An assessment of two popular Padé-based approaches for fast frequency sweeps of time-harmonic finite element problems

Multivariate Padé Approximants for Finite Element Solutions With Complex Parametric Dependence

“…(1) at a specific point x 0 in the parameter space. The WCAWE algorithm provides a robust generation of this sequence of successive derivatives, overcoming the inherently ill-conditioned transformation matrix emerging from the direct approach used in the component-wise Padé approximants [3,4], also used for the generation of Nested Padé approximants [7,8]. The resulting well-conditioned transformation matrix V N , consisting of N orthonormalized basis vectors, allows for a robust, non-stagnating convergence upon increasing the size of the subspace spanned.…”

“…In practice, the size of the reduction basis may be limited due to the increase of the computational cost of the procedure in Eq. (4) [3]. This implies that there may be a tradeoff between increasing the order of approximation, reflected in the size of the basis for a given reference point, and a multi-interval strategy.…”

“…In order to limit the high computational burden associated with the evaluation of multivariate Finite Element (FE) solutions, an approach based on a Reduced-Order-Model (ROM), derived after a univariate moment-matching method, is proposed. This univariate method, namely the Well-Conditioned Asymptotic Waveform Evaluation (WCAWE) method, proposed by Slone et al [1], is shown to allow for a robust generation of basis components with good convergence properties of the associated reduced systems [1][2][3]. It relies on a sequence of successive derivatives of the system of interest, leading to multiple Right-Hand-Side (RHS) problems to be solved sequentially, with intermediate orthonormalization steps and correction terms ensuring the robustness of the procedure and the good conditioning of the resulting transformation matrix.…”

“…This univariate basis generation procedure may be extended to a multivariate procedure. However, the cost associated with the orthonormalization and correction steps, may lead to a computationally prohibitive cost, particularly as the dimension of the variable space increases [3]. Additionally, not all cross-derivative terms may contribute significantly to the convergence of the ROM, and it may be critical to be able to reduce them to a minimum.…”

“…The foundation for a multi-interval strategy is also presented in view of the reconstruction of wide parametric ranges. The approach is shown to benefit from the superior robustness of the WCAWE compared to the so-called component-wise Padé approximants [3][4][5][6], previously used to derive the Nested Padé approximant method [7,8].…”