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A preconditioner defined by an algebraic multigrid cycle for a damped Helmholtz operator is proposed for the Helmholtz equation. This approach is well-suited for acoustic scattering problems in complicated computational domains and with varying material properties. The spectral properties of the preconditioned systems and the convergence of the GMRES method are studied with linear, quadratic, and cubic finite element discretizations. Numerical experiments are performed with two-dimensional problems describing acoustic scattering in a cross section of a car cabin and in a layered medium. Asymptotically the number of iterations grows linearly with respect to the frequency while for lower frequencies the growth is milder. The proposed preconditioner is particularly effective for low-frequency and mid-frequency problems.

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... The coefficient matrix of the linear system shows poor conditioning as the discretization is refined. 1 ...

... This class of Laplace preconditioners with a complex shift proposed and further studied in [43,46,47], where the solver requires a number of iterations that grows only linearly as the wave number increases. Inspired by this work, a number of generalizations appeared shortly afterwards in [42,93,17,1,44] together with applications in different industrial contexts in [120,3,94,89,90,114,2,87]. The real and imaginary shifts in these type of preconditioners determine the performance of the solver heavily. Therefore this attracts people to work on optimal choice of the shifts in preconditioner. ...

... As the complex shift introduces damping and renders the preconditioned system amenable to approximate inversion using multigrid or modified factorization methods [77]. More recently algebraic multigrid has been used to invert the preconditioner [17,1]. Inspired by advances in CSLP, a number of generalizations of the work appeared shortly afterwards in [42,93,17,1,44] together with applications in different industrial contexts in [120,3,94,89,90,114,2,87]. Along-with its variations, CSLP preconditioner has been observed as most effective and robust one for Helmholtz. ...

... The idea of adding weight to the diagonal of the ILU preconditioner forms the basis of [14]. The CSLP preconditioners were further developed in [15,16] and later generalized in [17][18][19][20][21]. This lead to a breakthrough in industrial applications [22][23][24]. ...

... For such spectra, the Krylov methods converge faster than for the spectra of the Helmholtz operator as extensively shown both by theory and experiments in e.g. [15,16,11,[17][18][19]. The term with β 2 introduces damping in the preconditioner and renders the solution of a linear system with M h,β 2 as coefficient matrix easy to compute. ...

... The term with β 2 introduces damping in the preconditioner and renders the solution of a linear system with M h,β 2 as coefficient matrix easy to compute. In the literature, approximate inversions using either MILU [14], geometric [20,34] or algebraic multigrid [19,18] have been studied. In this work we will use geometric multigrid with Galerkin coarsening on a sequence of uniformly coarsened meshes. ...

... with the same efficiency, but it turned out that this is a very difficult task. Textbooks mention that there are substantial difficulties, see [3, page 72], [11, page 212], [12, page 400], and also the review [7] for why in general iterative methods have difficulties when applied to the Helmholtz equation (1). Motivated by the early proposition in [2] to use the Laplacian to precondition the Helmholtz equation, the shifted Laplacian has been advocated over the past decade as a way of making multigrid work for the indefinite Helmholtz equation, see [6,10,1,5,4] and references therein. ...

... Textbooks mention that there are substantial difficulties, see [3, page 72], [11, page 212], [12, page 400], and also the review [7] for why in general iterative methods have difficulties when applied to the Helmholtz equation (1). Motivated by the early proposition in [2] to use the Laplacian to precondition the Helmholtz equation, the shifted Laplacian has been advocated over the past decade as a way of making multigrid work for the indefinite Helmholtz equation, see [6,10,1,5,4] and references therein. The idea is to shift the wave number into the complex plane to obtain a good preconditioner for a Krylov method when solving (1). ...

... Motivated by the early proposition in [2] to use the Laplacian to precondition the Helmholtz equation, the shifted Laplacian has been advocated over the past decade as a way of making multigrid work for the indefinite Helmholtz equation, see [6,10,1,5,4] and references therein. The idea is to shift the wave number into the complex plane to obtain a good preconditioner for a Krylov method when solving (1). The hope is that due to the shift, it becomes possible to use standard multigrid to invert the preconditioner, and if the shift is not too big, it is still an effective preconditioner for the Helmholtz equation with a real wave number. ...

... As the resulting system is non-Hermitian and indefinite, a good preconditioner is necessary for the iterative methods. Various preconditioners have been proposed [5][6][7][8][9][10][11], for example, a tensor product preconditioner [6], the incomplete factorization preconditioner [7] and the Laplacian preconditioner [8,9]. We will use the shifted-Laplacian preconditioner in this paper [9]. ...

... As the resulting system is non-Hermitian and indefinite, a good preconditioner is necessary for the iterative methods. Various preconditioners have been proposed [5][6][7][8][9][10][11], for example, a tensor product preconditioner [6], the incomplete factorization preconditioner [7] and the Laplacian preconditioner [8,9]. We will use the shifted-Laplacian preconditioner in this paper [9]. ...

... Various preconditioners have been proposed [5][6][7][8][9][10][11], for example, a tensor product preconditioner [6], the incomplete factorization preconditioner [7] and the Laplacian preconditioner [8,9]. We will use the shifted-Laplacian preconditioner in this paper [9]. ...

In this paper, wave simulation with the finite difference method for the Helmholtz equation based on the domain decomposition method is investigated. The method solves the problem by iteratively solving subproblems defined on smaller subdomains. Two domain decomposition algorithms both for nonoverlapping and overlapping methods are described. More numerical computations including the benchmark Marmousi model show the effectiveness of the proposed algorithms. This method can be expected to be used in the full-waveform inversion in the future.

... The efficient simulation of acoustic, electromagnetic and elastic wave phenomena is of fundamental importance in a wide range of engineering applications such as ultrasound tomography, wireless communication, or geophysical seismic imaging. When the problem of interest is linear and the time dependence is harmonic, the unknown wave field u typically satisfies the Helmholtz equation (1) −∆u − k 2 u = f in a bounded domain Ω ⊂ R d , d = 2, 3, supplemented with appropriate physical or radiation boundary conditions, which guarantee the well-posedness of the boundary value problem. Here the wave number k = ω/c represents the ratio of the (constant) angular frequency, ω, and the speed of propagation c. ...

... Alternatively, instead of applying a multigrid iteration directly to the Helmholtz equation (1), one can apply it to a different nearby problem, where it is more effective, and then use it to precondition an outer iteration. Due to its simplicity, this class of "shifted Laplacian" preconditioners has recently received much attention. ...

... where β is a free parameter. Both (1) and (4) are discretized with centered finite differences on the same equidistant grid and with identical boundary conditions. The complex shift induced by β moves the spectrum away from the origin and corresponds to damping in the time domain. ...

An algebraic multilevel (ML) preconditioner is presented for the Helmholtz equation in heterogeneous media. It is based on a multilevel incomplete LDL T factorization and preserves the inherent (complex) symmetry of the Helmholtz equation. The ML preconditioner incorporates two key components for efficiency and numerical stability: symmetric maximum weight matchings and an inverse-based pivoting strategy. The former increases the block-diagonal dominance of the system, whereas the latter controls ∥L-1∥for numerical stability. When applied recursively, their combined effect yields an algebraic coarsening strategy, similar to algebraic multigrid methods, even for highly indefinite matrices. The ML preconditioner is combined with a Krylov subspace method and applied as a "black-box" solver to a series of challenging two- and three-dimensional test problems, mainly from geophysical seismic imaging. The numerical results demonstrate the robustness and efficiency of the ML preconditioner, even at higher frequency regimes.

... With the Laplace preconditioners with a complex shift proposed and studied in [4][5][6], the solver requires a number of iterations that grows only linearly as the wave number increases. Inspired by this work, a number of generalizations appeared shortly afterwards in [1,[7][8][9][10] together with applications in different industrial contexts in [11][12][13][14][15][16][17][18]. The convergence of the shifted Laplace preconditioners is analyzed in [19][20][21]. ...

... The complex shift introduces damping and renders the preconditioned system amenable to approximate inversion using either geometric multigrid [1,27] or MILU [28]. More recently algebraic multigrid has been used to invert the preconditioner [8,9]. In this paper we limit ourselves to the exact inversion of M h,(β 1 ,β 2 ) . ...

... With an increase in k, the very small eigenvalues cause the convergence of the outer Krylov subspace iteration to slow down. This justifies to consider the matrix M h,(β 1 ,β 2 ) given by (9) to define a splitting of A h and to refer to the stationary iterative method with error propagation matrix S h,(β 1 ,β 2 ) given by ...

