 Research
 Open Access
Learning dominant wave directions for plane wave methods for highfrequency Helmholtz equations
 Jun Fang^{1},
 Jianliang Qian^{2},
 Leonardo ZepedaNúñez^{1}Email author and
 Hongkai Zhao^{1}
https://doi.org/10.1186/s4068701700989
© The Author(s) 2017
 Received: 16 August 2016
 Accepted: 5 February 2017
 Published: 1 May 2017
Abstract
We present a raybased finite element method for the highfrequency Helmholtz equation in smooth media, whose basis is learned adaptively from the medium and source. The method requires a fixed number of grid points per wavelength to represent the wave field; moreover, it achieves an asymptotic convergence rate of \(\mathcal {O}(\omega ^{\frac{1}{2}})\), where \(\omega \) is the frequency parameter in the Helmholtz equation. The local basis is motivated by the geometric optics ansatz and is composed of polynomials modulated by plane waves propagating in a few dominant ray directions. The ray directions are learned by processing a lowfrequency wave field that probes the medium with the same source. Once the local ray directions are extracted, they are incorporated into the local basis to solve the highfrequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and highfrequency wave fields iteratively. Finally, a fast solver is developed for solving the resulting linear system with an empirical complexity \(\mathcal {O}(\omega ^d)\) up to a polylogarithmic factor. Numerical examples in 2D are presented to corroborate the claims.
Keywords
 Helmholtz equation
 Numerical microlocal analysis
 RayFEM
1 Background

how to design a discretization that can achieve both accuracy and stability without oversampling; and

how to solve the resulting linear system in linear complexity, up to polylog factors, as the frequency becomes large.
Examples of nonadaptive discretization are: standard finite differences [63, 77], standard continuous or discontinuous finite elements [34, 52, 53, 83, 98], and spectral methods [79, 100, 101], among many others. They are very general in the sense that they can be used for a variety of different problems. However, in the case of the Helmholtz equation they yield either pollution error,^{1} inducing oversampled sparse discretizations [4, 6] whose associated linear systems can be solved in optimal complexity [28, 29, 91, 104, 109], or quasioptimal sparse discretizations whose associated linear systems are prohibitively expensive to solve [44, 101] in the highfrequency regime.^{2}
Adaptive methods, on the other hand, aim to leverage à priori knowledge of the solution of the Helmholtz equation, such as its known oscillatory behavior. In practice, adaptive methods have mostly focused on adaptivity to the medium, such as polynomial Galerkin methods with hp refinement [3, 70, 73, 96, 107, 111], specially optimized finite differences [23, 45, 92, 93, 102] and finite elements [4, 99], enriched finite elements [30–33], plane wave methods [5, 21, 42, 43, 46, 69, 74], generalized plane wave methods [54, 55], locally corrected finite elements [17, 38, 82], and discretizations with specially chosen basis functions [7, 8, 76], among many others. They have been especially successful on reducing the pollution effect by accurately capturing the dispersion relation. However, in the highfrequency regime, either they are not asymptotically quasioptimal for heterogeneous media or they yield linear systems that cannot be solved in quasilinear time with current algorithms.^{3}
Indeed, phasebased methods [41, 49, 78] are instances of fully adaptive discretizations. These methods use (2) to build an approximation space by modulating a polynomial basis with an oscillatory component using the phase functions, which need to be computed beforehand.
However, computing the appropriate global phase functions \(\phi _n(\mathbf {x})\) in the whole domain is a challenging task for a general medium with varying speed; different phase functions may be defined in different regions, whose boundaries are difficult to determine à priori; the error on the solution is proportional to the approximation error of the phase function times \(\omega \),^{4} implying that the phase functions need to be computed extremely accurately, thus, making the computation of the phase functions the bottleneck in such approaches.
The underpinning property used in this approach is that the phase functions are independent of the frequency, and the extraction of their gradient is a stable operation using signal processing algorithms, such as numerical microlocal analysis (NMLA) [10–12]. The resulting linear system is sparse, and it can be solved efficiently using stateoftheart preconditioners such as [28, 29, 91, 104, 109].
1.1 Results
The main result of this paper is an algorithm to solve the Helmholtz equation in the highfrequency regime with an optimal asymptotic cost \({\mathcal O}(\omega ^d)\), up to polylog factors, with respect to the number of intrinsic degrees of freedom.

we build a fully adaptive discretization based on the geometric optics ansatz and local linear approximation of the phase functions whose gradients are learned from a lowfrequency problem solved using standard finite elements; the resulting discretization is stable and asymptotically accurate; in particular, the error converges to zero as \({\mathcal O}(\omega ^{ \frac{1}{2}})\), as the frequency increases;

