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- Research
- Open Access
Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations
- Jun Fang1,
- Jianliang Qian2,
- Leonardo Zepeda-Núñez1Email author and
- Hongkai Zhao1
© The Author(s) 2017
- Received: 16 August 2016
- Accepted: 5 February 2017
- Published: 1 May 2017
We present a ray-based finite element method for the high-frequency 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 low-frequency 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 high-frequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and high-frequency 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 poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.
- Helmholtz equation
- Numerical micro-local analysis
- Ray-FEM
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 poly-log factors, as the frequency becomes large.
Examples of non-adaptive 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 quasi-optimal sparse discretizations whose associated linear systems are prohibitively expensive to solve [44, 101] in the high-frequency 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 high-frequency regime, either they are not asymptotically quasi-optimal for heterogeneous media or they yield linear systems that cannot be solved in quasi-linear time with current algorithms.3
Indeed, phase-based 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 micro-local analysis (NMLA) [10–12]. The resulting linear system is sparse, and it can be solved efficiently using state-of-the-art 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 high-frequency regime with an optimal asymptotic cost \({\mathcal O}(\omega ^d)\), up to poly-log 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 low-frequency 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 state-of-the-art preconditioners with linear complexity, up to poly-logarithmic factors.
In particular, we develop a simple ray-based finite element method (ray-FEM) 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 quasi-optimality on such mesh since \(\widetilde{\omega }^2h={\mathcal O}(1)\) [71]. Then NMLA [10–12] (see Sect. 3) is applied to the computed low-frequency 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 high-frequency 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 poly-logarithmic 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 high-frequency wave field iteratively. If necessary, the solution for the high-frequency 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 high-frequency 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, low-order compact finite-difference 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 (CIP-FEM) [107, 111], the interpolated optimized finite-difference 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 quasi-optimal 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 quasi-optimality [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 equi-spaced 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 ultra-weak variational formulation (UWVF) [18, 20, 21]. Trefftz-type methods [47] use local solutions of the Helmholtz equation as the basis functions, which in the case of piece-wise constant media are plane waves.
It is well known that these plane wave-based 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 ill-conditioned 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 high-order 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 high-frequency 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 non-oscillatory 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; phase-based 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 phase-based 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 ill-conditioning 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 second-kind integral equation by introducing the Green's function corresponding to the background, resulting in the so-called 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 ray-FEM 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 ray-FEM method
In this section we describe the ray-FEM 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 high-frequency 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 low-frequency 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 Ray-based FEM formulation
the standard FEM (S-FEM), where we use low-order \(\mathbb {P}1\) finite elements, i.e., piece-wise bilinear functions;
the ray-FEM, where we use \(\mathbb {P}1\) finite elements multiplied by plane waves as in (13).
2.4 Approximation property of ray-FEM with exact ray information
We provide a simple computation to estimate the approximation error of the ray-FEM 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 piece-wise 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 phase-based hybridizable DG method of [78] considers basis functions as products of polynomials and phase-based 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 ray-FEM 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 low-frequency wave. To be more specific, we first solve the Helmholtz equation (4) with the same speed function \(c(\mathbf {x})\), right-hand side \(f(\mathbf {x})\) and boundary conditions but with a relative low-frequency \(\widetilde{\omega } \sim \sqrt{\omega }\) on a mesh with size \(h={\mathcal O}(\widetilde{\omega }^{-2}) ={\mathcal O}(\omega ^{-1})\) with a standard finite element method, which is quasi-optimal in that regime. Then the local dominant ray directions are estimated based on the computed low-frequency wave field. The key point is that the low-frequency 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.
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 ray-FEM
From "Appendix B," the error estimation of dominant ray directions is \(\mathcal {O}(\omega ^{-1/2})\). The numerical ray-FEM 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 ray-FEM 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 low-frequency Helmholtz equation with the standard FEM;
- 2.
learning the dominant ray directions from the low-frequency-probed wave field by NMLA;
- 3.
solving the high-frequency Helmholtz equation in the ray-FEM 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 poly-logarithmic 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 quasi-linear 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 \(\mathbf u _{\widetilde{\omega }, h}\) = S-FEM (\(\widetilde{\omega }, h, c, f, g\)) denote the S-FEM solution of the low-frequency Helmholtz equation on \(\mathcal {T}_h\). Since \(\tilde{\omega }^2 h = \mathcal {O}(1)\), S-FEM is quasi-optimal 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 low-frequency 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 down-sampled 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 High-frequency 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 right-hand 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 ray-FEM method. Meanwhile, for vertices away from the source, we find the dominant ray directions by NMLA. Under this modification, ray-FEM 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 off-line complexity which was amortized among a large number of right-hand sides, which represents a typical situation in exploration geophysics.