A Helmholtz solver whose convergence is parameter independent can be obtained by combining the shifted Laplace preconditioner with multigrid deflation. To proof this claim, we develop a Fourier analysis of a two-level variant of the algorithm proposed in [1]. In this algorithm those eigenvalues that prevent the shifted Laplace preconditioner from being scalable are removed by deflation using multigrid vectors. Our analysis shows that the spectrum of the two-grid operator consists of a cluster surrounded by a few outliers, yielding a number of outer Krylov subspace iterations that remains
constant as the wave number increases. Our analysis furthermore shows that the imaginary part of the shift in the two-grid operator can be made arbitrarily large without affecting the convergence. This opens promising perspectives on obtaining a very good preconditioner at very low cost. Numerical tests for problems with constant and non-constant wave number illustrate our convergence theory.

... There, the scattering problems were posed in a rectangular domain and they were discretized using low-order finite differences. Our earlier study [34] extended this approach to general shaped two dimensional domains using linear, quadratic, and cubic finite element discretizations by applying an algebraic multigrid (AMG) instead of the geometric multigrid to approximate the inversion of the damped Helmholtz operator. In [35], this method was compared with the previously mentioned controllability method. ...

... In [35], this method was compared with the previously mentioned controllability method. In this paper, a generalization will be proposed to the preconditioner described in [34], an AMG-based damped preconditioner for time-harmonic wave propagation problems in elastic media, i.e. the Navier equation. This preconditioner will be called a damped Navier preconditioner. ...

... With sufficient damping, systems with F d can be solved much more easily than with F and the conditioning of FF −1 d can still be good. The use of different approximations for F −1 d have been studied in [30,21,38,34]. Here an algebraic multigrid approximation described in Section 5 is considered. ...

A physical damping is considered as a preconditioning technique for acoustic and elastic wave scattering. The earlier preconditioners for the Helmholtz equation are generalized for elastic materials and three-dimensional domains. An algebraic multigrid method is used in approximating the inverse of damped operators. Several numerical experiments demonstrate the behavior of the method in complicated two-dimensional and three-dimensional domains.

... In the finite element method, the weak formulation of the Helmholtz equation is used to form the discretized version of the equation. The weak form and corresponding spaces that are used here are identical to the ones described in [15]. The finite element discretization is made on a triangulation given by a set of non-overlapping triangles K h such that Ω h = τ ∈K h τ . ...

... By choosing β 1 = 1 and β 2 to be positive, B SL corresponds to damped Helmholtz operator. In [15], the algebraic multigrid method (AMG) was used to approximate inversion of B SL . We use this preconditioner here and denote it by B M G . ...

... One iteration of the Jacobi is used as a pre-and post-smoothener. The damping parameter β 2 is chosen to be 0.5, which was found to be a good choice in [15]. ...

Processes that can be modeled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmen-tal noise. We present two methods for making these calculations based on Helmoltz equation. The first method is based directly on the complex-valued Helmholtz equa-tion and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as a preconditioner. The second ap-proach returns to a transient wave equation, and finds the time-periodic solution using a controllability technique. We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well. Numerical ex-periments show that the control method takes more CPU time, whereas the shifted-Laplacian method has larger memory requirement.

... In [55] for example , the idea is highlighted. Further, a Finnish group at Jyväskylä has adopted the approach for their Helmholtz applications in [25, 26, 28, 1]. The preconditioner is also discussed and considered, for other Helmholtz applications in [12, 45, 7], and in different research areas that need to deal with indefinite problems, like electromagnetics or optics, in [6, 22, 59, 36, 42]. ...

... A fourthorder Helmholtz discretization enables us to use fewer grid points per wavelength compared to a second-order discretization. Finally, the overall solution method with these algorithmic improvements is not limited to structured Cartesian grids, as it can be set up fully algebraically (a similar goal has been pursued in [1]). Although our method extends to solving problems on unstructured grids, we focus here on heterogeneous Helmholtz problems on Cartesian grids. ...

... The number of wavelengths in a domain of size L equals L/λ f . A dimensionless wavenumber, k, on a non-dimensional [0, 1] 2 domain is defined by k = 2π f L/c, and a corresponding mesh size by h = λ f /(n w L), with n w the number of points per wavelength. The usual 5-point stencil related to a second-order accurate discretization reads: ...

We present an iterative solution method for the discrete high wavenumber Helmholtz equation. The basic idea of the solution method, already presented in [Y. A. Erlangga, C. W. Oosterlee and C. Vuik, SIAM J. Sci. Comput. 27, No. 4, 1471–1492 (2006; Zbl 1095.65109)], is to develop a preconditioner which is based on a Helmholtz operator with a complex-valued shift, for a Krylov subspace iterative method. The preconditioner, which can be seen as a strongly damped wave equation in Fourier space, can be approximately inverted by a multigrid method.

... Among the latter, a widely used method is the complex shifted Laplacian (CSL) introduced in [20] and further developed in [19,21]; see also [1,4,7,23,45]. In [18,25,49,50], multilevel Krylov and multilevel deflation techniques to accelerate the CSL preconditioner are introduced and analyzed. ...

... The set Ω = Ω(λ, φ, σ) in Theorem 3.2 has bounded boundary rotation; see [31,Lemma 3.5]. Then E = 1 2 Ω + 1 2 has the same boundary rotation, V (E) = V (Ω), which is easily seen from (A.1). Let us discuss V (Ω). ...

... In most cases, the large scale scattering problem occurs. 1 The application of acoustic scattering problems is considered by the wave equation. The problem of a time-harmonic wave is reduced by the Helmholtz equation in a frequency domain. ...

... For simplicity of illustration, without loss of generality, we replace (1 + i)k by k in (1). Hence, the Helmholtz equation (1) can be rewritten as ...

This paper deals with a new multigrid method with reduced phase error for solving 2D damped Helmholtz equations. The method is obtained by taking the high‐effective, reduced phase error 5‐point finite difference (FD) scheme as a coarse grid operator and the regular 5‐point FD scheme as a fine grid operator. It is found that the proposed method gives a faster convergent rate than the regular multigrid method. A local mode Fourier analysis confirms the validity of our proposed method. Finally, some numerical results demonstrate the efficiency of the method.

... Solving the Helmholtz equation (1) efficiently, especially with large wave number, is crucial in many physics and engineering problems. For example, in exploration seismology, the Helmholtz equation (1) with the pre-given wave speed c(x) is needed to be solved for hundreds of different sources f (x) in reverse time migration, and even more in full wave inversion. However, since the discrete Helmholtz problem with high wave number is highly indefinite, finding the efficient solver is quite challenging [26], and many methods has been proposed and studied, including the direct method [18], the multigrid method [25] and the domain decomposition method [15], all of them are briefly reviewed below. ...

... A complex shift is added to the Helmholtz operator in this method, resulting in an easier problem that could be solved with multigrid solver, which then can be used as an effective preconditioner for the original Helmholtz problem. The shifted Laplace method has been shown to be very effective, and followed by many research in literature, to name a few, [1,8,13,14,50,38,7,44,33]. The amount of the shift is a compromise, a larger shift leads to easier problem to solve in preconditioning but more iteration steps in the Krylov subspace solve while a smaller shift results in harder preconditioning but less iteration steps. ...

In this paper, we propose a novel diagonal sweeping domain decomposition method with source transfer for solving the high-frequency Helmholtz equation in $\mathbb{R}^n$ with ${n}=2$ or $3$. In this method, the computational domain is partitioned into structured subdomains along all spatial directions (i.e., checkerboard domain decomposition) and a set of diagonal sweeps over the subdomains are specially designed to solve the global system efficiently. We prove that the proposed method achieves the exact solution with $2^n$ sweeps in the constant medium case. Although the sweeping usually implies sequential subdomain solves, the number of sequential steps required for each sweep in the method is only proportional to the $n$-th root of the number of subdomains when the domain decomposition is quasi-uniform with respect to all directions, thus the method is very suitable to parallel computing for solving problems with multiple right-hand sides through the pipeline processing. Extensive numerical experiments in two and three dimensions are presented to show the effectiveness and efficiency of the proposed method.

... Here we adopt the preconditioned Bi-conjugate gradient stabilized (Bi-CGSTAB) to solve (11). The preconditioned is the so-called "shifted Laplace" preconditioner [18]. ...

... The sub problem on 1 or 2 together with condition (18) or (19) is now well posed. We describe the general domain decomposition algorithm in the following. ...

A new wave simulation technique for the elastic wave equation in the frequency domain based on a no overlapping domain decomposition algorithm is investigated. The boundary conditions and the finite difference discrimination of the elastic wave equation are derived. The algorithm of no overlapping domain decomposition method is given. The method solves the elastic wave equation by iteratively solving sub problems defined on smaller sub domains. Numerical computations both for homogeneous and inhomogeneous media show the effectiveness of the proposed method. This method can be used in the full-waveform inversion.