we solve the resulting linear system using stateoftheart preconditioners with linear complexity, up to polylogarithmic factors.
In particular, we develop a simple raybased finite element method (rayFEM) in 2D for smooth media as a proof of concept study of our proposed approach. We start with a finite element mesh with mesh size h satisfying \(wh={\mathcal O}(1)\), i.e., a few points per wavelength. First, the low frequency is chosen by \(\widetilde{\omega } \sim \sqrt{\omega }\) such that Eq. (4) is solved quasioptimality on such mesh since \(\widetilde{\omega }^2h={\mathcal O}(1)\) [71]. Then NMLA [10–12] (see Sect. 3) is applied to the computed lowfrequency wave field to estimate the local dominant wave directions.
The estimated dominant wave directions are then used to enrich the local finite element basis following (2) in order to discretize the highfrequency Helmholtz equation on the same mesh.
We develop an efficient preconditioner to solve the resulting linear system iteratively using GMRES [88]. The preconditioner is based on the method of polarized traces [109]. Numerical experiments show that it is possible to solve the linear system in \({\mathcal O}(N)\) complexity with a possible polylogarithmic factor for a smooth medium, where N is the total number of unknowns.
Moreover, once a more accurate wave field is computed, it can be used to get a better estimation of the dominant wave directions, which can be used to improve the highfrequency wave field iteratively. If necessary, the solution for the highfrequency Helmholtz equation can also be processed by NMLA to improve the estimation of local dominant wave directions which can be used to further improve the highfrequency solution.
1.2 Related work
In this section we briefly review related approaches to solve the Helmholtz equation, and we compare some of them with the approach proposed in this paper.
As stated in the prequel, it is difficult to design a sparse discretization that can achieve both accuracy and stability under the condition \(\omega h = {\mathcal O}(1)\) as \(\omega \) becomes large. This is mainly due to the pollution effect in error estimates for finite element methods [4, 6], i.e., the ratio between numerical error and best approximation error from a discrete finite element space is \(\omega \) dependent.
From a physical point of view, the wave field governed by the Helmholtz equation contains waves propagating in all directions and satisfying a specific dispersion relation. As a consequence, numerical errors due to dispersion or interpolation for these propagating modes will propagate as physical waves to pollute the whole computed wave field. In particular, a compact stencil on a mesh that is comparable to the wavelength cannot approximate the dispersion relations for propagating waves in all direction uniformly well as \(\omega \rightarrow \infty \) [6].
In order to minimize (or eliminate, if possible) the pollution effect, various approaches have been proposed lately in the literature. Approaches based on polynomial basis coupled with nonstandard variational formulations (such as [75]) have been proposed in order to approximate the Helmholtz operator so that the resulting discrete problems have better stability properties. For example, with an appropriate choice of coefficients, loworder compact finitedifference discretizations can effectively reduce the dispersion error [35, 58, 80]. Other instances of such approaches are the generalized finite element method (GFEM) [4] and continuous interior penalty finite element method (CIPFEM) [107, 111], the interpolated optimized finitedifference method (IOFD) [93, 94], Galerkin methods with hp refinement [70, 72, 73], among many others. These methods successfully reduce the pollution error; however, they require either a more restrictive condition on the mesh size or the degree of the polynomial approximation to be \(\omega \) dependent, resulting on a large increase in the size and interconnectivity of the associated linear systems as the frequency increases.
On the other hand, many approaches rely on specially designed basis in order to accurately represent the solution. One of such approaches is the multiscale Petrov–Galerkin method [17, 38, 82]; the method relies on local corrections, which are numerically computed in a fine mesh, to the basis functions. This method is stable and quasioptimal under the minimal resolution condition \(\omega H = {\mathcal O}(1)\) and \(m = {\mathcal O}(\log \omega )\) for the coarse mesh H and an oversampling parameter m. However, the condition on the fine mesh size, h, to solve the local subscale correction is the same as the standard FEM. It requires \(\omega ^{3/2} h = {\mathcal O}(1) \) for stability [107] and \(\omega ^2 h = {\mathcal O}(1)\) for quasioptimality [71].
Other instances of such approaches are methods that incorporate appropriate oscillatory behavior into the basis of Galerkin methods. The key issue for this strategy is how to design the oscillatory basis. Since the Helmholtz solutions locally behave like plane waves, one approach is to incorporate plane waves with a predetermined equispaced distribution in directions into the basis. For example, products of plane waves with local finite elements basis are used in the generalized finite element methods (GFEMs) [71], partition of unity finite element methods (PUFEM) [5], virtual element methods (VEM) [81], discontinuous Galerkin (DG) methods [36, 42, 46], and ultraweak variational formulation (UWVF) [18, 20, 21]. Trefftztype methods [47] use local solutions of the Helmholtz equation as the basis functions, which in the case of piecewise constant media are plane waves.
It is well known that these plane wavebased methods need fewer DOFs to achieve better accuracy than the conventional finite element methods [47, 64]. A comparison of these methods can be found in [37, 39, 51, 64]. However, these methods have two caveats: They normally perform poorly when the source is not zero, and it is not clear how to choose the number of plane wave directions à priori. In order to achieve a good accuracy, a fine, \(\omega \)dependent [47], resolution in the angle space is required. This refinement in the angle space will not only increase the DOFs significantly but also make the resulting linear system extremely illconditioned due to the numerical coherence of the elements of the basis.
Other basis functions can be utilized, such as Bessel functions [49, 67, 68] to improve the adaptivity to the curvature of the solution’s wavefront and also reduce the linear dependence of the basis. Moreover, generalized plane waves [54–56] in the form \(e^{P(\mathbf {x})}\) with an appropriate complex polynomial \(P(\mathbf {x})\) are developed to achieve highorder convergence for smooth heterogeneous media. Another instance of methods using other basis functions is the discontinuous enrichment method (DEM) [31–33, 97], which combines Lagrange multipliers on the mesh interfaces to enforce continuity of the solution with approximation spaces composed by sums of continuous polynomials and discontinuous plane waves, leading to a reduction of the number of DOFs.
A more adaptive approach to solve the highfrequency Helmholtz equation is based on the geometric optics ansatz of the wave field (2). In the ansatz, phases and amplitudes are independent of frequency and hence are nonoscillatory and smooth except at a measure zero set, e.g., focus points, caustics, corners in a smooth medium. Once the phase functions of the wave fronts are available, the oscillatory pattern of the wave field is known; phasebased numerical methods [14, 41, 49, 78] explicitly incorporate these known phases into the basis functions to significantly improve both stability and accuracy.
As discussed in the prequel, computing the global phase functions for general media is a challenging task. Meanwhile, a phase function can be locally approximated by a linear function with a leading term \(\widehat{\mathbf{d }}_n(\mathbf {x}_0) \cdot \mathbf {x}\), where \(\widehat{\mathbf{d }}_n(\mathbf {x}_0)\) is the local dominant wave direction and can be extracted stably by signal processing algorithms. With precomputed dominant wave directions by ray tracing [15, 16, 22, 49], the dominant plane wave method [14] incorporates them into the local basis to combine the advantages of phasebased methods and plane wave methods. Since only the dominant directions of wave fronts relevant to the problem are involved in this approach, the number of degrees of freedom can be kept minimal, and illconditioning of the resulting linear system due to redundancy can be reduced.
Finally, under the stronger assumption that the medium can be written as a homogeneous background plus a compactly supported perturbation, the Helmholtz equation can be converted to a secondkind integral equation by introducing the Green’s function corresponding to the background, resulting in the socalled Lippmann–Schwinger equation. Recent advances have shown that it is possible to solve the Lippmann–Schwinger equation, and hence the Helmholtz equation, in optimal time [110]. In this paper, however, we treat a more general case.
1.3 Outline of the paper
Here is an outline of this paper. We first describe the rayFEM using the geometric optics ansatz as the motivation and study its approximation property in Sect. 2. In Sect. 3 we introduce the NMLA with its stability and local ray direction error analyzed in “Appendices A and B.” Section 4 provides the full presentation of the numerical algorithm whose empirical complexity is given in Sect. 5. Numerical results are presented in Sect. 6. Conclusions and future works are summarized in Sect. 7.
2 The rayFEM method
In this section we describe the rayFEM method for the Helmholtz equation and its rationale. We explain briefly the geometric optic ansatz and how it is approximated locally via a superposition of plane waves propagating in a set of dominant directions. We then proceed to explain how these plane waves are incorporated into the finite element basis to improve both stability and accuracy of the numerical solution to the highfrequency Helmholtz equation.
In this section we suppose that the dominant directions are known exactly. In Sect. 3 we will describe how to learn the dominant wave directions by probing the medium using lowfrequency waves.
2.1 Geometric optics ansatz

