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Fast Least-Squares Padé approximation of problems with normal operators and meromorphic structure


Authors: Francesca Bonizzoni, Fabio Nobile, Ilaria Perugia and Davide Pradovera
Journal: Math. Comp. 89 (2020), 1229-1257
MSC (2010): Primary 41A21, 65D15, 35P15; Secondary 41A25, 35J05
DOI: https://doi.org/10.1090/mcom/3511
Published electronically: January 21, 2020
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Abstract: In this work, we consider the approximation of Hilbert space-valued meromorphic functions that arise as solution maps of parametric PDEs whose operator is the shift of an operator with normal and compact resolvent, e.g., the Helmholtz equation. In this restrictive setting, we propose a simplified version of the Least-Squares Padé approximation technique studied in [ESAIM Math. Model. Numer. Anal. 52 (2018), pp. 1261-1284] following [J. Approx. Theory 95 (1998), pp. 203-2124]. In particular, the estimation of the poles of the target function reduces to a low-dimensional eigenproblem for a Gramian matrix, allowing for a robust and efficient numerical implementation (hence the ``fast'' in the name). Moreover, we prove several theoretical results that improve and extend those in [ESAIM Math. Model. Numer. Anal. 52 (2018), pp. 1261-1284], including the exponential decay of the error in the approximation of the poles, and the convergence in measure of the approximant to the target function. The latter result extends the classical one for scalar Padé approximation to our functional framework. We provide numerical results that confirm the improved accuracy of the proposed method with respect to the one introduced in [ESAIM Math. Model. Numer. Anal. 52 (2018), pp. 1261-1284] for differential operators with normal and compact resolvent.


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Additional Information

Francesca Bonizzoni
Affiliation: Faculty of Mathematics, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
Email: francesca.bonizzoni@univie.ac.at

Fabio Nobile
Affiliation: CSQI – MATH, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland
Email: fabio.nobile@epfl.ch

Ilaria Perugia
Affiliation: Faculty of Mathematics, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
Email: ilaria.perugia@univie.ac.at

Davide Pradovera
Affiliation: CSQI – MATH, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland
Email: davide.pradovera@epfl.ch

DOI: https://doi.org/10.1090/mcom/3511
Received by editor(s): August 31, 2018
Received by editor(s) in revised form: August 15, 2019
Published electronically: January 21, 2020
Additional Notes: The first author acknowledges partial support from the Austrian Science Fund (FWF) through the project F 65, and has been supported by the FWF Firnberg-Program, grant T998.
The third author was funded by the Austrian Science Fund (FWF) through the projects F 65 and P 29197-N32, and by the Vienna Science and Technology Fund (WWTF) through the project MA14-006.
The fourth author was funded by the Swiss National Science Foundation (SNF) through project 182236.
Article copyright: © Copyright 2020 American Mathematical Society