... Note also that the absorption coefficient can be frequency dependent with the considered frequency domain formulation. An approximate solution for the partial differential equation Eq. (1) can be obtained using a finite element method [38][39][40]. ...

... Standard linear Lagrangian basis functions were used in finite elements. The solutions were computed with a damped Helmholtz preconditioner technique described in [39,40]. Each mesh corresponds to a different driver posture and they were generated so that there are at least 10 nodes per wavelength at the highest studied frequency (1000 Hz). ...

A new method is presented to obtain a local active noise control that is optimal in stochastic environment. The method uses numerical acoustical modeling that is performed in the frequency domain by using a sequence of finite element discretizations of the Helmholtz equation. The stochasticity of domain geometry and primary noise source is considered. Reference signals from an array of microphones are mapped to secondary loudspeakers, by an off-line optimized linear mapping. The frequency dependent linear mapping is optimized to minimize the expected value of error in a quiet zone, which is approximated by the numerical model and can be interpreted as a stochastic virtual microphone. A least squares formulation leads to a quadratic optimization problem. The presented active noise control method gives robust and efficient noise attenuation, which is demonstrated by a numerical study in a passenger car cabin. The numerical results demonstrate that a significant, stable local noise attenuation of 20–32 dB can be obtained at lower frequencies (< 500 Hz) by two microphones, and 8–36 dB attenuation at frequencies up to 1000 Hz, when 8 microphones are used.

... This reduction makes our iterative method extremely efficient and enables us to solve systems with billions of unknowns as our numerical examples demonstrate. With multigrid preconditioners like in [1,9,10] and with nonorthogonal grids in [28] such reduction cannot be made and the solution of the model problem would require much more memory and computation. With the separationof-variables preconditioner in [31] the reduction of the iterations onto a sparse subspace can be made with the model problem. ...

... Such extension is nontrivial and involves the use of higher order discretization in the layered media and efficient direct solver for the 3D separable preconditioner. Another preconditioning technique for scattering problems in layered media without an object has been considered in [1,10,28,31], for example, and with an object in [19]. ...

A finite element solution procedure is presented for accurately computing time-harmonic acoustic scattering by elastic targets buried in sediment. An improved finite element discretization based on trilinear basis functions leading to fourth-order phase accuracy is described. For sufficiently accurate discretizations 100 million to 1 billion unknowns are required. The resulting systems of linear equations are solved iteratively using the GMRES method with a domain decomposition preconditioner employing a fast direct solver. Due to the construction of the discretization and preconditioner, iterations can be reduced onto a sparse subspace associated with the interfaces. Numerical experiments demonstrate capability to evaluate the scattered field with hundreds of wavelengths.

... Our numerical results confirm our spectral analysis. of generalizations appeared shortly afterward in [1,[13][14][15][16] together with applications in different industrial contexts in [17][18][19][20][21][22][23][24]. More recent developments are given in [13,25]. ...

... The complex shift introduces damping and renders the preconditioned system amenable to approximate inversion using either geometric multigrid [1,36] or MILU [9]. More recently, algebraic multigrid has been used to invert the preconditioner [14,15]. The spectral properties of the shifted Laplace preconditioned Helmholtz operator M 1 h,.ˇ1,ˇ2/ ...

Deflating the shifted Laplacian with geometric multigrid vectors yields speedup. To verify this claim, we investigate a simplified variant of Erlangga and Nabben presented in [Erlangga and Nabben, ETNA, 2008;31:403–424]. We derive expressions for the eigenvalues of the two‐level preconditioner for the one‐dimensional problem. These expressions show that the algorithm analyzed is not scalable. They also show that the imaginary shift can be increased without delaying the convergence of the outer Krylov acceleration. An increase of the number of grid points per wavelength results in convergence acceleration. This contrasts to the use of the shifted Laplace preconditioner. Our analysis also shows that the use of deflation results in a spectrum more favorable to the convergence of the outer Krylov acceleration. The near‐null space components are still insufficiently well resolved, and the number of iterations increases with the wavenumber. In the two‐dimensional case, the number of near‐zero eigenvalues is larger than in the one‐dimensional case. We perform numerical computations with the two‐level and multilevel versions of the algorithm on constant and nonconstant wavenumber problems. Our numerical results confirm our spectral analysis. Copyright © 2013 John Wiley & Sons, Ltd.

... However, still being unable to efficiently solve very high wavenumber problems using this approach, the idea was reinvented and extended to the Complex Shifted Laplacian (CSL) by Erlangga, Vuik and Oosterlee in [18,19] and analyzed further in [20,38]. Leading to satisfactory convergence and scalability results on highly indefinite problems, the complex shifting of the original problem operator was generalized in [16,17,2,26]. ...

... (1) Stability property: ∀h, ∀λ l ∈ spec(A h ) : |p m (λ l )| < 1 (l = 1, . . . , n), (2) Smoothing property: p m (λ n ) = 0. ...

In this paper we construct and analyse a level-dependent coarsegrid
correction scheme for indefinite Helmholtz problems. This adapted multigrid
method is capable of solving the Helmholtz equation on the finest grid using a
series of multigrid cycles with a grid-dependent complex shift, leading to a
stable correction scheme on all levels. It is rigourously shown that the
adaptation of the complex shift throughout the multigrid cycle maintains the
functionality of the two-grid correction scheme, as no smooth modes are
amplified in or added to the error. In addition, a sufficiently smoothing
relaxation scheme should be applied to ensure damping of the oscillatory error
components. Numerical experiments on various benchmark problems show the method
to be competitive with or even outperform the current state-of-the-art
multigrid-preconditioned Krylov methods, like e.g. CSL-preconditioned GMRES or
BiCGStab.

... Thus, a functional of the solution of a stochastic PDE is computed directly without approximating the stochastic solution. This is a non-intrusive approach, that is, a solution method for non-stochastic problems like the one in [1,2] can be employed without any modications. ...

... Solving the system with a reasonable number of iterations is, however, challenging as the matrix A is badly conditioned and especially so when the calculation domain is large and the frequency is high. In the numerical example in Section 5, the solutions are computed after the systems are preconditioned by a damped Helmholtz preconditioner described in [1,2]. ...

A numerical method for optimizing the local control of sound in a stochastic domain is developed. A three-dimensional enclosed acoustic space, for example, a cabin with acoustic actuators in given locations is modeled using the nite element method in the frequency domain. The optimal local noise control signals minimizing the least square of the pres-sure eld in the silent region are given by the solution of a quadratic opti-mization problem. The developed method computes a robust local noise control in the presence of randomly varying parameters such as variations in the acoustic space. Numerical examples consider the noise experienced by a vehicle driver with a varying posture. In a model problem, a signi-cant noise reduction is demonstrated at lower frequencies.

... In practice, β = 0.5 was found to be effective [20,43,52]; for constant kh, the number of outer iterations then increases only linearly with k. Airaksinen et al. developed a similar algebraic multigrid preconditioner coupled with a full GMRES outer iteration [1]. Domain decomposition (DD) methods reduce the solution of (1.3) to a succession of smaller local problems, which can be solved by direct methods, for instance, and in parallel. ...

... We compare our ML preconditioner with the sparse direct solver Pardiso. 1 To do so, we monitor the number of SQMR iterations, the fill-in of the ML preconditioner relative to that of A, and both the set-up and the execution times. All numerical experiments were performed on a Intel Xeon server (2.2 GHz) with 32 GB of memory. ...

An algebraic multilevel (ML) preconditioner is presented for the Helmholtz equation in heterogeneous media. It is based on a multilevel incomplete $LDL^T$ factorization and preserves the inherent (complex) symmetry of the Helmholtz equation. The ML preconditioner incorporates two key components for efficiency and numerical stability: symmetric maximum weight matchings and an inverse-based pivoting strategy. The former increases the block-diagonal dominance of the system, whereas the latter controls $\|L^{-1}\|$ for numerical stability. When applied recursively, their combined effect yields an algebraic coarsening strategy, similar to algebraic multigrid methods, even for highly indefinite matrices. The ML preconditioner is combined with a Krylov subspace method and applied as a “black-box” solver to a series of challenging two- and three-dimensional test problems, mainly from geophysical seismic imaging. The numerical results demonstrate the robustness and efficiency of the ML preconditioner, even at higher frequency regimes.