A and \(\phi \) are independent of the frequency \(\omega \);

A and \(\phi \) depend on the medium, \(c(\mathbf {x})\), and the source distribution, \(f(\mathbf {x})\).
The coefficients \(\{A_l\}\) in the asymptotic expansion (6) satisfy a recursive system of transport equations [1, 2, 86] which are coupled with the eikonal equation. Under the assumption that the medium is smooth and no caustic occurs, one may solve the transport equations to estimate the coefficients \(\{A_l\}\) in different formulations [1, 2, 65]. Since the geometric optics term is oscillatory when \(\omega \ne 0\), it should be understood in the \(L^2\) sense rather than the \(L^{\infty }\) sense.
Assuming that the medium is smooth and no caustic occurs, the asymptotic expansion (7) will not fail as long as the frequency parameter \(\omega \) is not zero, but the resulting difference between the asymptotic expansion (7) and the exact solution may be large in the \(L^2\) norm as the frequency approaches zero [86]. Given an inhomogeneous medium, however, it is hard to pin down how large \(\omega \) should be so that the asymptotic expansion (7) is accurate up to a certain specified accuracy, as this is closely related to both fluctuations and correlation lengths of the normalized propagation speed of the medium [106] and the frequency parameter \(\omega \). We refer the reader to [27] for further details on the geometric optics ansatz.
2.2 Local plane wave approximation
From (9) and (13), we have that u can be approximated locally by a superposition of plane waves propagating in certain directions with affine complex amplitudes. Moreover, as \(\omega \rightarrow \infty \), such that \( \omega h={\mathcal O}(1)\), the asymptotic error for the local plane wave approximation (13) is \({\mathcal O}(\omega ^{1})\), which is of the same order as the asymptotic error for the original geometric optics ansatz (9). We use (13) as the motivation to construct local finite element basis with mesh size \(h={\mathcal O}(\omega ^{1})\), in which an affine function is multiplied by plane waves oscillating in those ray directions, resulting in local approximations similar to (13).
2.3 Raybased FEM formulation

the standard FEM (SFEM), where we use loworder \(\mathbb {P}1\) finite elements, i.e., piecewise bilinear functions;

