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## The method of polarized traces for the 2D Helmholtz equation. ArXiv e-prints (2014)

Citations: | 3 - 0 self |

### Citations

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Iterative Methods for Sparse Linear Systems
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Citation Context ...ompression was accelerated using the randomized SVD [51]. The inversion of D was implemented as a block back-substitution with compressed blocks. We used a simple GMRES algorithm11 ( Algorithm 6.9 in =-=[62]-=-). Given that the number of iteration remains, in practice, bounded by ten, neither low rank update to the Hessenberg matrix nor re-start were necessary. 7.2 Smooth Velocity Model For the smooth veloc... |

1048 |
Perfectly matched layer for the absorption of electromagnetic waves,”
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Citation Context ...mulas. 2.1 Discretization Let Ω = (0, Lx) × (0, Lz) be a rectangular domain. Throughout this paper, we pose Eq. 3 with absorbing boundary conditions on ∂Ω, realized as a perfectly matched layer (PML) =-=[6, 44]-=-. This approach consists of extending the computational domain with an absorbing layer, in which the differential operator is modified to efficiently damp the outgoing waves and reduce the reflections... |

501 | Strongly Elliptic Systems and Boundary Integral Equation,
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Citation Context ...pper half-plane Ωup. In case the wave speed becomes uniform past some large radius, the Sommerfeld radiation condition (SRC) puts to zero the contribution on D in Green’s representation formula (GRF) =-=[46, 52]-=- resulting in the incomplete Green’s formula u(x) = ∫ Γ ( ∂G ∂zy (x,y)u(y)−G(x,y) ∂u ∂zy (y) ) dSy, x ∈ Ωup \Γ. (1) On the contrary, if x approaches Γ from below, then we obtain the annihilation formu... |

445 |
Linear Integral Equations.
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Citation Context ...pper half-plane Ωup. In case the wave speed becomes uniform past some large radius, the Sommerfeld radiation condition (SRC) puts to zero the contribution on D in Green’s representation formula (GRF) =-=[46, 52]-=- resulting in the incomplete Green’s formula u(x) = ∫ Γ ( ∂G ∂zy (x,y)u(y)−G(x,y) ∂u ∂zy (y) ) dSy, x ∈ Ωup \Γ. (1) On the contrary, if x approaches Γ from below, then we obtain the annihilation formu... |

289 |
Rapid solution of integral equations of classical potential
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Citation Context ...sub-domains. In addition, the runtimes scales sub-linearly with respect to the number of volume unknowns. 11Some authors refer to GMRES algorithm as the Generalized conjugate residual algorithm [12], =-=[61]-=- 32 N ω/2pi [Hz] L = 8 L = 16 L = 32 L = 64 L = 128 195× 870 5.57 (3) 0.094 (3) 0.21 (3) 0.49 (3-4) 1.02 (3-4) 2.11 396× 1720 7.71 (3) 0.14 (3) 0.32 (3) 0.69 (3-4) 1.55 (3-4) 3.47 792× 3420 11.14 (3) ... |

266 |
Nested dissection of a regular finite element mesh.
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Citation Context ...f the 2D Laplacian, the sparse LU factorization with nested dissection is known to have O(N3/2) complexity, and the back-substitution is known to have linear complexity up to logarithmic factors, see =-=[38, 43]-=-. Given that each sub-domain has size O(N/L), the factorization cost is O((N/L)3/2). Moreover, given that the factorizations are completely independent, they can be computed simultaneously in L differ... |

103 |
Split-step Fourier migration:
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Citation Context ...e slab. With a local G, the conditions (Eq. 1 or 2) are only approximately polarizing. Other approximations of absorbing conditions, such as square-root operators, can be good in certain applications =-=[23, 63, 64, 34, 66]-=-, but are often too coarse to be useful in the context of a 3 fast domain decomposition solver. In particular, it should be noted that square-root operators do not approximate polarizing conditions to... |

98 | Domain decomposition method and the Helmholtz problem. In: Mathematical and numerical aspects of wave propagation phenomena
- Despres
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Citation Context ...n decomposition with transmission boundary conditions is perhaps to be found in the work of Despres [26], which led, in joint work with Cessenat and Benamou, to the Ultra Weak Variational Formulation =-=[5, 15, 16, 17]-=-; • Their work spawned a series of new formulations within the framework of Discontinous Galerkin methods, such as the Trefftz formulation of Perugia et al. [54, 40], and the discontinuous enrichment ... |

92 |
Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method.
- Davis
- 2004
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Citation Context ... of the Gauss-Seidel preconditioner. 7.1 Precomputation To extract the Green’s functions and to compute the local solutions, a pivoted sparse LU factorization was performed at each slab using UMFPACK =-=[21]-=-, and the LU factors were stored in memory. The LU factors for each slab are independent of the others, so they can be stored in different cluster nodes. The local solves are local to each node, enhan... |

