Convergence analysis of Padé approximations for Helmholtz frequency response problems

2018 ◽  
Vol 52 (4) ◽  
pp. 1261-1284 ◽  
Author(s):  
Francesca Bonizzoni ◽  
Fabio Nobile ◽  
Ilaria Perugia
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Padé Approximation ◽  
Approximation Error ◽  
Approximation Technique ◽  
Pade Approximation ◽  
Solution Map ◽  
Type Approximation ◽  
Numerical Tests ◽  
The Laplace Operator

The present work concerns the approximation of the solution map S associated to the parametric Helmholtz boundary value problem, i.e., the map which associates to each (real) wavenumber belonging to a given interval of interest the corresponding solution of the Helmholtz equation. We introduce a least squares rational Padé-type approximation technique applicable to any meromorphic Hilbert space-valued univariate map, and we prove the uniform convergence of the Padé approximation error on any compact subset of the interval of interest that excludes any pole. This general result is then applied to the Helmholtz solution map S, which is proven to be meromorphic in ℂ, with a pole of order one in every (single or multiple) eigenvalue of the Laplace operator with the considered boundary conditions. Numerical tests are provided that confirm the theoretical upper bound on the Padé approximation error for the Helmholtz solution map.

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

2018 ◽  
Author(s):  
Davide Pradovera ◽  
Fabio O. de Nobile ◽  
Francesca Bonizzoni ◽  
Ilaria Perugia
Keyword(s):  
Inner Product ◽  
Taylor Coefficient ◽  
Rational Map ◽  
Solution Map ◽  
Type Approximation ◽  
Negative Eigenvalues ◽  
Helmholtz Problem ◽  
The Laplace Operator ◽  
Region Of Convergence ◽  
Complex Valued

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 $z_0$.

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