On a refinement-free Calderón multiplicative preconditioner for the electric field integral equation
Introduction
The electric field integral equation (EFIE), which is used for solving electromagnetic scattering and radiation problems, results in an ill-conditioned linear system of equations when discretized with Rao–Wilton–Glisson (RWG) basis functions—the common choice in standard codes. The ill-conditioning stems from two issues: the low-frequency breakdown, which is due to the different scaling of the vector and the scalar potential in frequency, and the dense-discretization breakdown, which is due to the fact that the vector and the scalar potential are pseudo-differential operators of negative and positive order [1], a property that leads in general to ill-conditioned system matrices. Altogether, the condition number of the EFIE system matrix grows as , where k is the wavenumber and h is the average edge length of the mesh, leading to slowly or non-converging iterative solvers [2].
The low-frequency breakdown has been overcome in the past by using explicit quasi-Helmholtz decompositions such as the loop-star or the loop-tree decomposition [3], [4], [5], [6], [7]. While these decompositions cure the low-frequency breakdown, the dense-discretization breakdown persists and is even worsened to a scaling of the condition number in the case of the loop-star decomposition [8]. Some other low-frequency stable methods have been proposed in the past such as the augmented EFIE, the rearranged loop-tree decomposition, or the augmented EFIE with normally constrained magnetic field and static charge extraction [9], [10], [11], yet all of them suffer from h-ill-conditioning.
Different strategies have been presented to overcome the dense-discretization breakdown. A first class of techniques relies on algebraic strategies such as the incomplete LU factorization [12], [13], sparse approximate inverse [14], [15], or near-range preconditioners [16]. While they improve the conditioning, the condition number still grows with decreasing h.
In recent years more elaborate explicit quasi-Helmholtz decompositions have been presented, the so-called hierarchical basis preconditioners for both structured [17], [18], [19], [20] and unstructured meshes [21], [22] (in the mathematical community, the hierarchical basis preconditioners are typically termed multilevel or prewavelet preconditioners). The best of these methods can yield a condition number that only grows logarithmically in [20], [22] meaning that there is still space left for improvement.
In fact, Calderón identity-based preconditioners yield a condition number that has an upper bound independent from h [23], [24], [25], [26], [27]. In the static limit, however, the Calderón strategies stop working due to numerical cancellation in both the right-hand side excitation vector and the unknown current, since solenoidal and non-solenoidal components scale differently in k [28]. Explicit quasi-Helmholtz decompositions do not suffer from this cancellation since the solenoidal and non-solenoidal components are stored separately. To make Calderón preconditioners stable in the static limit, one could combine the Calderón multiplicative preconditioner (CMP) with an explicit quasi-Helmholtz decomposition.
Such an approach has a severe downside: if the geometry is multiply connected, then the quasi-harmonic global loop functions have to be added to the basis of the decomposition [6]. In contrast to loop, star, or tree functions, (or any of the hierarchical bases mentioned here), the construction of the global loops becomes costly if the genus g is proportional to the number of unknowns N resulting in the overall complexity , where N is the number of unknowns (see, for example, the discussion in [29]). In order to avoid the construction of the global loops, a modified CMP has been presented which leverages on an implicit quasi-Helmholtz decomposition based on projectors [30]. These projectors require the application of the inverse primal (i.e., cell-based) and the inverse dual (i.e., vertex-based) graph Laplacian, a task for which blackbox-like preconditioners such as algebraic multigrid methods can be used for rapidly obtaining the inverse.
In order to avoid the inversion of graph Laplacians, one could think that an alternative might be the Calderón preconditioner combined with an explicit loop-star quasi-Helmholtz decomposition as described in the penultimate paragraph, at least if no global loops are present. However, the inverse Gram matrices appearing in such a scheme are all spectrally equivalent to discretized Laplace–Beltrami operators. Different from the graph Laplacians, these Gram matrices are not symmetric since the loop-star basis is applied to a mixed Gram matrix, that is, Buffa–Christiansen (BC) functions are used as expansion and rotated RWG functions are used as testing functions [8], [30]. In general, this complication makes it more challenging to stably invert these Gram matrices compared with the graph Laplacians of the Helmholtz projectors since, for example, many algebraic multigrid preconditioners require symmetric matrices.
The work in [25], [30] demonstrated a Calderón scheme that can be relatively easily integrated in existing codes. Instead of discretizing the operator on the standard mesh with RWG and BC functions, only a single discretization with RWG functions on the barycentrically refined mesh is necessary. The disadvantage of this approach is that the memory consumption as well as the costs for a single-matrix vector product are increased by a factor of six.
In this work, we propose a refinement-free Calderón multiplicative preconditioner (RF-CMP) for the EFIE. In contrast to existing Calderón preconditioners, no BC functions are employed, so that a standard discretization of the EFIE with RWG functions can be used. What is more, we get a system matrix, which is Hermitian, positive definite (HPD). We obtain this result by leveraging on spectral equivalences between the single layer and the hypersingular operator known from electrostatics on the one hand and the Laplace–Beltrami operator on the other hand. Similar to [30], graph Laplacians need to be inverted. Since the new system matrix is HPD, we are allowed to employ the conjugate gradient (CG) solver. Different from other Krylov subspace methods, it guarantees convergence and has the least computational overhead. The numerical results corroborate the new formulation. Preliminary results have been presented at conferences [31], [32].
The paper is structured as follows. Section 2 sets the background and notation; Section 3 introduces the new formulation and provides the theoretical apparatus. Numerical results demonstrating the effectiveness of the new approach are shown in Section 4.
Section snippets
Notation and background
In the following, we denote quantities residing in , such as the electric field E, with an italic, bold, serif font. For any other vectors and matrices we use an italic, bold, sans-serif font, and we distinguish matrices from vectors by using capital letters for matrices and minuscules for vectors. The expression has to be read as , where is a constant independent of the mesh parameter h (i.e., the average edge length). Furthermore, means that and holds. Let and
New formulation
The formulation for which we are going to show that the system matrix is well-conditioned in the static limit reads where the outer matrix is with and the middle matrix is using and with the conjugate transpose . The unknown vector is recovered by .
The use of the imaginary unit in the definition of is motivated for the same reason as it was for : to prevent the numerical cancellation due to different scaling of the solenoidal and
Numerical results
For the implementation of the preconditioner, we did not use the wavenumber k directly to cure the low-frequency breakdown. Instead, we used the following definitions where These norms are estimated using the power iteration algorithm. Typically the condition number obtained by using norms is lower than using thereby, the number of iterations used by a Krylov subspace method is reduced (and this saving usually outweighs the costs for estimating the norms).
First, we considered a
Conclusion
We presented a preconditioner for the EFIE that yields a Hermitian, positive definite, and well-conditioned system matrix. Due to the applicability of the CG method, there is a—at least theoretically—guaranteed convergence and the complexity for obtaining such a solution is if a fast method is used to compress the EFIE system matrix (and if it is assumed that the algebraic multigrid methods allow to solve Laplacian systems in complexity). Preliminary results indicate that an
Acknowledgement
This work has been funded in part by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (ERC project 321, grant No. 724846).
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