... The idea of diagonal perturbation was first used by Kershaw [11] to eliminate unstable pivots during an incomplete cholesky factorization. In the specific case of the Helmholtz equation, earlier work [1,8,9,25,16] has shown that a simple complex shift added to the Laplace operator can yield an improved preconditioner. These papers all consider the problem at the operator level, i.e., they motivate the approach by considering the Partial Differential Operator and show evidence that shifting the Laplacean yields an effective preconditioner. ...

... Two classes of methods have attracted the interest of researchers in recent years. First, is the class of multigrid methods which do not work well for indefinite problems and for which various adaptations have been brought, see, e.g., [7,1,9]. Second, is the class of preconditioned Krylov subspace methods. ...

Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILU-based preconditioners ineffective. This paper presents a study of ways to enhance standard preconditioners by altering the diagonal by imaginary shifts. Prior work indicates that modifying the diagonal entries during the incomplete factorization process, by adding to it purely imaginary values can improve the quality of the preconditioner in a substantial way. Here we propose simple algebraic heuristics to perform the shifting and test these techniques with the ARMS and ILUT preconditioners. Comparisons are made with applications stemming from the diffraction of an acoustic wave incident on a bounded obstacle (governed by the Helmholtz Wave Equation).

... For the deterministic Helmholtz equation, preconditioning with the CSL preconditioner is a widely studied and successful technique for solving the discretized Helmholtz equation; see, e.g., [6,1,23,3,7] and [9], as well as references therein. See also [5] for a survey and [17] for recent developments. ...

We investigate the Helmholtz equation with suitable boundary conditions and uncertainties in the wavenumber. Thus the wavenumber is modeled as a random variable or a random field. We discretize the Helmholtz equation using finite differences in space, which leads to a linear system of algebraic equations including random variables. A stochastic Galerkin method yields a deterministic linear system of algebraic equations. This linear system is high-dimensional, sparse and complex symmetric but, in general, not hermitian. We therefore solve this system iteratively with GMRES and propose two preconditioners: a complex shifted Laplace preconditioner and a mean value preconditioner. Both preconditioners reduce the number of iteration steps as well as the computation time in our numerical experiments.

... One of the most common solvers for the discretized acoustic equation (1.1) is the shifted Laplacian multigrid method [20,19,55,36,53,10,1,14,13,40], where an attenuated version of (1.1) is used as a preconditioner for the true system inside a Krylov method. The attenuated system, which is the same system with a complex shift, can be easily solved by multigrid, if the attenuation is high enough. ...

The shifted Laplacian multigrid method is a well known approach for preconditioning the indefinite linear system arising from the discretization of the acoustic Helmholtz equation. This equation is used to model wave propagation in the frequency domain. However, in some cases the acoustic equation is not sufficient for modeling the physics of the wave propagation, and one has to consider the elastic Helmholtz equation. Such a case arises in geophysical seismic imaging applications, where the earth's subsurface is the elastic medium. The elastic Helmholtz equation is much harder to solve than its acoustic counterpart, partially because it is three times larger, and partially because it models more complicated physics. Despite this, there are very few solvers available for the elastic equation compared to the array of solvers that are available for the acoustic one. In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation, by combining the complex shift idea with approaches for linear elasticity. We demonstrate the efficiency and properties of our solver using numerical experiments for problems with heterogeneous media in two and three dimensions.

... One of the most common solvers for the discretized acoustic equation (1.1) is the shifted Laplacian multigrid method [20,19,55,36,53,10,1,14,13,40], where an attenuated version of (1.1) is used as a preconditioner for the true system inside a Krylov method. The attenuated system, which is the same system with a complex shift, can be easily solved by multigrid, if the attenuation is high enough. ...

The shifted Laplacian multigrid method is a well known approach for preconditioning the indefinite linear system arising from the discretization of the acoustic Helmholtz equation. This equation is used to model wave propagation in the frequency domain. However, in some cases the acoustic equation is not sufficient for modeling the physics of the wave propagation, and one has to consider the elastic Helmholtz equation. Such a case arises in geophysical seismic imaging applications, where the earth's subsurface is the elastic medium. The elastic Helmholtz equation is much harder to solve than its acoustic counterpart, partially because it is three times larger, and partially because it models more complicated physics. Despite this, there are very few solvers available for the elastic equation compared to the array of solvers that are available for the acoustic one. In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation, by combining the complex shift idea with approaches for linear elasticity. We demonstrate the efficiency and properties of our solver using numerical experiments for problems with heterogeneous media in two and three dimensions.

... In recent years, there has been a great effort to develop efficient solvers for systems arising from (1.1), using several different approaches to tackle the problem. One of the most common approaches is the shifted Laplacian multigrid preconditioner [11,15,16,17,12,18,19,20,21], which modifies the equation by adding complex values to the diagonal of the matrix. The modified system is then solved using a multigrid method, and is used as a preconditioner for the non-shifted system to obtain the solution of the problem. ...

The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complex-valued advection-diffusion-reaction (ADR) equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case, and hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our second order ADR discretization is more accurate than the standard second order discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach.

... La mayoría de las técnicas de precondicionamiento han sido diseñadas para la ecuación escalar de Helmholtz. Una de esas técnicas es la denominada desplazamiento del Laplaciano, originalmente propuesta por Bayliss, Goldstein y Turkel [5] y recientemente generalizada por Erlangga, Vuik y Oosterlee [20,19] para diferencias nitas y por Airaksinen, Heikkola, Pennanen y Toivanen en [1] para el método clásico de elemento nito. El precondicionador es simétrico positivo de nido y se obtiene sumando un Laplaciano debidamente escalado al operador original. ...

A numerical study of a preconditioner for the vector Helmholtz equation based on the shifted Laplacian preconditioning technique is presented. The Local Discontinuous Galerkin (LDG) method is used as spatial dis-
cretization technique. Scalability of the preconditioner is validated on a series of numerical experiments in polyhedral domains for high order approximations on low frecuency problems in the real case.

... Since the preconditioner is algebraic, in principle this combination can solve the problem for any mesh. For the Helmholtz linear system in (2.5), we have implemented a variant of the shifted Laplacian preconditioner [15,30,1] using a geometric multigrid framework on a regular grid. This problem is considered harder than the previous one, as it is indefinite. ...

Estimating parameters of Partial Differential Equations (PDEs) from noisy and indirect measurements requires solutions of ill-posed inverse problems. Such problems arise in a variety of applications such as geophysical, medical imaging, and nondestructive testing. These so called parameter estimation or inverse medium problems, are computationally intense since the underlying PDEs need to be solved numerous times until the reconstruction of the parameters is sufficiently accurate. Typically, the computational demand grows significantly when more measurements are available, which poses severe challenges to inversion algorithms as measurement devices become more powerful. In this paper we present jInv, a flexible framework and open source software that provides parallel algorithms for solving parameter estimation problems with many measurements. Being written in the expressive programing language Julia, jInv is portable, easy to understand and extend, cross-platform tested, and well-documented. It provides parallelization schemes that exploit the inherent structure in many parameter estimation problems and can be used to solve multiphysics inversion problems as is demonstrated using numerical experiments motivated by geophysical imaging.

... In the medium and high frequency regimes, the discretization of the Helmholtz equation leads to very large indefinite systems of linear equations that are difficult to solve efficiently. For traditional finite element discretizations, various iterative techniques have been proposed for the solution of such systems, including domain decomposition (DD) methods [7,10,11,15,19,24], fictitious domain methods [16,17], and multigrid methods [1,2,5,6]. In the case of DEM, the algebraic system of equations turns out to be very ill-conditioned and therefore quite challenging for standard iterative solvers. ...

A nonoverlapping domain decomposition (DD) method is proposed for the iterative solution of systems of equations arising from the discretization of Helmholtz problems by the discontinuous enrichment method. This discretization method is a discontinuous Galerkin finite element method with plane wave basis functions for approximating locally the solution and dual Lagrange multipliers for weakly enforcing its continuity over the element interfaces. The primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface Lagrange multipliers only. As in the FETI-H and FETI-DPH DD methods for continuous Galerkin discretizations, this system of Lagrange multipliers is iteratively solved by a Krylov method equipped with both a local preconditioner based on subdomain data, and a global one using a coarse space. Numerical experiments performed for two- and three-dimensional acoustic scattering problems suggest that the proposed DD-based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains. Copyright © 2009 John Wiley & Sons, Ltd.

... Solving the system with a reasonable number of iterations is, however, challenging as the matrix A is badly conditioned and especially so when the calculation domain is large and the frequency is high. In the numerical example in Section 5, the solutions are computed after the systems are preconditioned by a damped Helmholtz preconditioner described in [12, 13]. ...

