Performance guarantees for a variational “multi-space” decoder

Abstract

Model-order reduction methods tackle the following general approximation problem: find an “easily-computable” but accurate approximation $\hat {\boldsymbol {h}}$ of some target solution h^⋆. In order to achieve this goal, standard methodologies combine two main ingredients: (i) a set of partial observations of h^⋆ and (ii) some “simple” prior model on the set of target solutions. The most common prior models encountered in the literature assume that the target solution h^⋆ is “close” to some low-dimensional subspace. Recently, triggered by the work by Binev et al. (SIAM/ASA Journal on Uncertainty Quantification 5(1), 1–29, 2017), several contributions have shown that refined prior models (based on a set of embedded approximation subspaces) may lead to enhanced approximation performance. In this paper, we focus on a particular decoder exploiting such a “multi-space” information and evaluating $\hat {\boldsymbol {h}}$ as the solution of a constrained optimization problem. To date, no theoretical results have been derived to support the good empirical performance of this decoder. The goal of the present paper is to fill this gap. More specifically, we provide a mathematical characterization of the approximation performance achievable by this variational “multi-space” decoder and emphasize that, in some specific setups, it has provably better recovery guarantees than its standard “single-space” counterpart. We also discuss the similarities and differences between this decoder and the one proposed in Binev et al. (SIAM/ASA Journal on Uncertainty Quantification 5(1), 1–29, 2017).

Appendix: Proof of (42)

In this appendix, we show that the cost function $f({{\boldsymbol {h}}})\triangleq {\sum }_{j=1}^{{n}} {\left ({y}_{j}- \left \langle {{\boldsymbol w}_{j},{{\boldsymbol {h}}}}\right \rangle \right )}^{2}$ can be rewritten as in (42) when h ∈ V_n and σ_n > 0.¹ First, using the definition of y_j, we have

$$ f({{\boldsymbol{h}}})= \sum\limits_{j=1}^{{n}} {\left( \left\langle{{{\boldsymbol{w}}}_{j},{\boldsymbol{h}}^{\star}}\right\rangle - \left\langle{{{\boldsymbol{w}}}_{j},{{\boldsymbol{h}}}}\right\rangle\right)}^{2}. $$

(73)

Moreover, using the particular bases introduced in Section 5.1.1, we obtain

$$ \begin{array}{@{}rcl@{}} f({{\boldsymbol{h}}}) &=& \sum\limits_{j=1}^{{n}} {\left( \left\langle{{{\boldsymbol{w}}}_{j}^{*},{\boldsymbol{h}}^{\star}}\right\rangle - \left\langle{{{\boldsymbol{w}}}_{j}^{*},{{\boldsymbol{h}}}}\right\rangle\right)}^{2},\\ &=& \sum\limits_{j=1}^{{n}} {\left( \left\langle{{{\boldsymbol{w}}}_{j}^{*},{\boldsymbol{h}}^{\star}}\right\rangle - {\sigma}_{j} \left\langle{{\boldsymbol v}_{j}^{*},{{\boldsymbol{h}}}}\right\rangle\right)}^{2}, \end{array} $$

(74)

where the first equality follows from the fact that $\{{{\boldsymbol {w}}}_{j}\}_{j=1}^{{n}}$ and $\{{{\boldsymbol {w}}}_{j}^{*}\}_{j=1}^{{n}}$ differ up to an orthogonal transformation; the second is a consequence of (55) and our hypothesis h ∈ V_n.

Since ${{\hat {\boldsymbol {h}}}_{\text {SS}}}$ corresponds to the minimum of f(h) over V_n (see (17)), we simply have

$$ \left\langle{{\boldsymbol v}_{j}^{*},{\hat{{\boldsymbol{h}}}_{\text{SS}}}}\right\rangle = \frac{\left\langle{{{\boldsymbol{w}}}_{j}^{*},{\boldsymbol{h}}^{\star}}\right\rangle}{{\sigma}_{j}}, $$

(75)

if σ_n > 0. Hence, under this assumption, (2) can also be rewritten as in (42).