In the context of the present paper, the linear systems issued from the ray-based FEM depend on the source distribution, making it impossible to amortize a superlinear off-line cost. In order to reduce the off-line cost we use a matrix-free 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 \(h = {\mathcal O}(\omega ^{-1})\)
Iterative Ray-FEM
\(\sqrt{\omega } \)
\(\sqrt{\omega } \text{ or } \omega \)
\(\omega \)
\(\omega \)
\({\mathcal O}(\omega ^d \log ^3{\omega })\)
\({\mathcal O}(\omega ^d)\)
\({\mathcal O}(\omega ^d \log {\omega })\)
\({\mathcal O}(\omega ^d \log ^3{\omega })\)
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 \(h ={\mathcal O}(\omega ^{-1})\)
\(\varvec{M}_{\varvec{\omega }}\)
\(\varvec{C}_{\varvec{ray}, \varvec{h}_{\varvec{c}}}\)
\(\varvec{C}_{\varvec{ray, h}}\)
\(\omega \)
\(\omega ^{-\frac{1}{2}} \)
\(\omega ^{\frac{d-1}{2}} \)
\(\omega ^{\frac{d-1}{2}} \log \omega \)
\(\omega ^{d-\frac{1}{2}} \log \omega \)
\(\omega ^d\)
\(\omega ^d\)
\(\widetilde{\omega } \sim \sqrt{\omega }\)
\(\widetilde{\omega }^{-\frac{1}{2}} \)
\(\widetilde{\omega }^{\frac{d-1}{2}} \)
\(\widetilde{\omega }^{\frac{d-1}{2}} \log \widetilde{\omega } \)
\(\widetilde{\omega }^{d-\frac{1}{2}} \log \widetilde{\omega } \)
\(\omega ^d\)
\(\omega ^d\)
5.1 Ray learning
As described in Sect. 4.2, Algorithm 5 applies NMLA to computed wave fields with low-frequency \(\widetilde{\omega } \sim \sqrt{\omega }\) or high-frequency \(\omega \). It first estimates ray directions at vertices on a down-sampled 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 high-frequency 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 high-frequency and low-frequency 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 under-sampled 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 ray-FEM algorithm is solving the linear systems after discretization of the Helmholtz equation. Algorithm 7 solves both \(\mathbf u _{\widetilde{\omega }, h} = S-FEM (\widetilde{\omega }, h, c, f, g)\) and \(\mathbf u _\mathbf{d _{\omega }, h} = Ray-FEM (\omega , h, c, f, g, \lbrace \varvec{ d}_{\omega ,h}^j \rbrace _{j=1}^{N_h} )\) 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 factorized5 using a multifrontal method [26] coupled with a nested dissection ordering [40] in \({\mathcal O}(\sqrt{N_{h}})\) time for the high-frequency problem (or \({\mathcal O}(\sqrt{N_{h}} \log ^3 {N_{h}})\) time for the low-frequency 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 low-frequency 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 low-frequency 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 low-frequency solves for smooth media. Hence, the empirical overall complexity is \({\mathcal O}(N_{h} \log {N_{h}})\) for the high-frequency solve and \({\mathcal O}(N_{h} \log ^3{N_{h}})\) for the low-frequency one, which as stated before in Table 1.
6 Numerical experiments
In this section we provide several numerical experiments to test the proposed ray-FEM 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 right-hand side in (16), and the \(L^2\) errors of the ray-FEM solutions are numerically computed by using a high-order 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 E5-2670 with 384 GB of RAM.