... Iterative solution methods for complex-valued indefinite systems based on Krylov subspace methods [31] are typically generalizations of the conjugate-gradient (CG) method. The Bi-conjugate gradient stabilized (Bi-CGSTAB) algorithm [34] is one of the better known Krylov subspace algorithms for non-Hermitian problems, which has been used for Helmholtz problems, for example, in [1,10]. One of the advantages of Bi-CGSTAB, compared to full GMRES, is its limited memory requirements. ...

In this paper, an iterative solution method for a fourth-order accurate discretization of the Helmholtz equation is presented. The method is a generalization of that presented in (SIAM J. Sci. Comput. 2006; 27:1471–1492), where multigrid was employed as a preconditioner for a Krylov subspace iterative method. The multigrid preconditioner is based on the solution of a second Helmholtz operator with a complex-valued shift. In particular, we compare preconditioners based on a point-wise Jacobi smoother with those using an ILU(0) smoother, we compare using the prolongation operator developed by de Zeeuw in (J. Comput. Appl. Math. 1990; 33:1–27) with interpolation operators based on algebraic multigrid principles, and we compare the performance of the Krylov subspace method Bi-conjugate gradient stabilized with the recently introduced induced dimension reduction method, IDR(s). These three improvements are combined to yield an efficient solver for heterogeneous problems. Copyright © 2009 John Wiley & Sons, Ltd.

... While this preconditioner is effective at low wave-numbers, it is not as effective at higher wave-numbers. Progress has been made in extending the shifted-Laplace approach to an algebraic setting [13] and for a wider range of frequencies [14]. This recent work establishes an effective multilevel method for the Helmholtz problem, but further underscores the need for more robust and algebraic solvers. ...

We outline a smoothed aggregation algebraic multigrid method for 1D and 2D scalar Helmholtz problems with exterior radiation boundary conditions. We consider standard 1D finite difference discretizations and 2D discontinuous Galerkin discretizations. The scalar Helmholtz problem is particularly difficult for algebraic multigrid solvers. Not only can the discrete operator be complex-valued, indefinite, and non-self-adjoint, but it also allows for oscillatory error components that yield relatively small residuals. These oscillatory error components are not effectively handled by either standard relaxation or standard coarsening procedures. We address these difficulties through modifications of SA and by providing the SA setup phase with appropriate wave-like near null-space candidates. Much is known a priori about the character of the near null-space, and our method uses this knowledge in an adaptive fashion to find appropriate candidate vectors. Our results for GMRES preconditioned with the proposed SA method exhibit consistent performance for fixed points-per-wavelength and decreasing mesh size. Copyright © 2010 John Wiley & Sons, Ltd.

... In Sections 5.1 and 5.2, we compute the gradient of the functional , an essential point of the method, using the adjoint state technique. The algebraic multigrid method [26,27] is used for preconditioning the conjugate gradient algorithm in Section 5.3. Numerical experiments concerning the propagation of time-harmonic waves show the efficiency of the algorithm in Section 6. ...

The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions.The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situated at the Gauss–Lobatto quadrature points of the elements, and the Gauss–Lobatto quadrature rule is used so that the mass matrix becomes diagonal. This method is combined with the second-order central finite difference or the fourth-order Runge–Kutta time discretization. As a consequence of these choices, only matrix–vector products are needed in time dependent simulation. This makes the controllability method computationally efficient.

... Some preconditioning techniques have been successful in solving many electromagnetic and acoustic scattering problems. For example, a shifted-Laplacian preconditioner [12,13] was used for heterogeneous Helmholtz problems. A domain decomposition method was proposed in [14] for acoustic scattering by elastic objects in layered media. ...

The paper focuses on the numerical study of electromagnetic scattering from two-dimensional (2D) large partly covered cavities, which is described by the Helmholtz equation with a nonlocal boundary condition on the aperture. The classical five-point finite difference method is applied for the discretization of the Helmholtz equation and a linear approximation is used for the nonlocal boundary condition. We prove the existence and uniqueness of the numerical solution when the medium in the cavity is yy-direction layered or the number of the mesh points on the aperture is large enough. The fast algorithm proposed in Bao and Sun (2005) [2] for open cavity models is extended to solving the partly covered cavity problem with (vertically) layered media. A preconditioned Krylov subspace method is proposed to solve the partly covered cavity problem with a general medium, in which a layered medium model is used as a preconditioner of the general model. Numerical results for several types of partly covered cavities with different wave numbers are reported and compared with those by ILU-type preconditioning algorithms. Our numerical experiments show that the proposed preconditioning algorithm is more efficient for partly covered cavity problems, particularly with large wave numbers.

... where Z 0 and Z 1 are solutions of the adjoint state equation at time t = 0 and t = ∆t, respectively. For preconditioning the algorithm, we use the algebraic multigrid (AMG) method [16] (see also [27,19] ). As a smoother for the AMG we use the successive over relaxation (SOR) method with relaxation factor equal to 1.2. ...

The classical way of solving the time-harmonic linear acousto-elastic wave problem is to discretize the equations with finite elements or finite differences. This approach leads to large-scale indefinite complex-valued linear systems. For these kinds of systems, it is difficult to construct efficient iterative solution methods. That is why we use an alternative approach and solve the time-harmonic problem by controlling the solution of the corresponding time dependent wave equation.In this paper, we use an unsymmetric formulation, where fluid-structure interaction is modeled as a coupling between pressure and displacement. The coupled problem is discretized in space domain with spectral elements and in time domain with central finite differences. After discretization, exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method.

... As acoustical problems are often considered in rather large and complex geometries , solving acoustic problem may demand a considerable amount of time and memory. This motivates the development of efficient simulation methods, that we have reported, e.g., in [2] [3] [4] [8] [14]. ...

Computational acoustical models allow automated optimization of tractor design with respect to acoustic properties, which could speed up significantly the design process of tractor cabin prototypes. This article gives insightful prospec-tives to the tractor design process by considering modern computational acoustics technology. Mathematical formulation for a system consisting of vibrating elastic tractor structure and air-filled acoustic enclosure are given and a related numerical solution technique with finite element method (FEM) is presented. Simulation results produced with commercially available software are reviewed.

In this paper, we propose a new finite difference scheme for the 3D Helmholtz problem, which is compact and fourth-order in accuracy. Different from a standard compact fourth-order one, the new scheme is specially established based on minimizing the numerical dispersion, by approximating the zeroth-order term of the equation with a weighted-average for the values at 27 points. To determine optimal weight parameters, an optimization problem is formulated and then dealt with the singular value decomposition method based on the dispersion equation. For the proposed scheme, by skillfully splitting the 3D error equation into several 1D difference problems, the solution's uniqueness and convergence are derived with an effort. To solve the resulting linear system stemming from difference discretization, which is sparse and large-sized, we develop a Bi-CGSTAB iterative solver based on the preconditioning of shifted-laplacian and 3D full-coarsening multigrid. The shifted-laplacian is used to generate the preconditioner with a discretization by the proposed compact fourth-order scheme, while the full-coarsening multigrid with matrix-based prolongation operators is built to approximate the inverse of the preconditioner. Finally, numerical examples are presented to demonstrate the efficiency of the new difference scheme and the preconditioned solver.

Over the past years, the shifted Laplacian has been advocated as a way of making multigrid work for the indefinite Helmholtz equation. The idea is to use a shift into the complex plane of the wave number in the operator, and then to use the shifted operator as a preconditioner for a Krylov method. The hope is that due to the shift, it becomes possible to use standard multigrid to invert the preconditioner, and if the shift is not too big, it is still an effective preconditioner for the Helmholtz equation with a real wave number. There are however two conflicting requirements here: the shift should be not too large for the shifted preconditioner to be a good preconditioner, and it should be large enough for multigrid to work. It was rigorously proved last year that the preconditioner is good if the shift is at most of the size of the wavenumber. We prove here rigorously that if the shift is less than the size of the wavenumber squared, multigrid will not work. It is therefore not possible to solve the shifted Laplace preconditioner with multigrid in the regime where it is a good preconditioner.

In recent work we showed that the performance of the complex shifted Laplace preconditioner for the discretized Helmholtz equation can be significantly improved by combining it multiplicatively with a deflation procedure that employs multigrid vectors. In this chapter we argue that in this combination the preconditioner improves the convergence of the outer Krylov acceleration through a new mechanism. This mechanism allows for a much larger damping and facilitates the approximate solve with the preconditioner. The convergence of the outer Krylov acceleration is not significantly delayed and occasionally even accelerated. To provide a basis for these claims, we analyze for a one-dimensional problem a two-level variant of the method in which the preconditioner is applied after deflation and in which both the preconditioner and the coarse grid problem are inverted exactly. We show that in case that the mesh is sufficiently fine to resolve the wave length, the spectrum after deflation consists of a cluster surrounded by two tails that extend in both directions along the real axis. The action of the inverse of the preconditioner is to shrink the length of the tails while at the same time rotating them and shifting the center of the cluster towards the origin. A much larger damping parameter than in algorithms without deflation can be used.