the rayFEM, where we use \(\mathbb {P}1\) finite elements multiplied by plane waves as in (13).
2.4 Approximation property of rayFEM with exact ray information
We provide a simple computation to estimate the approximation error of the rayFEM space. In particular, we compute an asymptotic bound on \(\inf _{u_h \in V_{Ray}(\mathcal {T}_h)}  u  u_h _{L^2(\varOmega )}\), where u is the solution to the Helmholtz equation (1). We achieve the bound by estimating the interpolating error using \(V_{Ray}(K)\) as a basis.
Remark 1
The ray information can be incorporated into other Galerkin basis in the same fashion. For example, in the hybrid numerical asymptotic method of [41], the basis functions are constructed by multiplying nodal piecewise bilinear functions to oscillating functions with phase factors; the plane wave DG method of [14] employs the products of small degree polynomials and dominant plane waves as basis functions; the phasebased hybridizable DG method of [78] considers basis functions as products of polynomials and phasebased oscillating functions. Moreover, the phase or ray information in these methods is obtained from solving the eikonal equation with ray tracing and related techniques.
3 Learning local dominant ray directions
In Sect. 2 we use geometric optics to provide the motivation for the rayFEM by building an adaptive approximation space that incorporates ray information specific to the underlying Helmholtz equation. However, the ray directions, which depend on the medium and source distribution, are unknown quantities themselves; hence, they need to be computed or estimated. One way is to compute the global phase function, by either ray tracing or solving the eikonal equation, and take its gradient. As discussed in the introduction, computing the global phase function in a general varying medium can be extremely difficult.
In the present paper, we propose a totally different approach. This novel approach is based on learning the dominant ray directions by probing the same medium with the same source but using a relative lowfrequency wave. To be more specific, we first solve the Helmholtz equation (4) with the same speed function \(c(\mathbf {x})\), righthand side \(f(\mathbf {x})\) and boundary conditions but with a relative lowfrequency \(\widetilde{\omega } \sim \sqrt{\omega }\) on a mesh with size \(h={\mathcal O}(\widetilde{\omega }^{2}) ={\mathcal O}(\omega ^{1})\) with a standard finite element method, which is quasioptimal in that regime. Then the local dominant ray directions are estimated based on the computed lowfrequency wave field. The key point is that the lowfrequency wave has probed the medium specific to the problem globally while only local dominant ray directions need to be learned, which allows us to handle multiple arrivals of wave fronts locally. In particular, we use NMLA, which is simple, stable, and robust, to extract the dominant ray directions locally. However, this is a signal processing task that can be accomplished using other methods such as Prony’s method [19], Pisarenko’s method [84], MUSIC [89], matrix pencil [50], wavefront tracking methods [103], among many others. The main advantage of NMLA is that it was explicitly designed for capturing the dominant directions; in particular, NMLA was designed to be more robust to perturbations of the underlying model.
3.1 NMLA
In this subsection, for the sake of completeness, we provide a brief introduction to NMLA developed in [11, 12]. If we suppose that a wave field is locally a weighted superposition of plane waves having the same wave number and propagating in different directions, then the aim of NMLA is to extract the directions and the weights by sampling and processing the wave field locally. In the sequel, we use a 2D example to illustrate the method, which can be easily extended to 3D cases [12].
However, for applications, the measured data are never a perfect superposition of plane waves; therefore, we provide, for completeness, stability and error estimates for NMLA from [11] in “Appendix A.” In principle, for a single wave, as long as the perturbation is relatively small with respect to the true plane wave signal, say the relative noise level do not surpass \(25 \%\), the estimation error is \({\mathcal O}(\frac{1}{kr})\). In other words, the larger the radius of the circle compared to wavelength the more accurate the estimation is.

the geometric optics ansatz has an asymptotic error of order \({\mathcal O}(\omega ^{1})\) [see (9)];

