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

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 $z_0$.