Augmented AMG‐shifted Laplacian preconditioners for indefinite Helmholtz problems

Tsuji, Paul H.; Tuminaro, Raymond S.

doi:10.1002/nla.1997

Cited by 10 publications

(4 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we want to focus on an idea which received a lot of attention over the last decade, namely, to use a shifted Helmholtz operator as a preconditioner; see [13,21,31,5,25,27,32,28,33] and references therein. The latter is defined as a discretization of the operator H k for k 2 = k 2 + iε, where ε > 0 is the so-called shift.…”

Section: A439mentioning

confidence: 99%

How Large a Shift is Needed in the Shifted Helmholtz Preconditioner for its Effective Inversion by Multigrid?

Cocquet¹,

Gander²

2017

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

The shifted Helmholtz operator has received a lot of attention over the past decade as a preconditioner for the iterative solution of the Helmholtz equation. The idea is that if one uses a small complex shift, the shifted Helmholtz operator is still close to the original Helmholtz operator and could thus be an effective preconditioner. It was shown in [M. J. Gander, I. G. Graham, and E. A. Spence, Numer. Math., 53 (2015), pp. 573-579] that the shift can be at most O(k) to prove rigorously wave number independent convergence of the preconditioned system solved with GMRES, provided the preconditioner is inverted exactly. In practice, however, the preconditioner is inverted only approximately, and if one shifts enough, this can be done effectively by standard multigrid methods. We show in this paper that for a finite element discretization, the shift has to be at least O(k 2) to be able to invert the shifted Helmholtz preconditioner using multigrid. There is therefore a gap between being a good preconditioning operator (shift at most O(k)) and being able to effectively invert the preconditioner by multigrid (shift at least O(k 2)). So what shift should be chosen in practice, and when is the preconditioner not inverted exactly? By studying the numerical range of the preconditioned operator, we show that one cannot obtain analytical results for this case with currently available tools. We thus test the preconditioner extensively numerically for a wave guide type square domain in the range of shifts between O(√ k) and O(k 2) with approximate inversion by one multigrid V-cycle. We find in our experiments that preconditioned GMRES iteration numbers will then inevitably grow like O(k 2). We also see that in contrast to common practice where shifts of O(k 2) are used, it might be beneficial for the wave guide to use a smaller shift, e.g., O(k 3/2), especially when several smoothing steps are used.

show abstract

Section: A439mentioning

confidence: 99%

How Large a Shift is Needed in the Shifted Helmholtz Preconditioner for its Effective Inversion by Multigrid?

Cocquet¹,

Gander²

2017

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

An Additive Overlapping Domain Decomposition Method for the Helmholtz Equation

Leng¹,

Ju²

2019

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

In this paper, the diagonal sweeping domain decomposition method (DDM) [26] for solving the high-frequency Helmholtz equation in R n is re-proposed with the trace transfer method [37], where n is the dimension. The diagonal sweeping DDM [26] uses 2 n sweeps of diagonal directions on the checkerboard domain decomposition based on the source transfer method [7], it is sequential in nature yet suitable for parallel computing, since the number of sequential steps is quite small compared to the number of subdomains. The advantages of changing the basic transfer method from source transfer to trace transfer are as follows: first, no overlapping region is required in the domain decomposition; second, the sweeping algorithm becomes simpler since the transferred traces have only n cardinal directions, whereas the transferred sources have all 3 n − 1 directions. We proved that the exact solution is obtained with the proposed method in the constant medium case, and also in the two-layered medium case provided the source is on the same side with the first swept subdomain. The efficiency of the proposed method is demonstrated using numerical experiments in two and three dimensions, and it is found that numerical differences of the diagonal sweeping DDM with the two transfer methods are very small using second-order finite difference discretization.where k(x) is the wave number and defined by k(x) := ω/c(x) with ω denoting the angular frequency and c(x) the wave speed. The Helmholtz equation has varies applications, including acoustics, elasticity, electromagnetics and geophysics. The main computation of these applications involves solving the discrete system of the Helmholtz equation, however, it is hard to find an efficient and robust solver for the Helmholtz equation especially for large wave number, since the discrete system is highly indefinite and the Green's function of the Helmholtz operator is quite oscillatory [16].Many numerical methods have been developed to solve the discrete system of the Helmholtz equation, including the direct methods [13,22] with the sparsity of the matrix exploited [36], the multigrid method with the shifted Laplace as the preconditioner [19,17,18,34,30,2], the domain decomposition method with different transmission conditions [10,9,21,11,4,29,33], and in this paper we focus on the DDM for the Helmholtz problem.The sweeping type DDM is first proposed by Engquist and Ying in [14,15], and then further developed in [31,35,37,7,20]. The sweeping type DDMs are very effective to solve the Helmholtz problem using two ingredients, the first is employing the PML boundary condition on each subdomain, and the second is the correct transmission conditions between subdomains, which are the main differences between these DDMs. In the sweeping type DDMs, the domain is partitioned into layers in one direction, and the layered subdomain problems are solved one after another in the forward and backward sweeps. The subdomain problem is preferred to be solved with the direct method, which greatly reduces the com...

show abstract

“…[13][14][15] In recent years, there has been a great effort to develop efficient solvers for systems arising from (1), using several different approaches to tackle the problem. One of the most common approaches is the shifted Laplacian multigrid preconditioner, 12,13,[16][17][18][19][20][21][22] 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 nonshifted system.…”

Section: Introductionmentioning

confidence: 99%

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

Treister

Haber

2018

Numerical Linear Algebra App

View full text Add to dashboard Cite

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 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 advection-diffusion-reaction 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.

show abstract

Augmented AMG‐shifted Laplacian preconditioners for indefinite Helmholtz problems

Cited by 10 publications

References 21 publications

How Large a Shift is Needed in the Shifted Helmholtz Preconditioner for its Effective Inversion by Multigrid?

How Large a Shift is Needed in the Shifted Helmholtz Preconditioner for its Effective Inversion by Multigrid?

An Additive Overlapping Domain Decomposition Method for the Helmholtz Equation

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

Contact Info

Product

Resources

About