The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic partial differential equations. Abstractly, it can be viewed as determining a "good" n dimensional space H n to be used in approximating the elements of a compact set F in a Hilbert space H. One, by now popular, computational approach is to find H n through a greedy strategy. It is natural to compare the approximation performance of the H n generated by this strategy with that of the Kolmogorov widths d n (F) since the latter gives the smallest error that can be achieved by subspaces of fixed dimension n. The first such comparisons, given in [1], show that the approximation error, σ n (F) := dist(F, H n ), obtained by the greedy strategy satisfies σ n (F) ≤ Cn2 n d n (F). In this paper, various improvements of this result will be given. Among these, it is shown that whenever d n (F) ≤ M n −α , for all n > 0, and some M, α > 0, we also have σ n (F) ≤ C α M n −α for all n > 0, where C α depends only on α. Similar results are derived for generalized exponential rates of the form M e −an α . The exact greedy algorithm is not always computationally feasible and a commonly used computationally friendly variant can be formulated as a "weak greedy algorithm". The results of this paper are established for this version as well.

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Measuring picometre-scale shifts in the positions of individual atoms in materials provides new insight into the structure of surfaces, defects and interfaces that influence a broad variety of materials' behaviour. Here we demonstrate sub-picometre precision measurements of atom positions in aberration-corrected Z-contrast scanning transmission electron microscopy images based on the non-rigid registration and averaging of an image series. Non-rigid registration achieves five to seven times better precision than previous methods. Non-rigidly registered images of a silica-supported platinum nanocatalyst show pm-scale contraction of atoms at a (1 11)/( 1 11) corner towards the particle centre and expansion of a flat (1 11) facet. Sub-picometre precision and standardless atom counting with o1 atom uncertainty in the same scanning transmission electron microscopy image provide new insight into the threedimensional atomic structure of catalyst nanoparticle surfaces, which contain the active sites controlling catalytic reactions.

This paper considers the problem of optimal recovery of an element u of a Hilbert space H from measurements of the form ℓ j (u), j = 1, . . . , m, where the ℓ j are known linear functionals on H. Problems of this type are well studied [18] and usually are carried out under an assumption that u belongs to a prescribed model class, typically a known compact subset of H. Motivated by reduced modeling for solving parametric partial differential equations, this paper considers another setting where the additional information about u is in the form of how well u can be approximated by a certain known subspace V n of H of dimension n, or more generally, in the form of how well u can be approximated by each of a sequence of nested subspaces V 0 ⊂ V 1 · · · ⊂ V n with each V k of dimension k. A recovery algorithm for the one-space formulation was proposed in [16]. Their algorithm is proven, in the present paper, to be optimal. It is also shown how the recovery problem for the one-space problem, has a simple formulation, if certain favorable bases are chosen to represent V n and the measurements. The major contribution of the present paper is to analyze the multi-space case. It is shown that, in this multi-space case, the set of all u that satisfy the given information can be described as the intersection of a family of known ellipsoids in H. It follows that a near optimal recovery algorithm in the multi-space problem is provided by identifying any point in this intersection. It is easy to see that the accuracy of recovery of u in the multi-space setting can be much better than in the one-space problems. Two iterative algorithms based on alternating projections are proposed for recovery in the multi-space problem and one of them is analyzed in detail. This analysis includes an a posteriori estimate for the performance of the iterates. These a posteriori estimates can serve both as a stopping criteria in the algorithm and also as a method to derive convergence rates. Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for u.