In this paper, we apply a control algorithm and higher-order discretization methods for solving a time-harmonic elasto-acoustic problem. The scalar-valued Helmholtz equation models the propagation of acoustic waves and the vector-valued Navier-Cauchy equation concerns the propagation of waves in an elastic medium. These fundamental equations occur in a number of physical applications, such as acoustics, medical ultrasonics, and geophysics. We concentrate on the mathematical model expressing the interaction between velocity potential in the fluid domain and displacement in the structure domain. Traditionally, the complex-valued time-harmonic equations and low-order finite elements are used for solving the time-harmonic problems. This leads to large-scale indefinite systems, for which it is challenging to develop efficient iterative solution methods. Taking account of these difficulties, we turn to time-dependent equations. It is known that time-dependent equations can be simulated with respect to time until a time-harmonic solution is reached, but the approach suffers from poor convergence. Thus, we accelerate the convergence rate by employing the exact controllability method. The problem is formulated as a least-squares optimization problem, which is solved with the conjugate gradient (CG) algorithm. Computation of the gradient of the functional is done directly for the discretized problem. A graph-based multigrid method is used for preconditioning the CG algorithm.

A local active noise control method that uses stochastic numerical acoustical modeling is introduced. The frequency domain acoustical simulations are performed by a sequence solutions to Helmholtz equations approximated by FEM. The proposed ANC method maps microphone measurements linearly to the output signals of antinoise actuators. The matrix defining the linear mapping is optimized for each frequency to minimize expected value of the noise. The paper concentrates on defining the quadratic least-squares optimization problem for the minimization of the sound pressure field in the silent region. The formulation leads to a robust and accurate noise control in stochastic domains that has a stochastic noise source. The method is demonstrated numerically by an experiment in a car cabin, and significant noise reduction is demonstrated at lower frequencies.

In the last two decades, substantial effort has been devoted to solve large systems of linear equations with algebraic multigrid (AMG) method. Usually, these systems arise from discretizing partial differential equations (PDE) which we encounter in engineering problems. The main principle of this methodology focuses on the elimination of the so-called algebraic smooth error after the smoother has been applied. Smoothed aggregation style multigrid is a particular class of AMG method whose coarsening process differs from the classic AMG. It is also a very popular and effective iterative solver and preconditioner for many problems. In this paper, we present two kinds of novel methods which both focus on the modification of the aggregation algorithm, and both lead a better performance while apply to several problems, such as Helmholtz equation.

Extensions of deflation techniques previously developed for the Poisson equation to static elasticity and acoustics are presented. Compared to the (scalar) Poisson equation,4, 52, 58 the elasticity equations represent a system of equations, giving rise to more complex low frequency modes.73 In particular, the straightforward extension from the scalar case does not provide generally satisfactory convergence. However, a simple modification allows to recover the remarkable acceleration in convergence and CPU time reached in the scalar case. For the Helmholtz equation, numerous difficulties arise compared to the previous case. After discretization, the matrix is now indefinite without Sommerfeld boundary conditions, or complex with them. It is generally symmetric complex but not Hermitian, discarding optimal short recurrences from an iterative solver viewpoint.66 Furthermore, the kernel of the operator in an infinite space typically does not belong to the discrete space, and its size grows with the frequency. Numerous examples and timings are provided and show the dramatic improvements of up to two orders of magnitude in CPU time compared to the non deflated version for the elasticity. For the Helmholtz equation, the gains are noticeable only for low wave number.

We consider a controllability method for the time-periodic solution of the two-dimensional scalar wave equation with a first order absorbing boundary condition describing the scattering of a time-harmonic incident wave by a sound-soft obstacle. Solution of the time-harmonic equation is equivalent to finding a periodic solution for the corresponding time-dependent wave equation. We formulate the problem as an exact controllability one and solve the wave equation in time-domain. In a mixed formulation we look for solutions u = (v, p)T. The use of mixed formulation allows us to set the related controllability problem in (L2(Ω))d+1, a space of square-integrable functions in dimension d + 1. No preconditioning is needed when solving this with conjugate gradient method. We present numerical results concerning performance and convergence properties of the method.

An algebraic multigrid (AMG) with aggregation technique to coarsen is applied to construct a better preconditioner for solving Helmholtz equations in this paper. The solution process consists of constructing the preconditioner by AMG and solving the preconditioned Helmholtz problems by Krylov subspace methods. In the setup process of AMG, we employ the double pairwise aggregation (DPA) scheme firstly proposed by Y. Notay (2006) as the coarsening method. We compare it with the smoothed aggregation algebraic multigrid and meanwhile show shifted Laplacian preconditioners. According to numerical results, we find that DPA algorithm is a good choice in AMG for Helmholtz equations in reducing time and memory. Spectral estimation of system preconditioned by the three methods and the influence of second-order and fourth-order accurate discretizations on
the three techniques are also considered.

Extensions of deflation techniques developed for the Poisson and Navier equations (Aubry et al., 2008; Mut et al., 2010; Löhner et al., 2011; Aubry et al., 2011) , , and are presented for the Helmholtz equation. Numerous difficulties arise compared to the previous case. After discretization, the matrix is now indefinite without Sommerfeld boundary conditions, or complex with them. It is generally symmetric complex but not Hermitian, discarding optimal short recurrences from an iterative solver viewpoint (Saad, 2003) [5]. Furthermore, the kernel of the operator in an infinite space typically does not belong to the discrete space. The choice of the deflation space is discussed, as well as the relationship between dispersion error and solver convergence. Similarly to the symmetric definite positive (SPD) case, subdomain deflation accelerates convergence if the low frequency eigenmodes are well described. However, the analytic eigenvectors are well represented only if the dispersion error is low. CPU savings are therefore restricted to a low to mid frequency regime compared to the mesh size, which could be still relevant from an application viewpoint, given the ease of implementation.

The numerical study of exterior acoustics problems is usually carried out in the frequency domain. Finite element analyses often require the solution of large-scale algebraic linear systems. For very large problems, sometimes the time domain is used. Implicit time integration requires linear system solves, but these are often far easier than those from the frequency domain. This paper shows a connection between a spectral transformation preconditioner and a frequency shift time integration. This preconditioner is close to the shifted Laplace preconditioner. The preconditioned iterative method appears to be faster than time integration. Copyright © 2008 John Wiley & Sons, Ltd.

We introduce a novel strategy for constructing symmetric positive definite
(SPD) preconditioners for linear systems with symmetric indefinite matrices.
The strategy, called absolute value preconditioning, is motivated by the
observation that the preconditioned minimal residual method with the inverse of
the absolute value of the matrix as a preconditioner converges to the exact
solution of the system in at most two steps. Neither the exact absolute value
of the matrix nor its exact inverse are computationally feasible to construct
in general. However, we provide a practical example of an SPD preconditioner
that is based on the suggested approach. In this example we consider a model
problem with a shifted discrete negative Laplacian, and suggest a geometric
multigrid (MG) preconditioner, where the inverse of the matrix absolute value
appears only on the coarse grid, while operations on finer grids are based on
the Laplacian. Our numerical tests demonstrate practical effectiveness of the
new MG preconditioner, which leads to a robust iterative scheme with minimalist
memory requirements.

In this paper, we analyse and implement preconditioners for efficiently solving the discrete Helmholtz equation. The preconditioners are based on definite versions of the Helmholtz equation. The computational performance with Bi-CGSTAB is presented for a 2D homogeneous model problem.

Elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the well-posedness of the problem. Examples of practical applications include the Helmholtz equation and Laplace's equation. Computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface. Direct imposition of the condition at infinity along the finite boundary results in large errors. A sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain. Estimates of the error due to the finite boundary are obtained for several cases. Computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions. The examples are based on a finite element formulation but finite difference methods can also be used.

State-of-the-art finite-element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the field are specifically addressed: the effective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, infinite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales, and requiring a large computational effort; efficient algebraic equation solving methods for the resulting complex-symmetric non-Hermitian matrix systems including sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite-element methods are described. Stabilized, multiscale, and other wave-based discretization methods developed to reduce this error are reviewed. A review of finite-element methods for acoustic inverse problems and shape optimization is also given. © 2006 Acoustical Society of America.