88 | Optimization of Collective Communication Operations in MPICH.
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Citation Context ...4) (a more representative figure in Table 4), then the same argument leads to δ = 1/8 and an overall online runtime of O(N7/8). 6.2 Communication cost We use a commonly-used communication cost model (=-=[2, 25, 58, 69]-=-) to perform the cost analysis. The model assumes that each process is only able to send or receive a single message at a time. When the messages has size m, the time to communicate that message is α+... |

79 | A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations
- Elman, Ernst, et al.
- 1999
(Show Context)
Citation Context ...liminated by a ray-cycle, that exploits a geometric optics approximation of the Green’s function; • Elman and Ernst use a relaxed multigrid method as a preconditioner for an outer Krylov iteration in =-=[27]-=-; • Haber and McLachlan proposed an alternative formulation for the Hemholtz equation [41], which reduces the problem to solve an eikonal equation and an advection-diffusion-reaction equation, which c... |

71 | A review on the inverse of symmetric tridiagonal and block tridiagonal matrices,
- Meurant
- 1992
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Citation Context ...g of the unknowns implies that every block in H correspond to degrees of freedom with fixed depth (fixed z). In order to prove the equivalent of the rank-one property in the 2D case we follow [7] and =-=[53]-=-. Definition 10. Let ∆i and Σi be defined by the following recurrence relations ∆1 = H1, ∆i = Hi −Ci (∆i−1)−1 Cti; (125) Σnz = Hnz , Σi = Hi −Cti+1 (Σi+1)−1 Ci+1. (126) Proposition 3. H is proper12, a... |

68 |
Application of an ultra weak variational formulation of elliptic PDEs to the twodimensional Helmholtz problem.
- Cessenat, Després
- 1998
(Show Context)
Citation Context ...n decomposition with transmission boundary conditions is perhaps to be found in the work of Despres [26], which led, in joint work with Cessenat and Benamou, to the Ultra Weak Variational Formulation =-=[5, 15, 16, 17]-=-; • Their work spawned a series of new formulations within the framework of Discontinous Galerkin methods, such as the Trefftz formulation of Perugia et al. [54, 40], and the discontinuous enrichment ... |

65 | A Novel Multigrid Based Preconditioner for Heterogeneous Helmholtz Problems,”
- Erlangga, Oosterlee, et al.
- 2006
(Show Context)
Citation Context ...efficiently by a Krylov method using a multigrid method as a preconditioner; and 7 • Grote and Schenk [10] proposed an algebraic multi-level preconditioner for the Helmholtz equation. Erlangga et al. =-=[31]-=- showed how to implement a simple, although suboptimal, complex-shifted Laplace preconditioner with multigrid, which shines for its simplicity. Chen et al. implemented a 3D solver using the complex-sh... |

55 | Minimizing communication in numerical linear algebra,
- Ballard, Demmel, et al.
- 2012
(Show Context)
Citation Context ...4) (a more representative figure in Table 4), then the same argument leads to δ = 1/8 and an overall online runtime of O(N7/8). 6.2 Communication cost We use a commonly-used communication cost model (=-=[2, 25, 58, 69]-=-) to perform the cost analysis. The model assumes that each process is only able to send or receive a single message at a time. When the messages has size m, the time to communicate that message is α+... |

50 | A discontinuous enrichment method.
- Harari, Franca
- 2001
(Show Context)
Citation Context ... series of new formulations within the framework of Discontinous Galerkin methods, such as the Trefftz formulation of Perugia et al. [54, 40], and the discontinuous enrichment method of Farhat et al. =-=[33]-=-; • Hiptmair et al. proposed the multi-trace formulation [42], which involves solving an integral equation posed on the interfaces of a decomposed domain, which is naturally well suited for operator p... |

46 | Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers,” Multiscale Model.
- Engquist, Ying
- 2011
(Show Context)
Citation Context ...ach can be traced back to the AILU method of Gander and Nataf in 2001 [35, 37]. The first linear complexity claim was perhaps made in the work of Engquist and Ying on sweeping preconditioners in 2011 =-=[28, 29]-=- – a special kind of domain decomposition into grid-spacing-thin layers. In 2013, Stolk [65] restored the flexibilty to consider coarser layerings, with a domain decomposition method that realizes int... |

