Abstract
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). 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.
Highlights
Fourier transforms lie at the heart of applications in optics, electric engineering, chemistry, probability, partial differential equations, statistics and finance
We focus on the case of parametric Fourier transforms in the subsequent section 4
We introduce the magic point empirical interpolation method for parametric integration to approximate parametric integrals of the form (1)
Summary
At the basis of a large variety of mathematical applications lies the computation of parametric integrals of the form (1). IM (h)(p, z) := hp(zm∗ )θmM (z) m=1 is constructed to approximate all functions from the set {hp| p ∈ P} well in the L∞-norm This allows us to define the magic point integration operator (6). The second step of the offline phase is a greedy procedure, where the discrete empirical interpolation (DEIM) extracts empirical integration points from the reduced basis Both approaches differ as well: Antil et al (2013) employ a twostage procedure. The two approaches, empirical interpolation and low-rank tensor decompositions, are closely interrelated as discussed in Bebendorf et al (2014) In contrast to these approaches, magic point integration uses the actual function set of interest as an input into the learning procedure. We use the magic point integration method to evaluate densities of a parametric class of distributions that are defined through their Fourier transforms
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