in the geometric optics ansatz the wave field at a point is a superposition of curved wave fronts. In particular, the curvature of the wave fronts results in a compromise in the choice of the radius of the sampling circle to be of order \({\mathcal O}(\omega ^{\frac{1}{2}})\) for the NMLA in order to achieve the stability and the minimal error of order \({\mathcal O}(\omega ^{\frac{1}{2}})\).
3.2 Approximation property of numerical rayFEM
From “Appendix B,” the error estimation of dominant ray directions is \(\mathcal {O}(\omega ^{1/2})\). The numerical rayFEM space \(V_{Ray}^h(\mathcal {T}_h)\) is defined similar to \(V_{Ray}(\mathcal {T}_h)\) with the exact ray directions \(\lbrace \widehat{\mathbf{d }}_j \rbrace \) replaced by the ones \(\lbrace \widehat{\mathbf{d }}_j^h \rbrace \) estimated by NMLA and \(\vert \widehat{\mathbf{d }}_j  \widehat{\mathbf{d }}_j^h \vert \sim \mathcal {O}(\omega ^{1/2})\).
4 Algorithms
In this section we provide the full algorithm for the rayFEM including a fast iterative solver based on a modification of the method of polarized traces for the resulting linear systems. In order to streamline the presentation and to make the algorithm easier to understand, we introduce several subroutines.
 1.
probing the medium by solving a relatively lowfrequency Helmholtz equation with the standard FEM;
 2.
learning the dominant ray directions from the lowfrequencyprobed wave field by NMLA;
 3.
solving the highfrequency Helmholtz equation in the rayFEM space.
We remind the reader that the ultimate objective of the algorithm presented in this paper (i.e., Algorithm 7) is to solve the Helmholtz equation (1) at frequency \(\omega \) with a total \({\mathcal O}(\omega ^d)\) (up to polylogarithmic factors) computational complexity. In order to achieve this objective, we discretize the PDE with a mesh size \(h = {\mathcal O}(\omega ^{1})\), which leads to a total of \({\mathcal O}(\omega ^d)\) number of degrees of freedom and a sparse linear system with \({\mathcal O}(\omega ^d)\) number of nonzeros. Then we develop a fast iterative solver with quasilinear complexity to solve the resulting linear system after discretization. Below is a more detailed description of the three stages. Finally, following the notation defined in the prequel, we denote the triangular mesh by \(\mathcal {T}_h\).
4.1 Probing
Let \(\mathbf u _{\widetilde{\omega }, h}\) = SFEM (\(\widetilde{\omega }, h, c, f, g\)) denote the SFEM solution of the lowfrequency Helmholtz equation on \(\mathcal {T}_h\). Since \(\tilde{\omega }^2 h = \mathcal {O}(1)\), SFEM is quasioptimal in the norm \(\Vert \cdot \Vert _{ \mathcal {H}} := \Vert \nabla \cdot \Vert _{L^2} + k \Vert \cdot \Vert _{L^2} \) [71], and it has an optimal \(L^2\) error estimate [107].
4.2 Learning
Once the lowfrequency problem has been solved, we extract the dominant ray directions from \(\mathbf u _{\widetilde{\omega }, h}\) using NMLA as described in Sect. 3.1 around each mesh node. We utilize the smoothness of the phase functions, and hence the smoothness of the ray directions field to reduce the computational cost. The reduction is achieved by restricting the learning of the dominant ray directions to vertices of a coarse mesh downsampled from \(\mathcal {T}_h\). Such remeshed coarse mesh is denoted by \(\mathcal {T}_{h_c} = \lbrace K^c \rbrace \), where \(h_c={\mathcal O}(\sqrt{h})\). The resulting dominant ray directions are then linearly interpolated onto the fine mesh \(\mathcal {T}_h\).

numerical errors of \(\mathbf u _{\widetilde{\omega }, h}\);

model errors in the geometric optics ansatz;

interpolation errors.
4.3 Highfrequency solver
Remark 2
Extensive numerical experiments and “Appendix A” indicate that the NMLA process in learning dominant ray directions stage is remarkably stable even for noisy plane wave data. Hence, the iterative process in Algorithm 7 usually needs very few iterations to reach the desired accuracy. Typically, we only need one or two iterations in our numerical tests.
Remark 3
Since NMLA can not be used to estimate ray directions near the point source, a slight modification of Algorithm 7 is used for solving a point source inside domain problem. First, we approximate the righthand side with the associated column of the mass matrix (normalized by mesh size h). Moreover, we use a standard finite element basis function at the source point. For vertices near the source, we apply the radial directions (exact ray directions in homogeneous medium) in the construction of the approximation space for the rayFEM method. Meanwhile, for vertices away from the source, we find the dominant ray directions by NMLA. Under this modification, rayFEM can capture the phase accurately and it will be demonstrated numerically in Sect. 6.2.
4.4 Fast linear solver

layered domain decompositions;

absorbing boundary conditions between subdomains implemented via PML [13];

transmission conditions issued from a discrete Green’s representation formula;

efficient preconditioners arising from localization of the waves via an incomplete Green’s formula.
The method of polarized traces aims at solving the global linear system in (38) by solving the local systems \(\mathbf {H}^{\ell }\), which are the discrete version of (39).
In order to solve the global system, or in this case, to find a good approximate solution, we need to “glue” the subdomains together. This is achieved via a discrete Green’s integral formula deduced by imposing discontinuous solutions.
In the original formulation of the method of polarized traces [109], the Green’s representation formula was used to build a global surface integral equation (SIE) at the interfaces between slabs. The SIE was solved using an efficient preconditioner coupled with a multilevel compression of the discrete kernels to accelerate the online stage of the algorithm. The original algorithm had an embarrassingly parallel superlinear offline complexity which was amortized among a large number of righthand sides, which represents a typical situation in exploration geophysics.
In the context of the present paper, the linear systems issued from the raybased FEM depend on the source distribution, making it impossible to amortize a superlinear offline cost. In order to reduce the offline cost we use a matrixfree formulation (see Chapter 2 in [108]) with a domain decomposition in thin layers. In this case, the cost per iteration is linear with respect to the number of degrees of freedom, depending on the growth or the auxiliary degrees of freedom corresponding to the PML’s. Finally, the convergence is normally achieved in \(\mathcal {O}(\log {\omega })\) iterations, as it will be shown in the sequel.
5 Complexity
Overall computational complexities with respect to \(\omega \) given that the mesh size scaled as \(h = {\mathcal O}(\omega ^{1})\)
Methods  SFEM  Learning  RayFEM  Iterative RayFEM 