44 | Wave-ray multigrid method for standing wave equations.
- Brandt, Livshits
- 1997
(Show Context)
Citation Context ... and higher dimensions: • Bhowmik and Stolk recently proposed new rules of optimized coarse grid corrections for a two level multigrid method [67]; • Brandt and Livshits developed the wave-ray method =-=[11]-=-, in which the oscillatory error components are eliminated by a ray-cycle, that exploits a geometric optics approximation of the Green’s function; • Elman and Ernst use a relaxed multigrid method as a... |

42 |
Hierarchical matrices: a means to efficiently solve elliptic boundary value problems. In:
- Bebendorf
- 2008
(Show Context)
Citation Context ... calling this structure partitioned low-rank (PLR). An early reference for PLR matrices is the work of Jones, Ma, and Rokhlin in 1994 [45]. PLR matrices are a special case of H-matrices5. It is known =-=[3]-=-, that the blocks of matrices such as M can have low rank, provided they obey an admissibility condition that takes into account the distance between blocks. In regimes of high frequencies and rough h... |

42 |
Advances in Iterative Methods and Preconditioners for the Helmholtz Equation,”
- Erlangga
- 2008
(Show Context)
Citation Context ... by Plessix [56], Riyanti et al. [60] and, more recently, Calandra et al. [13]. A good review of iterative methods for the Helmholtz equation, including complex-shifted Laplace preconditioners, is in =-=[30]-=-. Another review paper that discussed the difficulties generated by the high-frequency limit is [32]. Finally, a beautiful mathematical exposition of the Helmholtz equation and a good review is [55]. ... |

40 | Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: Analysis of the p-version. SAM-ETHZurich Report
- Hiptmair, Moiola, et al.
- 2009
(Show Context)
Citation Context ...eak Variational Formulation [5, 15, 16, 17]; • Their work spawned a series of new formulations within the framework of Discontinous Galerkin methods, such as the Trefftz formulation of Perugia et al. =-=[54, 40]-=-, and the discontinuous enrichment method of Farhat et al. [33]; • Hiptmair et al. proposed the multi-trace formulation [42], which involves solving an integral equation posed on the interfaces of a d... |

38 | Why it is difficult to solve Helmholtz problems with classical iterative methods.
- Ernst, Gander
- 2012
(Show Context)
Citation Context ...rative methods for the Helmholtz equation, including complex-shifted Laplace preconditioners, is in [30]. Another review paper that discussed the difficulties generated by the high-frequency limit is =-=[32]-=-. Finally, a beautiful mathematical exposition of the Helmholtz equation and a good review is [55]. 1.5 Organization The discrete integral reformulation of the Helmholtz equation is presented in secti... |

34 |
Generalization of the phase-screen approximation for the scattering of acoustic waves:
- Hoop, Rousseau, et al.
- 2000
(Show Context)
Citation Context ...e slab. With a local G, the conditions (Eq. 1 or 2) are only approximately polarizing. Other approximations of absorbing conditions, such as square-root operators, can be good in certain applications =-=[23, 63, 64, 34, 66]-=-, but are often too coarse to be useful in the context of a 3 fast domain decomposition solver. In particular, it should be noted that square-root operators do not approximate polarizing conditions to... |

30 |
Complexity bounds for regular finite difference and finite element grids
- Hoffman, Martin, et al.
- 1973
(Show Context)
Citation Context ...f the 2D Laplacian, the sparse LU factorization with nested dissection is known to have O(N3/2) complexity, and the back-substitution is known to have linear complexity up to logarithmic factors, see =-=[38, 43]-=-. Given that each sub-domain has size O(N/L), the factorization cost is O((N/L)3/2). Moreover, given that the factorizations are completely independent, they can be computed simultaneously in L differ... |

28 | Wave propagation using bases for bandlimited functions
- Beylkin, Sandberg
- 2005
(Show Context)
Citation Context ...ompressed using the recursive Alg. 5. The result of this algorithm is a quadtree structure on the original matrix, where the leaves are maximally large square submatrices with fixed -rank. We follow =-=[8]-=- in calling this structure partitioned low-rank (PLR). An early reference for PLR matrices is the work of Jones, Ma, and Rokhlin in 1994 [45]. PLR matrices are a special case of H-matrices5. It is kno... |

27 |
Algebraic multilevel preconditioner for the Helmholtz equation in heterogeneous media
- Bollhöfer, Grote, et al.
(Show Context)
Citation Context ...em to solve an eikonal equation and an advection-diffusion-reaction equation, which can be solved efficiently by a Krylov method using a multigrid method as a preconditioner; and 7 • Grote and Schenk =-=[10]-=- proposed an algebraic multi-level preconditioner for the Helmholtz equation. Erlangga et al. [31] showed how to implement a simple, although suboptimal, complex-shifted Laplace preconditioner with mu... |

