A quasi-optimal coarse problem and an augmented Krylov solver for the Variational Theory of Complex Rays

The Variational Theory of Complex Rays (VTCR) is an indirect Trefftz method designed to study systems governed by Helmholtz-like equations. It uses wave functions to represent the solution inside elements, which reduces the dispersion error compared to classical polynomial approaches but the resulting system is prone to be ill conditioned. This paper gives a simple and original presentation of the VTCR using the discontinuous Galerkin framework and it traces back the ill-conditioning to the accumulation of eigenvalues near zero for the formulation written in terms of wave amplitude. The core of this paper presents an efficient solving strategy that overcomes this issue. The key element is the construction of a search subspace where the condition number is controlled at the cost of a limited decrease of attainable precision. An augmented LSQR solver is then proposed to solve efficiently and accurately the complete system. The approach is successfully applied to different examples.Comment: International Journal for Numerical Methods in Engineering, Wiley, 201

A Universal Framework for Holographic MIMO Sensing

This paper addresses the sensing space identification of arbitrarily shaped continuous antennas. In the context of holographic multiple-input multiple-output (MIMO), a.k.a. large intelligent surfaces, these antennas offer benefits such as super-directivity and near-field operability. The sensing space reveals two key aspects: (a) its dimension specifies the maximally achievable spatial degrees of freedom (DoFs), and (b) the finite basis spanning this space accurately describes the sampled field. Earlier studies focus on specific geometries, bringing forth the need for extendable analysis to real-world conformal antennas. Thus, we introduce a universal framework to determine the antenna sensing space, regardless of its shape. The findings underscore both spatial and spectral concentration of sampled fields to define a generic eigenvalue problem of Slepian concentration. Results show that this approach precisely estimates the DoFs of well-known geometries, and verify its flexible extension to conformal antennas

Vekua theory for the Helmholtz operator

Vekua operators map harmonic functions defined on domain in $${\mathbb R^{2}}$$ to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907-1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N≥2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane wave

Vekua theory for the Helmholtz operator

Vekua operators map harmonic functions defined on domain in \mathbb R2R2 to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves