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An efficient algorithm for Padé-type approximation of the frequency response for the Helmholtz problem

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Abstract

We consider the map $\mathcal{S}:\mathbb{C}\to H^1_0(\Omega)=\{v\in H^1(D), v|_{\partial\Omega}=0\}$, which associates a complex value z with the weak solution of the (complex-valued) Helmholtz problem $-\Delta u-zu=f$ over $\Omega$ for some fixed $f\in L^2(\Omega)$. We show that $\mathcal{S}$ is well-defined and meromorphic in $\mathbb{C}\setminus\Lambda$, $\Lambda=\{\lambda_\alpha\}_{\alpha=1}^\infty$ being the (countable, unbounded) set of (real, non-negative) eigenvalues of the Laplace operator (restricted to $H^1_0(\Omega)$). In particular, it holds $\mathcal{S}(z)=\sum_{\alpha=1}^\infty\frac{s_\alpha}{\lambda_\alpha-z}$, where the elements of $\{s_\alpha\}_{\alpha=1}^\infty\subset H^1_0(\Omega)$ are pair-wise orthogonal with respect to the $H^1_0(\Omega)$ inner product. We define a Pad\'e-type approximant of any map as above around $z_0\in\mathbb{C}$: given some integer degrees of the numerator and denominator respectively, $M,N\in\mathbb{N}$, the exact map is approximated by a rational map $\mathcal{S}_{[M/N]}:\mathbb{C}\setminus\Lambda\to H^1_0(\Omega)$. We define such approximant within a Least-Squares framework, through the minimization of a suitable functional based on samples of the target solution map and of its derivatives at $z_0$. In particular, the denominator of the approximant is the minimizer (under some normalization constraints) of the $H^1_0(\Omega)$ norm of a Taylor coefficient of $Q\mathcal{S}$, as Q varies in the space of polynomials with degree $\leq N$. The numerator is then computed by matching as many terms as possible of the Taylor series of $\mathcal{S}$ with those of $\mathcal{S}_{[M/N]}$, analogously to the classical Pad\'e approach. The resulting approximant is shown to converge, as $M+N$ goes to infinity, to the exact map $\mathcal{S}_{[M/N]}$ in the $H^1_0(\Omega)$ norm for values of the parameter sufficiently close to $z_0$ (a sharp bound on the region of convergence is given). Moreover, it is proven that the approximate poles converge exponentially (as M goes to infinity) to the N elements of $\Lambda$ closer to $z_0$.

Author and article information

Affiliations
[1 ]MATH-CSQI, EPFL, Switzerland
[2 ]Department of Mathematics, University of Vienna, Austria
[* ]Correspondence: davide.pradovera@ 123456epfl.ch
Journal
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ScienceOpen
27 April 2018
10.14293/P2199-8442.1.SOP-MATH.HWEDJF.v1
Copyright © 2018

This work has been published open access under Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com.

 ScienceOpen disciplines: Applied mathematics, Applications, Statistics, Data analysis, Mathematics, Mathematical modeling & Computation Keywords: parametric PDEs, rational approximation, meromorphic functions, vector Padé approximation, frequency response