6.1 Convergence tests
First, a probing wave with low-frequency \(\widetilde{\omega } = \sqrt{\omega }\) is solved by the standard FEM. Then NMLA is applied to the low-frequency 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 ray-FEM to produce the first numerical solution \(u_\mathbf{d _{\widetilde{w}}}\) to the high-frequency Helmholtz equation.
We employ one more iteration in the framework of the iterative ray-FEMs 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 ray-FEM to get a more accurate numerical solution \(u_\mathbf{d _w}\) to the high-frequency Helmholtz equation.
Table 3 and the left column of Fig. 1 show that the NMLA and ray-FEM algorithm are stable, and the error for both the ray estimation and the numerical solution by the ray-FEM 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 ray-FEM 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 ray-FEM, 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 high-frequency waves, respectively; \(u_\mathbf{d _{\widetilde{\omega }}}\), \(u_\mathbf{d _{\omega }} \), and \(u_\mathbf{d _{ex}} \) are ray-FEM solutions using low-frequency ray estimation \(\mathbf d _{\widetilde{\omega }}\), high-frequency ray estimation \(\mathbf d _{\omega }\), and exact ray \(\mathbf d _{ex}\), respectively
\(\omega /2\pi \)
1 / h
\(\Vert \theta (\mathbf d _{\widetilde{\omega }}) - \theta _{ex}\Vert _{L^2}\)
\(\Vert \theta (\mathbf d _{\omega }) - \theta _{ex}\Vert _{L^2}\)
\(\Vert u_\mathbf{d _{\widetilde{\omega }}} - u_{ex}\Vert _{L^2}\)
\(\Vert u_\mathbf{d _{\omega }} - u_{ex} \Vert _{L^2}\)
\(\Vert u_\mathbf{d _{ex}} - u_{ex}\Vert _{L^2}\)
6.2 Phase errors
Here we show that the ray-FEM 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 ray-FEM. A key step of the algorithms presented is solving the sparse linear systems generated by the ray-FEM using iterative methods with a performant preconditioner, e.g., domain decomposition techniques coupled with high-quality 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 ray-FEM 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 ray-FEM, 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 high-frequency 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 low-frequency 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 low-frequency 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 low-frequency problem.
The main issue with the low-frequency 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 poly-logarithmic factors as shown in our complexity study. The low-frequency 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 ray-FEM, for the high-frequency Helmholtz equation in smooth media based on learning problem-specific basis functions to represent the wave field. The key information, local ray directions, is extracted from a relatively low-frequency 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 high-frequency wave field. Moreover, both local ray directions and the high-frequency 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 ray-FEM 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 ray-FEM away from the source in our future work.
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 wave-type Trefftz methods with wave directions in equi-spaced distribution [47], usually yield extremely ill-conditioned 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 right-hand side of the linear system and the \(L^2\) error of the ray-FEM solution can only be computed numerically for a general source term \(f(\mathbf {x})\).
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In his last blog Hans Lodder advocated that the role of the CIO should focus more on an effective ICT operation and less on an efficient ICT operation. Does this contradict with my quest for a Production Architect? I believe not: they could even be the biggest buddies!
Consultant vs Advisor vs Coach
Since we started to offer personal coaching for IT-executives, I got a lot of questions about our format and the differences with more common consultancy models. All formats serve a purpose but personal coaching as we see it is clearly something different.
Consultant is probably one of the most often misused titles in IT. Originally meant to describe a content matter expert who would advise on a topic in a certain setting. A pure independent, conceptual advice given observations and parameters, like a consult you could have at your specialist: you tell the symptoms and the specialist determines the illness and suggests a treatment. Nowadays it seems almost every (external) resource hired in IT is called a consultant, from the tools specialist implementing the software you bought (more truly an engineer) to the person analyzing and defining business processes (more truly an information analyst).
Then there is the special breed of "armchair consultants". Unlike the above mentioned doctor who gives you advice on the treatment most likely to cure, these consultants pride themselves in writing thick reports in which they show that they know at least 42 theoretical variants that could apply. No sound advice though for your specific situation (more truly researchers).
A true consultant will analyze logically, describe potential solution scenario's, rank them on relevance given the specific problem at hand and will be able to tell you the risks involved, so that you can take a sound decision.