Frequency  \(\sqrt{\omega } \)  \(\sqrt{\omega } \text{ or } \omega \)  \(\omega \)  \(\omega \) 
Complexity  \({\mathcal O}(\omega ^d \log ^3{\omega })\)  \({\mathcal O}(\omega ^d)\)  \({\mathcal O}(\omega ^d \log {\omega })\)  \({\mathcal O}(\omega ^d \log ^3{\omega })\) 
Computational complexities of estimating ray directions on a coarse mesh \(\mathcal {T}_{h_c}\) with \(h_c ={\mathcal O}(\omega ^{\frac{1}{2}})\) and a fine mesh \(\mathcal {T}_{h}\) with \(h ={\mathcal O}(\omega ^{1})\)
Frequency  \(\varvec{r}\)  \(\varvec{M}_{\varvec{\omega }}\)  \(\varvec{C}_{\varvec{NMLA}}\)  \(\varvec{C}_{\varvec{ray}, \varvec{h}_{\varvec{c}}}\)  \(\varvec{C}_{\varvec{Int}}\)  \(\varvec{C}_{\varvec{ray, h}}\) 

\(\omega \)  \(\omega ^{\frac{1}{2}} \)  \(\omega ^{\frac{d1}{2}} \)  \(\omega ^{\frac{d1}{2}} \log \omega \)  \(\omega ^{d\frac{1}{2}} \log \omega \)  \(\omega ^d\)  \(\omega ^d\) 
\(\widetilde{\omega } \sim \sqrt{\omega }\)  \(\widetilde{\omega }^{\frac{1}{2}} \)  \(\widetilde{\omega }^{\frac{d1}{2}} \)  \(\widetilde{\omega }^{\frac{d1}{2}} \log \widetilde{\omega } \)  \(\widetilde{\omega }^{d\frac{1}{2}} \log \widetilde{\omega } \)  \(\omega ^d\)  \(\omega ^d\) 
5.1 Ray learning
As described in Sect. 4.2, Algorithm 5 applies NMLA to computed wave fields with lowfrequency \(\widetilde{\omega } \sim \sqrt{\omega }\) or highfrequency \(\omega \). It first estimates ray directions at vertices on a downsampled coarse mesh \(\mathcal {T}_{h_c}\) and then interpolates the ray directions to the vertices on a fine mesh \(\mathcal {T}_{h}\). We remind the reader the following scalings: \(h={\mathcal O}(\omega ^{1})\), \(h_c={\mathcal O}(\sqrt{h})={\mathcal O}(\omega ^{\frac{1}{2}})\). These scalings allow us to strike a balance among the number of observation points at which NMLA is used to estimate ray directions, the radius of the sampling circle, and the corresponding number of sampling points on the circle so as to resolve the wave field to reach the optimal accuracy of NMLA with desired total computational complexity.
The dominant computational cost of the ray learning is coming from the application of NMLA to the highfrequency wave field. Here we analyze its complexity in 2D case. As shown in “Appendix B,” the least error that can be achieved by NMLA is \({\mathcal O}(\omega ^{\frac{1}{2}})\) when the radius r of the sampling circle centered at an observation point is \({\mathcal O}(\omega ^{\frac{1}{2}})\). Hence, the number of points sampled on the circle to resolve the wave field with frequency \(\omega \) is \(M_{\omega }= {\mathcal O}(\omega r) = {\mathcal O}(\omega ^{\frac{1}{2}})\). Since NMLA is a linear filter based on the Fourier transform in the angle space, the corresponding computational complexity is \({\mathcal O}(M_{\omega }\log M_{\omega })\) [12]. The number of observation points that we need to perform NMLA is the number of vertices on the coarse mesh which is \({\mathcal O}(h_c^{2})={\mathcal O}(\omega )\). Hence, the computational cost to obtain the ray directions at the vertices on the coarse mesh by NMLA is \({\mathcal O}(\omega ^{\frac{3}{2}}\log \omega )\). Finally, the ray directions estimated at the vertices on the coarse mesh by NMLA are linearly interpolated onto the fine mesh \(\mathcal {T}_{h}\). Interpolation is a linear operation, and hence, its computational complexity is \({\mathcal O}(\omega ^2)\).
Table 2 provides the complexity of ray learning stage for both highfrequency and lowfrequency wave fields, where d is the dimension and \(C_{NMLA}\), \(C_{ray, h_c}\), \(C_{Int}\), and \(C_{ray, h}\) are the computation complexity of NMLA at a single vertex, NMLA on the undersampled coarse mesh, interpolation of local ray directions to the fine mesh, and the full algorithm for learning local ray directions at frequency \(\omega \) on the fine mesh \(\mathcal {T}_{h}\), respectively.
5.2 Helmholtz solver
The most computationally intensive component in the whole rayFEM algorithm is solving the linear systems after discretization of the Helmholtz equation. Algorithm 7 solves both \(\mathbf u _{\widetilde{\omega }, h} = SFEM (\widetilde{\omega }, h, c, f, g)\) and \(\mathbf u _\mathbf{d _{\omega }, h} = RayFEM (\omega , h, c, f, g, \lbrace \varvec{ d}_{\omega ,h}^j \rbrace _{j=1}^{N_h} )\) on the same mesh \(\mathcal {T}_{h}\). Each solver is composed of three steps: the assembling step, the setup step, and the iterative solve step.
Since the basis functions are locally supported, the resulting matrix is sparse. The complexity of the assembling step is of the same order as the degrees of freedom \(N_h={\mathcal O}(\omega ^d)\).
In the setup stage, the computational domain is decomposed into subdomains of thin layers whose width is comparable to the characteristic wavelength. The local problems in each subdomain are factorized^{5} using a multifrontal method [26] coupled with a nested dissection ordering [40] in \({\mathcal O}(\sqrt{N_{h}})\) time for the highfrequency problem (or \({\mathcal O}(\sqrt{N_{h}} \log ^3 {N_{h}})\) time for the lowfrequency problem, depending on the width of the auxiliary PML for each subdomain in terms of the wavelength). Given that the layers are \({\mathcal O}(1)\) elements thick, we have to factorize \({\mathcal O}(\sqrt{N_{h}})\) subsystems, which results in a total \({\mathcal O}(N_{h})\) (or \({\mathcal O}(N_{h} \log ^3{N_{h}})\) for the lowfrequency problem) asymptotic complexity for the setup step.