In many numerical procedures one wishes to improve the basic approach either to improve efficiency or else to improve accuracy. Frequently this is based on an analysis of the properties of the discrete system being solved. Using a linear algebra approach one then improves the algorithm. We review methods that instead use a continuous analysis and properties of the differential equation rather than the algebraic system. We shall see that frequently one wishes to develop methods that destroy the physical significance of intermediate results. We present cases where this procedure works and others where it fails. Finally we present the opposite case where the physical intuition can be used to develop improved algorithms.

In this paper, the algorithms of the automatic mesh generator NETGEN are described. The domain is provided by a Constructive
Solid Geometry (CSG). The whole task of 3D mesh generation splits into four subproblems of special point calculation, edge
following, surface meshing and finally volume mesh generation. Surface and volume mesh generation are based on the advancing
front method. Emphasis is given to the abstract structure of the element generation rules. Several techniques of mesh optimization
are tested and quality plots are presented.

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year's Report, St. Hugh's College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called “shifted Laplace” preconditioners of the form Δφ−αk2φ with . Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.

The Helmholtz equation (Δ+K2n2)u=f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. Such problems can be solved numerically by first truncating the given unbounded domain, imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization (for example, using a finite element method). In practical applications, the mesh size h and the wave number K are not independent but are constrained by the accuracy of the desired computation. It will be shown that the number of points per wavelength, measured by (Kh)−1, is not sufficient to determine the accuracy of a given discretization. For example, the quantity K3h2 is shown to determine the accuracy in the L2 norm for a second-order discretization method applied to several propagation models.

An iterative algorithm for the solution of the Helmholtz equation is developed. The algorithm is based on a preconditioned conjugate gradient iteration for the normal equations. The preconditioning is based on an SSOR sweep for the discrete Laplacian. Numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm.

An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtz-type differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with Fourier analysis. Multigrid analysis results are verified by numerical experiments. High wave number Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner.

In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation. Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artifical boundaries. These boundary conditions not only guarantee stable difference approximations, but also minimize the (unphysical) artificial reflections that occur at the boundaries.

The numerical solution of the Helmholtz equation subject to nonlocal radiation boundary conditions is studied. The specific problem is the propagation of hydroacoustic waves in a two-dimensional curvilinear duct. The problem is discretized with a second-order accurate finitedifference method, resulting in a linear system of equations. To solve the system of equations, a preconditioned Krylov subspace method is employed. The preconditioner is based on fast transforms, and yields a direct fast Helmholtz solver for rectangular domains. Numerical experiments for curved ducts demonstrate that the rate of convergence is high. Compared with band Gaussian elimination the preconditioned iterative method shows a significant gain in both storage requirement and arithmetic complexity.

In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation. Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artificial boundaries. These boundary conditions not only guarantee stable difference approximations but also minimize the (unphysical) artificial reflections which occur at the boundaries.

Standard multigrid algorithms have proven ineffective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's effectiveness through theoretical analysis of a model problem and experimental results. In particular, we show thai the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200. For fixed wave numbers, it displays grid-independent convergence rates and has costs proportional to the number of unknowns.

RESUMEN RESUMEN
The treatment of domains with corners in the problem of absorbing boundary conditions for the wave equation is very important from a practical point of view. A technical difficulty appears as soon as conditions of order greater than or equal to 2 are considered. A solution is proposed for the two-dimensional case when second-order conditions are used. This solution consists of prescribing an adequate corner condition. The problem thus obtained is analyzed theoretically and the condition is proved to be optimal. The results obtained here are illustrated by numerical simulations. Some extensions to higher-space dimensions and higher-order conditions are proposed.

This paper discusses an iterative method for solving the Helmholtz equation with the perfectly matched layer (PML). The method consists of an outer and inner itera- tion process. The inner iteration is used to approximately solve a preconditioner, which in this case is based on a modified PML equation. The outer iteration is a Krylov subspace method (Bi-CGSTAB). The method explained here is identical with the method already discussed and proposed, e.g., in (Erlangga, Oosterlee, Vuik, SIAM J. Sci. Comput., 27 (2006), pp. 1471-1492). We show that the extension of the method to the PML equa- tion is straight-forward, and the performance for this type of problem does not degrade as compared to Helmholtz problems with, e.g. Engquist and Majda's second order boundary condition.

A cognitive journey towards the reliable simulation of scattering problems using finite element methods, with the pre-asymptotic analysis of Galerkin FEM for the Helmholtz equation with moderate and large wave number forming the core of this book. Starting from the basic physical assumptions, the author methodically develops both the strong and weak forms of the governing equations, while the main chapter on finite element analysis is preceded by a systematic treatment of Galerkin methods for indefinite sesquilinear forms. In the final chapter, three dimensional computational simulations are presented and compared with experimental data. The author also includes broad reference material on numerical methods for the Helmholtz equation in unbounded domains, including Dirichlet-to-Neumann methods, absorbing boundary conditions, infinite elements and the perfectly matched layer. A self-contained and easily readable work.

The development of numerical methods for solving the Helmholtz equation, which behaves robustly with respect to the wave number, is a topic of vivid research. It was observed that the solution of the Galerkin finite element method (FEM) differs significantly from the best approximation with increasing wave number. Many attempts have been presented in the literature to eliminate this lack of robustness by various modifications of the classical Galerkin FEM.
However, we will prove that, in two and more space dimensions, it is impossible to eliminate this so-called pollution effect. Furthermore, we will present a generalized FEM in one dimension which behaves robustly with respect to the wave number

this paper is to preconditionthis linear system with a new preconditioner and then solve it iteratively usinga Krylov subspace method. Numerical analysis shows the preconditionerto be eective on a simple 1D test problem, and results are presented showingconsiderable convergence acceleration for a number of dierent Krylovmethods for more complex problems in 2D, as well as for the more generalproblem of harmonic disturbances to a non-stagnant steady ow

Fictitious domain methods for the numerical solution of two-dimensional scattering problems are considered. The original exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary. First-order, second-order, and exact nonreflecting boundary conditions are tested on rectangular and circular boundaries. The finite element discretizations of the corresponding approximate boundary value problems are performed using locally fitted meshes, and the discrete equations are solved with fictitious domain methods. A special finite element method using nonmatching meshes is considered. This method uses the macro-hybrid formulation based on domain decomposition to couple polar and cartesian coordinate systems. A special preconditioner based on fictitious domains is introduced for the arising algebraic saddle-point system such that the subspace of constraints becomes invariant with respect to the preconditioned iterative procedure. The performance of the new method is compared to the fictitious domain methods both with respect to accuracy and computational cost.

The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps. These conditions are, however, nonlocal in nature, and we introduce local approximations which we optimize for performance of the Schwarz method. This leads to an algorithm in the class of optimized Schwarz methods. We present an asymptotic analysis of the optimized Schwarz method for two types of transmission conditions, Robin conditions and transmission conditions with second order tangential derivatives. Numerical results illustrate the effectiveness of the optimized Schwarz method on a model problem and on a problem from industry.

The first iterative methods used for solving large linear systems were based on relaxation of the coordinates. Beginning with a given approximate solution, these methods modify the components of the approximation, one or a few at a time and in a certain order, until convergence is reached. Each of these modifications, called relaxation steps, is aimed at annihilating one or a few components of the residual vector. Now these techniques are rarely used separately. However, when combined with the more efficient methods described in later chapters, they can be quite successful. Moreover, there are a few application areas where variations of these methods are still quite popular.

Preconditioning techniques based on incomplete factorization of matrices are investigated, to solve highly indefinite complex-symmetric linear systems. A novel preconditioning is introduced. The real part of the matrix is made positive definite, or less indefinite, by adding properly defined perturbations to the diagonal entries, while the imaginary part is unaltered. The resulting preconditioning matrix, which is obtained by applying standard methods to the perturbed complex matrix, turns out to perform significantly better than classical incomplete factorization schemes. For realistic values of the GMRES restart parameter, spectacular reduction of iteration counts is observed. A theoretical spectral analysis is provided, in which the spectrum of the preconditioner applied to indefinite matrix is related to the spectrum of the same preconditioner applied to a Stieltjes matrix extracted from the indefinite matrix. Results of numerical experiments are reported, which display the efficiency of the new preconditioning. Copyright © 2001 John Wiley & Sons, Ltd.