An advisor is in essence also a consultant but with a more elevated view on the subject and seniority wise at (more) equal level with the person who is being advised. Normally we see elements like maturity, seniority, experience and independence in an advisor. Certainly the trusted advisor can give also advice even when not asked for.
Also the advisor name is misused. The trusted advisor status is the heavenly stage any commercial person wants to achieve (and is told so by all sales methodologies). But they are never independent, there's always an agenda in the background. I would call them "Vendorvisor".
In sports we see two typical forms of coaches. First the highly visible conductor type in team sports who defines the strategy, decides the line-up and corrects at the side line. Secondly the rather invisible, sometimes completely unknown coach of individual athletes who organizes training scheme's, prepares for competitions and gives mental support at an individual level. Of course from this specialized forms have grown like mental coaching. In all cases the athletes and teams in the end have to perform although the coach is the first one to be throw out if the results are not coming.
The Results2Match concept of MyPersonalCoach is based on the individual needs of senior executives in the IT (CIO, VP/Director of IT, etc.). The personal coach encompasses the analytical conceptual skills of a senior consultant, the independence and maturity of the advisor. This is all centralized around the specific coachee and his or hers position. The personal coach is there for the coachee and the coachee only. So support and advice can be far more intimate and truly personal. The manager is still the one making the decisions. The personal coach is there, if-and-when required, as a knowledgeable sounding board, mirror, conscience and sometimes father figure.
In the rather lonely world of IT-executives, squeezed between the pressure from board, customers, vendors and their own organization, a personal coach can be a logical insurance premium with a certain payback.
If you want to comment or curious about the MyPersonalCaoch concept, please let me know.
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Cubic Yards to Square Yards
by patriot05 on 08/22/05 at 13:28:03
I need to place 2000 SY of 18 -24" crushed concrete (rip-rap). I pay a certain price per truck load which is in CY (approx 13 - 15 per truckload.) How many truck loads will I need?
Re: Cubic Yards to Square Yards
by Robert Fogt on 08/22/05 at 22:03:10
The formula for volume is:Volume = Area * HeightThey just all need to be the same base unit, such as yards.You know the area, 2000 square yards, so you just need to know how thick you want to lay it, in yards.For example, if you want the rock layed 1 foot thick, you convert 1 foot to yards then multiply:2000 square yards * (1/3) yards = 666.67 cubic yards666.67 cubic yards / 13 cubic yards/truck = 51.3 trucks666.67 cubic yards / 15 cubic yards/truck = 44.4 trucksSo you would need between 45 and 52 truck loads if you wanted the rock 1 foot thick. Replace the actual thickness in the formula above.
Re: Cubic Yards to Square Yards
by patriot05 on 09/14/05 at 17:32:36
If I buy the rip-rap in tons, how many tons will it take to get to 2400 sq yds. I think I need to know the specific gravity of concrete?Help?????
Re: Cubic Yards to Square Yards
by Robert Fogt on 09/15/05 at 01:01:38
Yes, you would need to know the density.I searched but could not find anything specific to crushed concrete.I have data on concrete gravel, and crushed stone, but I couldn't say how accurate those values would be for your product.
Re: Cubic Yards to Square Yards
by patriot05 on 09/15/05 at 08:35:41
Thanks for your help. Concrete gravel seems like it would be a better match for density than crushed stone, what do you think? The density of concrete is 2300kg/m3. (I got that off the web.) I used that as a starting point to determine how much to order. What is the density of concrete gravel and just as a comparison, crushed stone?Rip Rap is bigger than either of those. The chunks of concrete are about 18" to 24". The shapes vary so it's difficult to determine volume but I used both 6" and 12" converted to cubic meters and then multiplied by the density of concrete to arrive at kg. I then converted that to short tons. If I also calculate using the density of concrete gravel and crushed stone, I'll have a better estimate.
Re: Cubic Yards to Square Yards
by Robert Fogt on 09/15/05 at 19:19:22
Crushed stone has a density of 1600 kg/m³ and concrete gravel has a density of 2400 kg/m³Your density data is probably for concrete asphalt, which I show has a density of 2240 kg/m³. There is also Portland limestone concrete with a density of 2370 kg/m³Anyhow, your rip rap will probably be less than the density of solid concrete because of gaps between the pieces, the bigger the pieces the bigger the gaps. I would guess it is somewhere between 1600 and 2000 kg/m³.You could calculate for both values and give yourself a minimum and maximum estimate.
