Parametric Integration by Magic Point Empirical Interpolation

We derive analyticity criteria for explicit error bounds and an exponential rate of convergence of the magic point empirical interpolation method introduced by Barrault et al. (2004, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math.,339, 667–672). Furthermore, we investigate its application to parametric integration. We find that the method is well-suited to Fourier transforms and has a wide range of applications in such diverse fields as probability and statistics, signal and image processing, physics, chemistry and mathematical finance. To illustrate the method, we apply it to the evaluation of probability densities by parametric Fourier inversion. Our numerical experiments display convergence of exponential order, even in cases where the theoretical results do not apply. The magic point integration performs considerably well, when compared with Clenshaw–Curtis quadrature and with Chebyshev interpolation in the parameter space