Abstract
The present work deals with rational model order reduction methods based on the single-point Least-Square (LS) Padé approximation techniques introduced in Bonizzoni et al. (ESAIM Math. Model. Numer. Anal., 52(4), 1261–1284 2018, Math. Comput. 89, 1229–1257 2020). Algorithmical aspects concerning the construction of rational LS-Padé approximants are described. In particular, we show that the computation of the Padé denominator can be carried out efficiently by solving an eigenvalue-eigenvector problem involving a Gramian matrix. The LS-Padé techniques are employed to approximate the frequency response map associated with two parametric time-harmonic acoustic wave problems, namely a transmission-reflection problem and a scattering problem. In both cases, we establish the meromorphy of the frequency response map. The Helmholtz equation with stochastic wavenumber is also considered. In particular, for Lipschitz functionals of the solution and their corresponding probability measures, we establish weak convergence of the measure derived from the LS-Padé approximant to the true one. 2D numerical tests are performed, which confirm the effectiveness of the approximation methods.
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Funding
Open access funding provided by Austrian Science Fund (FWF). F. Bonizzoni 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.
I. Perugia has been funded by the Vienna Science and Technology Fund (WWTF) through the project MA14-006, and by the Austrian Science Fund (FWF) through the projects P 29197-N32 and F 65.
D. Pradovera has been funded by the Swiss National Science Foundation (SNF) through project 182236.
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Communicated by: Anthony Nouy
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Bonizzoni, F., Nobile, F., Perugia, I. et al. Least-Squares Padé approximation of parametric and stochastic Helmholtz maps. Adv Comput Math 46, 46 (2020). https://doi.org/10.1007/s10444-020-09749-3
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DOI: https://doi.org/10.1007/s10444-020-09749-3
Keywords
- Hilbert space-valued meromorphic maps
- Padé approximants
- Convergence of Padé approximants
- Parametric Helmholtz equation
- PDE with random coefficients