Re: Cubic Yards to Square Yards
by patriot05 on 09/16/05 at 08:39:32
Thanks so much for your help!
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NQT - Saltley, Saltley, Birmingham.
Contract type: Full Time
Contract term: Fixed Term
Suitable for NQTs: Yes
ABC Teachers are currently seeking a Newly Qualified Teacher for a role starting in September 2018 at a well-established school in Saltley.
The position available is for a full time Year 3 class teacher and all aspects of the full teaching role such as planning and assessment will be required. The school itself has a dedicated ethos focusing on high achievement and a positive learning environment. Comments have been made on the friendly staff team and the excellent school resources.
The Head Teacher has expressed that for the chosen candidate they will offer them every support they need in order to excel in their induction year. They will also be able to go in into the school throughout July through our agency for paid supply work, in order to get to know the staff and the children so they feel more assured come September.
The school is one that we have worked closely with for a number of years now so this role can only be applied for through ABC Teachers.
If you are interested in this position, please contact Alexander Waldron today on 0121 274 0345 or 'Apply now' to submit your details.
As a preferred supplier of Birmingham City Council, you can be safe in the knowledge that we will be confident of finding the work that you want.
Not found what you are looking for? Keep up to date with us on our Website, Twitter and Facebook pages for details of all our current jobs and upcoming registration days!
Website : http://www.abc-teachers.co.uk
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ABC Teachers are an REC Gold Standard Awarded Education Agency, part of TES. Candidates pay rates are competitive and in line with the AWR policy which means that after a qualifying period, supply teachers can be paid to the comparable scale offered by the school.
We are committed to safeguarding and promoting the welfare of children and expect all staff to share this commitment. All staff will hold an enhanced DBS disclosure (or be prepared to allow us to process one for you), need to provide professional or academic references and attend a face to face interview. A prohibition from teaching check will be completed for all applicants. Telephone calls to and/or from ABC Teachers may be recorded for training, safeguarding and compliance purposes.
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text:
LEA 260 Police Photography
3 Credit Hours • 60 Contact Hours (15 Lecture, 45 Lecture/Lab Combination)
Focuses on current methods and techniques of police photography. Includes the use, nomenclature, and operation of 35mm and 4x4 cameras at simulated crime scenes and traffic accidents. Incorporates the development, printing, and enlargement of photos.


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The Spinward Arm Crusade
Only in Death, Does Duty End
On the fringes of Imperial Space, surrounded by a group of raging warp-storms known as The Great Reef, lies the Spinward Arm. The worlds of the Arm were brought into the Imperium of Man a thousand years ago by Saint Drusus, who led the armies of the Imperium to victory in the Arm.
Centuries later, the systems of the Arm were cut off, unheard from for over a hundred years, when the safe passages through The Great Reef became engulfed by the warp-storms. Those storms have begun to recede, and the forces of the Imperium can once again reach into the Arm.
The High Lords of Terra have elevated Lord General Slodan to the position of Warmaster, and he has assembled a crusade of awesome power to retake the worlds of the Spinward Arm for the Emperor of Man. One of the hundreds of Imperial Guard Regiments included in his forces is the 10th Dokkalgard Dragoons- this is the Crusade from their point of view.


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marriage in Bahamas, reconiged in USA, what do I need to do
Category: Family and Home > Relationships
Asked by: andre2070-ga
List Price: $20.00
20 Mar 2005 15:27 PST
Expires: 19 Apr 2005 16:27 PDT
Question ID: 497716
I am planning a marrage in the Bahamas. What do I need to legally do in the USA or in my state of Maryland to have my marriage legal and reconizable in the states. Do I need to file paperwork or Just present when asked, the document from the Bahamas. IE name change, driver's permit name changes.
