The combination of the Reduced Basis (RB) and the Empirical Interpolation Method (EIM) approaches have produced outstanding results in gravitational wave (GW) science and in many other disciplines. In GW science in particular, these results range from building non-intrusive surrogate models for gravitational waves to fast parameter estimation adding the use of Reduced Order Quadratures. These surrogates have the salient feature of being essentially indistinguishable from or very close to supercomputer simulations of the Einstein equations  but can be evaluated in less than a second on a laptop. In this note we analyze in detail how the EIM at each iteration attempts to choose the interpolation nodes so as to make the related Vandermonde-type matrix " as invertible as possible " as well as attempting to optimize the conditioning of its inversion and minimizing a Lebesgue-type constantfor accuracy. We then compare through numerical experiments the EIM performance with fully optimized nested variations. We also discuss global optimal solutions through Fekete nodes. We find that in these experiments the EIM is actually close to optimal solutions but can be improved with small variations and relative low computational cost .
The combination of the Reduced Basis and the Empirical Interpolation Method (EIM) approaches have produced outstanding results in many disciplines. In particular, in gravitational wave (GW) science these results range from building non-intrusive surrogate models for GWs to fast parameter estimation adding the use of Reduced Order Quadratures. These surrogates have the salient feature of being essentially indistinguishable from or very close to supercomputer simulations of the Einstein equations , but can be evaluated in the order of a milisecond per multipole mode on a standard laptop. In this article we clarify a common misperception of the EIM as originally introduced and used in practice in GW science. Namely, we prove that the EIM at each iteration chooses the interpolation nodes so as to make the related Vandermonde-type matrix as invertible as possible ; not necessarily optimizing its conditioning or accuracy of the interpolant as is sometimes thought. In fact, we introduce two new variations of the EIM, nested as well, which do optimize with respect to conditioning and the Lebesgue constant, respectively, and compare them through numerical experiments with the original EIM using GWs. Our analyses and numerical results suggest a subtle relationship between solving for the original EIM, conditioning, and the Lebesgue constant, in consonance with active research in rigorous approximation theory and related fields .
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the EIM performance with fully optimized nested variations. We also discuss global optimal solutions through Fekete nodes. We find that in these experiments the EIM is actually close to optimal solutions but can be improved with small variations and relative low computational cost
with the original EIM using GWs. Our analyses and numerical results suggest a subtle relationship between solving for the original EIM, conditioning, and the Lebesgue constant, in consonance with active research in rigorous approximation theory and related fields