We consider acoustic scattering about a general body. This is described by the Helmholtz equation exterior to the body. In order to truncate the infinite domain we use the BGT absorbing boundary condition. The resultant problem in a finite domain is solved by a finite element procedure. This yields a large sparse system of linear equations which is neither symmetric nor positive definite. We solve the system by an iterative Krylov space type method. To increase the rate of convergence a preconditioner is introduced. This preconditioner is based on a different Helmholtz equation with complex coefficients. This preconditioned system is again solved by a Krylov space method with an ILU preconditioner. Computations are presented to show the efficiency of this technique.

We introduce an algorithm for the efficient numerical solution of exterior boundary value problems for the Helmholtz equation.
The problem is reformulated as an equivalent one on a bounded domain using an exact non-local boundary condition on a circular
artificial boundary. An FFT-based fast Helmholtz solver is then derived for a finite-element discretization on an annular
domain. The exterior problem for domains of general shape are treated using an imbedding or capacitance matrix method. The
imbedding is achieved in such a way that the resulting capacitance matrix has a favorable spectral distribution leading to
mesh independent convergence rates when Krylov subspace methods are used to solve the capacitance matrix equation.

We present two different but related Lagrange multiplier based domain decomposition (DD) methods for solving iteratively large-scale systems of equations arising from the finite element discretization of high-frequency exterior Helmholtz problems. The proposed methods are essentially two distinct extensions of the regularized finite element tearing and interconnecting (FETI) method to indefinite or complex problems. The first method employs a single Lagrange multiplier field to glue the local solutions at the subdomain interface boundaries. The second method employs two Lagrange multiplier fields for that purpose. The key ingredients of both of these FETI methods are the regularization of each subdomain matrix by a complex lumped mass matrix defined on the subdomain interface boundary, and the preconditioning of the global interface problem by a coarse second-level problem constructed with planar waves. We show numerically that both methods are scalable with respect to the mesh size, the subdomain size, and the wavenumber, but that the FETI method with a single Lagrange multiplier field – labeled FETI-H (H for Helmholtz) in this paper – delivers superior computational performances. We apply the FETI-H method to the parallel solution on a 24-processor Origin 2000 of an acoustic scattering problem with a submarine shaped obstacle, and report performance results that highlight the unique efficiency of this DD method for the solution of high frequency acoustic scattering problems.

This paper contains a brief review of the formulation of the finite element method for structural-acoustic analysis of an enclosed cavity, and illustrations are given of the application of this analytical method at General Motors Corporation to investigate the acoustics of the automobile passenger compartment. Low frequency noise in the passenger compartment (in approximately the 20–200 Hz frequency range) is of primary interest, and particularly that noise which is generated by the structural vibration of the wall panels of the compartment. The topics which are covered in the paper include the computation of acoustic modes and resonant frequencies of the passenger compartment, the effect of flexible wall panels on the cavity acoustics, the methods of direct and modal coupling of the structural and acoustic vehicle systems, and forced vibration analysis illustrating the techniques for computing panel-excited noise and for identifying critical panels around the passenger compartment. The capabilities of the finite element method are illustrated by applications to the production automobile, and experimental verifications of the various techniques are presented to illustrate the accuracy of the method.

We discuss in this article the application of controllability techniques to the computation of the time-periodic solutions of evolution equations. The basic principles of the computational methods are presented in a fairly general context where the time discretization aspect is also discussed. Then this general methodology is applied to the solution of scattering problems for harmonic planar waves by two- and three-dimensional purely reflecting nonconvex obstacles. Numerical results obtained by the above method and comparisons with the results obtained by more classical methods show the superiority of the former ones.

We formulate the Helmholtz equation as an exact controllability problem for the time-dependent wave equation. The problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements. This approach leads to high accuracy in spatial discretization. Moreover, the spectral element method results in diagonal mass matrices, which makes the time integration of the wave equation highly efficient. After discretization, the exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method. We illustrate the method with some numerical experiments, which demonstrate the significant improvements in efficiency due to the higher order spectral elements. For a given accuracy, the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method. In addition, by using higher order polynomial basis the influence of the pollution effect is reduced.

We developed a fast iterative solver for computing time-harmonic acoustic waves scattered by an elastic object in layered media. The discretization of the problem was performed using a finite element method with linear elements based on a locally body-fitted uniform triangulation. We used a domain decomposition preconditioner in the iterative solution of the resulting system of linear equations. The preconditioner was based on a cyclic reduction type fast direct solver. The solution procedure reduces GMRES iterates onto a sparse subspace which decreases the storage and computational requirements essentially. The numerical results demonstrate the effectiveness of the proposed approach for two-dimensional domains that are hundreds of wavelengths wide and require the solution of linear systems with several millions of unknowns.

A preconditioned iterative method based on separation-of-variables for solving the Helmholtz equation in an inhomogeneous medium is tested. The preconditioner is constructed by approximating the wavenumber by a sum of two terms, one depending only on one spatial coordinate, say x, and the other depending on the remaining coordinates. The Helmholtz equation can be solved efficiently if the wavenumber has such a separable form. First, an eigenvalue-eigenvector decomposition is applied in the x-direction. With these eigenvectors, a change of variables is performed in order to obtain a set of independent systems with one dimension less than the original one. For smooth models and low frequencies, the convergence rate with this preconditioner is satisfactory. Unfortunately, it rapidly deteriorates when the roughness of the model or the frequency increases. Examples from seismic modeling are given to illustrate this behaviour. Moreover, numerical evidence is presented that suggests that the decomposition of the wavenumber in the sum of two terms cannot be improved with this approach.

Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

Recently the Conjugate Gradients-Squared (CG-S) method has been proposed as an attractive variant of the Bi-Conjugate Gradients (Bi-CG) method. However, it has been observed that CG-S may lead to a rather irregular convergence behaviour, so that in some cases rounding errors can even result in severe cancellation effects in the solution. In this paper, another variant of Bi-CG is proposed which does not seem to suffer from these negative effects. Numerical experiments indicate also that the new variant, named Bi-CGSTAB, is often much more efficient than CG-S.

. The application of fictitious domain methods to the three-dimensional Helmholtz equation with absorbing boundary conditions is considered. The finite element discretization is performed by using locally fitted meshes, and algebraic fictitious domain methods with separable preconditioners are used in the iterative solution of the resultant linear systems. These methods are based on embedding the original domain into a larger one with a simple geometry. With this approach, it is possible to realize the GMRES iterations in a low-dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner. An efficient parallel implementation of the iterative algorithm is introduced. Results of numerical experiments demonstrate good scalability properties on distributed-memory parallel computers and the ability to solve high frequency acoustic scattering problems. Key words. Helmholtz equation, acoustic scattering, fictitious domain method, separable prec...

. This paper is devoted to the construction of Algebraic Multi-Grid (AMG) methods, which are especially suited for the solution of large sparse systems of algebraic equations arising from the finite element discretization of second-order elliptic boundary value problems on unstructured, fine meshes in two or three dimensions. The only information needed is recovered from the stiffness matrix. We present two types of coarsening algorithms based on the graph of the stiffness matrix. In some special cases of nested mesh refinement, we observe, that some geometrical version of the multi-grid method turns out to be a special case of our AMG algorithms. Finally, we apply our algorithms on two and three dimensional heat conduction problems in domains with complicated geometry (e.g. micro-scales), as well as to plane strain elasticity problems with jumping coefficients. 1 Introduction In this paper we are interested in efficient techniques for the solution of largescale finite element equatio...

. Multigrid methods are known for their high efficiency in the solution of definite elliptic problems. However, difficulties that appear in highly indefinite problems, such as standing wave equations, cause a total loss of efficiency in the standard multigrid solver. The aim of this paper is to isolate these difficulties, analyze them, suggest how to deal with them, and then test the suggestions with numerical experiments. The modified multigrid methods introduced here exhibit the same high convergence rates as usually obtained for definite elliptic problems, for nearly the same cost. They also yield a very efficient treatment of the radiation boundary conditions. Key words. Helmholtz equations, multigrid methods, wave-ray approach, radiation boundary conditions. AMS subject classifications. 65N55, 65N06, 65N22, 65B99. 1. Introduction. What are the properties of highly indefinite problems that make their solution by standard multigrid methods inefficient? First, there is the well-k...

Application of an algebraic multigrid method to incompressible flow problems

- J Martikainen
- A Pennanen
- T Rossi

J. Martikainen, A. Pennanen, T. Rossi, Application of an algebraic multigrid method to incompressible flow problems, Tech. Rep. B2/2006, Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland, 2006.

- W L Briggs
- V E Henson
- S F Mccormick

W. L. Briggs, V. E. Henson, S. F. McCormick, A multigrid tutorial, 2nd Edition,
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA,
2000.

FETI-DPH: a dual-primal domain decomposition method for acoustic scattering

- Farhat