Re: marriage in Bahamas, reconiged in USA, what do I need to do
Answered By: hummer-ga on 20 Mar 2005 17:30 PST
Hi andre2070, Not to worry, all is well. An original marriage certificate, issued by the appropriate local authorities where the marriage took place (in your case the Registrar's Office in the Bahamas) will be accepted in the U.S. as a legal document, no problem. Obtaining a legal marriage license abroad is always determined by where the marriage takes place. Once back in the U.S. you will proceed with changing your name, etc., by "just presenting when asked, the document from the Bahamas". How to get married in The Bahamas 10. Would my marriage licence be legal in my country? Yes, the Bahamas marriage licence is a legal document issued by the Bahamas government. http://www.go-abacos.com/news/conian/new12-00/abaco_wedding.html The Registrar of Marriages performs all civil ceremonies. "If you are living in the US, Britain, or Canada, the marriage license is enough for proof of marriage. In other areas or in certain legal situations, an "Apostille" license may be required." http://220.127.116.11/search?q=cache:vuRo6kij3akJ:www.intheislands.com/process.html+caribbean+marriage+license+legal+in+u.s.&hl=en "Your marriage will be legal and binding worldwide" http://www.caribbean-wedding.com/ Regulations for marriages in the Bahamas 6. "If either party is unmarried and is a citizen or resident of a country other than the U.S.A., a declaration certifying this fact must be sworn before a Notary Public or other person authorized to administer oaths in that country and must accompany the application for marriage license. Said declaration may also be obtained at a Notary Public in Marsh Harbour. Would my marriage license be legal in my country? "Yes, the Bahamas marriage license is a legal document issued by the Bahamas government." http://usmarriagelaws.com/search/caribbean/bahamas/ # Bahamas # Fee charged # Residency requirement: 24 hours # Necessary documents: Passports; birth certificates; proof of divorce or death certificate of former spouse/s (if applicable); declaration certifying both parties are unmarried U.S. citizens, sworn before a U.S. Consul at the American Embassy in Nassau, or a marriage license from the Commissioner's Office on other islands; proof of arrival in the Bahamas. # Knot Note: Both parties must apply in person. # More info: Bahamas Ministry of Tourism Romance Director at (888) NUPTIAL or http://www.bahamas.com http://www.theknot.com/ch_article.html?Object=AI980914211954&keywordID=156&keywordType=2&parentID=527/ Bahamas * Declaration certifying both are unmarried U.S. citizens sworn before a U.S. Consul at the American Embassy, Nassau, or marriage license obtained from Commissioner's Office for weddings on any Bahamas' island outside of New Providence * Both parties must apply in person for the marriage license * Proof of arrival in the Bahamas * Passport or birth certificate & photo ID * Proof of divorce or death certificate, if applicable * Parental consent for parties under 18 years of age http://www.villageinc.com/marriage%20caribbean.htm "A Bahamian marriage license is a legal document issued by the Bahamian government and is recognized in other countries" http://www.graycliff.com/index.php?option=displaypage&Itemid=277&op=page&SubMenu= Additional Links of Interest The Bahamas: Consular Information Sheet http://travel.state.gov/travel/cis_pa_tw/cis/cis_989.html Tips for Travelers to the Caribbean http://travel.state.gov/Publications/tips_caribbean.html I was glad to find this for you. If you have any questions, please post a clarification request *before* closing/rating my answer and I'll be happy to reply. Thank you and Congratulations! hummer Google Search Terms Used: bahamas marriage license bahamas u.s. "getting married abroad" legally u.s. bahamas site:travel.state.gov caribbean marriage licence legal u.s.
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Hey, when I screw up, I screw up.....
Well, when I question whether the Quad City Times is relevant, I should take the information in the rag as suspicious. So getting back in town from a weekend away, I learn the QC Times screwed up the facts regarding the settlement between the city and the guy trying to open the Hispanic bar/venue in the Village of East Davenport. There is a difference between $5,000 and $500,000. I apologize for relying upon the Quad City Times. My Bad….
I still believe Alderman Lynn and Howard screwed this up. I know first hand Alderman Howard walked the applicant through city hall and tried to grease the process. Alderman Lynn in turn, joined the group of Village businesses that wanted to violate city laws and processes because of skin color and heritage. And lets be honest, if this was a country western bar, nothing would have come of it.
Anyways, I apologize to the City Hall folks for spreading misinformation. Unfortunately, most readers just assumed something stupid like this would come from city hall. And that says something as well.