma_h^{-1} \|B\|) \inf_{w_h \in X_h} \|u - w_h\|_X. $$

*Proof.* The map $B_h$ is linear and injective by (1). It is bijective due to finite $\dim X_h = \dim Y_h = \dim Y'_h$. Thus a unique $u_h := B_h^{-1}(Bu)|_{Y_h}$ exists and $u \mapsto u_h$ is linear. By (1), $\gamma_h \|u_h\|_X \le \|B_h u_h\|_{Y'_h} = \|Bu\|_{Y'_h}$. From $\|u - u_h\|_X \le \|u - w_h\|_X + \|w_h - u_h\|_X$ and $\gamma_h \|w_h - u_h\|_X \le \|B(u_h - w_h)\|_{Y'_h} = \|B(u - w_h)\|_{Y'_h} \le \|B\| \|u - w_h\|_{Y'_h}$ we obtain (3). $\square$

**Proposition 2.** The set $U_h := \operatorname{argmin}_{w_h \in X_h} \|Bu - Bw_h\|_{Y'_h} \subset X_h$ of residual minimizers is nonempty, convex and bounded. Any $u_h \in U_h$ satisfies the quasi-optimality estimate

$$ (4) \qquad \|u - u_h\|_X \le (1 + 2\gamma_h^{-1} \|B\|) \inf_{w_h \in X_h} \|u - w_h\|_X. $$

*Proof.* The first statement is elementary: consider the metric projection of $(Bu)|_{Y_h} \in Y'_h$ onto $B_h X_h \subset Y'_h$. Quasi-optimality is obtained as above, except that $\|B(u_h - w_h)\|_{Y'_h} \le \|B(u - u_h)\|_{Y'_h} + \|B(u - w_h)\|_{Y'_h} \le 2\|B(u - w_h)\|_{Y'_h}$. $\square$

The set $U_h$ of minimizers is a singleton if the unit ball of $Y'_h$ is strictly convex. Since $Y_h$ is finite-dimensional, this is the case if and only if the norm of $Y_h$ is Gâteaux differentiable.

The constants in (3) and (4) are sharp: Take $X = Y = \mathbb{R}^2$ with the $|\cdot|_1$ norm. Then $|\cdot|_\infty$ is the norm of $Y'$. Take $u := (0, 1)$ and $B(w_1, w_2) := (w_1 + w_2, w_2)$. Set $X_h := \mathbb{R} \times \{0\}$ ($\rightsquigarrow$ B is identity on $X_h$). Observe $\|B\| = 1$.

* For (3) let $Y_h := \mathbb{R} \times \{0\}$. Then $\|Bw_h\|_{Y'_h} = \|w_h\|_X$ for all $w_h \in X_h$ gives $\gamma_h = 1$. Now, $u_h = (1, 0) \in X_h$ solves (2). In the quasi-optimality estimate we have $\|u - u_h\|_X = 2$ while $\|u - w_h\|_X = 1$ for $w_h = 0$.

* For (4) let $Y_h := Y$. Again, $\gamma_h = 1$. Since $Bu = (1, 1)$, the set of minimizers $U_h$ is the segment $[0, 2] \times \{0\}$. For $u_h := (2, 0) \in U_h$ we have $\|u - u_h\|_X = 3$ while $\|u - w_h\|_X = 1$ for $w_h = 0$. With a slight perturbation of the norms, say, we can achieve $U_h = \{u_h\}$ without essentially changing the distances.

If $X$ and $Y$ are Hilbert spaces and $B: X \to Y'$ is bounded by $\|B\|$ then in both propositions the mapping $P_h: X \to X$, $u \mapsto u_h$, is a well-defined bounded linear projection with $\|P_h\| \le \gamma_h^{-1} \|B\|$. The argument of

[1] J. Xu and L. Zikatanov. Some observations on Babuška and Brezzi theories. Numer. Math., 94(1), 2003.

then improves the quasi-optimality estimate to $\|u - u_h\|_X \le \|P_h\| \inf_{w_h \in X_h} \|u - w_h\|_X$.

<footnote>†UNIVERSITÉ PARIS DIDEROT, SORBONNE PARIS CITÉ, LJLL (UMR 7598 CNRS), F-75205, PARIS, FRANCE.</footnote>

<footnote>E-mail address: roman.andreev@upmc.fr</footnote>

<footnote>Date: June 27, 2016. MSC (2010): 65N30. Support: Swiss NSF #164616.</footnote>