Finally, for the iterative solve step, each application of the preconditioner involves 6 local solves per layer, each one performed with \({\mathcal O}(\sqrt{N_{h}})\) ( or \({\mathcal O}(\sqrt{N_{h}} \log ^2{N_{h}})\)) complexity. Given that we have \({\mathcal O}(\sqrt{N_{h}})\) layers, we have an overall \({\mathcal O}(N_{h})\) (or \({\mathcal O}(N_{h} \log {N_{h}})\) for the lowfrequency problem) complexity per iteration. Extensive numerical experiments suggest that the number of iterations to converge is \({\mathcal O}(\log {N_{h}})\) for both high and lowfrequency solves for smooth media. Hence, the empirical overall complexity is \({\mathcal O}(N_{h} \log {N_{h}})\) for the highfrequency solve and \({\mathcal O}(N_{h} \log ^3{N_{h}})\) for the lowfrequency one, which as stated before in Table 1.
6 Numerical experiments
In this section we provide several numerical experiments to test the proposed rayFEM and corroborate our claims. For all cases, the domain of interest is \(\varOmega = (1/2,1/2)^2\) with different source terms and boundary conditions. \(\varOmega \) is discretized using a standard triangular mesh. The integrals to assemble the mass and stiffness matrices in (15), the righthand side in (16), and the \(L^2\) errors of the rayFEM solutions are numerically computed by using a highorder Gaussian quadrature rule.^{6}
The algorithm described in this paper was implemented in MATLAB, and the numerical experiments were executed using MATLAB 2015b in a dual socket server with 2 Intel Xeon E52670 with 384 GB of RAM.
6.1 Convergence tests
First, a probing wave with lowfrequency \(\widetilde{\omega } = \sqrt{\omega }\) is solved by the standard FEM. Then NMLA is applied to the lowfrequency probing wave to get an estimation of the local dominant ray directions \(\mathbf d _{\widetilde{w}}\). Instead of using the regular NMLA for the plane wave decomposition, we use NMLA with curvature correction (see details in Algorithm 3 and “Appendix C”) to estimate the ray information of a circular wave front. The estimated local ray directions are then used in the rayFEM to produce the first numerical solution \(u_\mathbf{d _{\widetilde{w}}}\) to the highfrequency Helmholtz equation.
We employ one more iteration in the framework of the iterative rayFEMs by applying NMLA to \(u_\mathbf{d _{\widetilde{w}}}\) to get an improved local ray direction estimation \(\mathbf d _w\) and then use it again in the rayFEM to get a more accurate numerical solution \(u_\mathbf{d _w}\) to the highfrequency Helmholtz equation.
Table 3 and the left column of Fig. 1 show that the NMLA and rayFEM algorithm are stable, and the error for both the ray estimation and the numerical solution by the rayFEM with fixed NPW, i.e., \(\omega h = {\mathcal O}(1)\), asymptotically decreases as the frequency increases. Moreover, they show that one more iteration using the iterative rayFEM can improve the accuracy of final numerical solution to the order of \({\mathcal O}(\omega ^{1})\), which is of the same order when the exact ray direction \(\mathbf d _{ex}\) is used in the rayFEM, due to the asymptotic error of the geometric optics ansatz.
Errors of one point source problem for fixed NPW = 6. \(\theta _{ex}\) is the exact ray angle; \(\theta (\mathbf d _{\widetilde{\omega }})\) and \(\theta (\mathbf d _{\omega })\) are ray angle estimations using low and highfrequency waves, respectively; \(u_\mathbf{d _{\widetilde{\omega }}}\), \(u_\mathbf{d _{\omega }} \), and \(u_\mathbf{d _{ex}} \) are rayFEM solutions using lowfrequency ray estimation \(\mathbf d _{\widetilde{\omega }}\), highfrequency ray estimation \(\mathbf d _{\omega }\), and exact ray \(\mathbf d _{ex}\), respectively
\(\omega /2\pi \)  20  40  80  160 
1 / h  120  240  480  960 
\(\Vert \theta (\mathbf d _{\widetilde{\omega }})  \theta _{ex}\Vert _{L^2}\)  7.50e04  4.26e04  1.96e04  1.07e04 
\(\Vert \theta (\mathbf d _{\omega })  \theta _{ex}\Vert _{L^2}\)  1.82e04  7.99e05  4.43e05  2.10e05 
\(\Vert u_\mathbf{d _{\widetilde{\omega }}}  u_{ex}\Vert _{L^2}\)  4.36e05  1.92e05  9.03e06  4.69e06 
\(\Vert u_\mathbf{d _{\omega }}  u_{ex} \Vert _{L^2}\)  3.15e05  1.47e05  7.57e06  3.73e06 
\(\Vert u_\mathbf{d _{ex}}  u_{ex}\Vert _{L^2}\)  2.97e05  1.49e05  7.47e06  3.74e06 
6.2 Phase errors
Here we show that the rayFEM method can capture the phase and satisfy the dispersion relation more accurately. We test our algorithm with a point source inside the domain, given its importance in many practical applications, in particular, in exploration geophysics, in which the sources are often modeled as point sources. Moreover, in applications oriented toward inverse and imaging problems, having a numerical method that produces the correct phase in the far field is of great importance in order to properly locate features in the image.
