Reduced modelling and approximation from point evaluations

Talk in Mathematics
In approximation of functions, one often seeks a reduced model, that is, a linear space of moderate dimension that best approaches the considered class of functions. Once such a space is fixed, the question remains as to how to reconstruct a given function in this space, based on a limited number of measurements. In particular, when only point evaluations can be queried, we show that weighted least-squares can achieve near-optimal recovery error, with a sample size close to the reduced model dimension. These results are obtained through a combination of randomized sampling strategies andlinear algebra for sums of rank-one matrices.