On the stability and accuracy of the Empirical Interpolation Method and Gravitational Wave Surrogates
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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 constant for 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.
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