In this experiment we focus our attention on the far field since our current method can not deal with singularities in amplitude and phase at source points. We test a point source located at \(\mathbf {x}_0 = (0.4, 0.4)\) with frequency \(\omega = 80 \pi \) in a homogeneous medium. Following Remark 3, we use radial directions (exact directions in homogeneous medium) for vertices \(\mathbf {x}\) near the source with \(\vert \mathbf {x} \mathbf {x}_0 \vert \le 0.1\) and estimate ray directions for other vertices; see the left part of Fig. 2 for the ray direction field.
6.3 Complexity tests
In this subsection we test the computational complexity for the rayFEM. A key step of the algorithms presented is solving the sparse linear systems generated by the rayFEM using iterative methods with a performant preconditioner, e.g., domain decomposition techniques coupled with highquality absorbing/transmission boundary conditions. In our tests, we use a modification of the method of polarized traces to solve the linear systems resulting from both the standard FEM and rayFEM as described in Sect. 4.4.
We solve the Helmholtz equation with a point source in both a homogeneous and heterogeneous medium. We compute for many different frequencies, using Algorithm 7 with only one iteration of the rayFEM, the solution to the Helmholtz equation posed on \(\varOmega \) with absorbing boundary conditions implemented via PML. For each frequency we report the execution time of the low and highfrequency problems and the time spent in processing the data using NMLA to extract the dominant ray information.
As explained in Sect. 4, in order to process the data using NMLA we need to solve the lowfrequency problem in a slightly larger domain. The size of the larger domain is given by the sampling radius of the NMLA. For the sake of simplicity, we use a lowfrequency subdomain, \(\varOmega _\mathrm{low} = (1,1)\times (1,1)\), i.e., four times bigger than the original domain. The size can be reduced in order to lower the computational cost for the lowfrequency problem.
The main issue with the lowfrequency solver in our case is related to the PML, since the PML may not be very effective given that each thin slab contains less than one wavelength across. In order to decrease the number of iterations to converge, we increase the number of PML points logarithmically with the frequency. This implies a slightly more expensive setup cost and solve cost as shown in Figs. 6 left and 7 left.
Figure 6 shows the runtime for solving the Helmholtz equation with a point source inside a homogeneous medium. We can observe that the overall cost is \({\mathcal O}(N)\) up to polylogarithmic factors as shown in our complexity study. The lowfrequency solver has a slightly higher asymptotic cost in this case, given the ratio between the width of the PML and the characteristic wavelength inside the domain.
7 Conclusion
In this work we present a numerical method, the rayFEM, for the highfrequency Helmholtz equation in smooth media based on learning problemspecific basis functions to represent the wave field. The key information, local ray directions, is extracted from a relatively lowfrequency wave field that has probed the whole domain. These local ray directions are then incorporated into the basis to improve both stability and accuracy in the computation for a highfrequency wave field. Moreover, both local ray directions and the highfrequency wave field can be further improved through more iterations. Numerical tests suggest that our method only requires a fixed number of points per wavelength with an asymptotic convergence as the frequency becomes large. By designing a fast solver for the discretized linear systems an overall complexity of order \({\mathcal O}(\omega ^d\log ^3 \omega )\) is achieved.
However, the rayFEM cannot handle singularities of both the amplitude and phase on a given mesh. We will develop a hybrid method that combines a local asymptotic expansion near the source and the rayFEM away from the source in our future work.
Acknowledgements
Zhao is partially supported by NSF Grant (1418422 and 1622490). Qian is partially supported by NSF Grants (1522249 and 1614566).
The ratio between numerical error and best approximation error from a discrete finite element space is \(\omega \) dependent.
Recent advances such as [79, 100], have lowered the complexity of global spectral methods; however, they still have a superlinear cost.
Some of the discretizations mentioned above, in particular plane wavetype Trefftz methods with wave directions in equispaced distribution [47], usually yield extremely illconditioned systems due to loss of numerical orthogonality in the basis. In general, the resulting linear system need to be solved using pivoted QR factorization in superlinear time.
If we suppose that the approximation error of computing \(\phi _n\) is \(\delta \phi _n\), then the approximation error of the solution is given by \(\vert e^{i \omega \phi _n}  e^{i \omega (\phi _n + \delta \phi _n)} \vert \sim \omega \delta \phi _n\), which is \(\omega \) dependent.
Given the expression of the mass and stiffness matrices, which are polynomials times a plane wave, it is possible to compute the integral analytically [81]. However, the righthand side of the linear system and the \(L^2\) error of the rayFEM solution can only be computed numerically for a general source term \(f(\mathbf {x})\).
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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