27 |
A randomized algorithm for the decomposition of matrices
- Rokhlin, Martinsson, et al.
(Show Context)
Citation Context ...ressed in sparse PLR form, following Alg. 5. M and R were implemented as a block matrices, in which each block is a PLR matrix in sparse form. The compression was accelerated using the randomized SVD =-=[51]-=-. The inversion of D was implemented as a block back-substitution with compressed blocks. We used a simple GMRES algorithm11 ( Algorithm 6.9 in [62]). Given that the number of iteration remains, in pr... |

27 |
Separation-of-variables as a preconditioner for an iterative Helmholtz solver
- Plessix, Mulder
- 2003
(Show Context)
Citation Context ...terfaces of a decomposed domain, which is naturally well suited for operator preconditionning in the case of piece-wise constant medium; • Plessix and Mulder proposed a method similar to AILU in 2003 =-=[57]-=-; • More recently, Geuzaine and Vion explored approximate transmission boundary conditions [73, 72] coupled with a multiplicative Schwartz iteration, to improve on traditional domain decomposition met... |

26 | A fast butterfly algorithm for the computation of Fourier integral operators, Multiscale Model
- Candès, Demanet, et al.
(Show Context)
Citation Context .... This choice is neither original nor optimal in the high-frequency regime, but it gives rise to elementary code. More sophisticated approaches have been proposed elsewhere, including by one of us in =-=[14]-=-, but the extension of those ideas to the kernel-independent framework is not immediate. This section is therefore added for the sake of algorithmic completeness, and for clarification of the ranks an... |

24 |
Using Plane Waves as Base Functions for Solving Time Harmonic Equations with the Ultra Weak VariaInternational
- Cessenat, Després
- 2003
(Show Context)
Citation Context ...n decomposition with transmission boundary conditions is perhaps to be found in the work of Despres [26], which led, in joint work with Cessenat and Benamou, to the Ultra Weak Variational Formulation =-=[5, 15, 16, 17]-=-; • Their work spawned a series of new formulations within the framework of Discontinous Galerkin methods, such as the Trefftz formulation of Perugia et al. [54, 40], and the discontinuous enrichment ... |

21 |
Décomposition de domaine et problème de Helmholtz
- Després
- 1990
(Show Context)
Citation Context ...on and sweeping-style methods for the Helmholtz equation: • The earliest suggestion to perform domain decomposition with transmission boundary conditions is perhaps to be found in the work of Despres =-=[26]-=-, which led, in joint work with Cessenat and Benamou, to the Ultra Weak Variational Formulation [5, 15, 16, 17]; • Their work spawned a series of new formulations within the framework of Discontinous ... |

19 |
A fast direct solver for scattering problems involving elongated structures
- Martinsson, Rokhlin
- 2007
(Show Context)
Citation Context ... adequate in our case. The “nearby interaction” blocks (from one interface to itself) have a singularity along the diagonal, but tend to otherwise have large low-rank blocks in media close to uniform =-=[50]-=-. The “remote interaction” blocks (from one interface to the next) do not have the diagonal problem, but have a wave vector diversity that generates higher ranks. Adaptivity is therefore a very natura... |

17 |
Application d’une nouvelle formulation variationnelle aux équations d’ondes harmoniques.
- Cessenat
- 1996
(Show Context)
Citation Context |

17 |
On 3D Modeling of Seismic Wave Propagation Via a Structured Parallel Multifrontal Direct Helmholtz Solver,” Geophys.
- Wang, Hoop, et al.
- 2011
(Show Context)
Citation Context ...ion conditions with domain decomposition. In a different direction, much progress has been made on making direct methods efficient for the Helmholtz equation. Such is the case of Wang et al.’s method =-=[22]-=-, which couples multifrontal elimination with H-matrices. Another example is the work of Gillman and Martinsson on computing impedance-to-impedance maps in a multiscale fashion [39]. It is not yet cle... |

15 |
AILU for Helmholtz problems: a new preconditioner based on the analytic parabolic factorization,”
- Gander
- 2001
(Show Context)
Citation Context ... is the right mix of ideas for simulating propagating high-frequency waves in a heterogeneous medium. To a great extent, the approach can be traced back to the AILU method of Gander and Nataf in 2001 =-=[35, 37]-=-. The first linear complexity claim was perhaps made in the work of Engquist and Ying on sweeping preconditioners in 2011 [28, 29] – a special kind of domain decomposition into grid-spacing-thin layer... |

15 |
A parallel multigrid-based preconditioner for the 3d heterogeneous high frequency Helmholtz equation. J.Comput
- Riyanti, Kononov, et al.
(Show Context)
Citation Context ...lver using the complex-shifted Laplacian with optimal grids to minimize pollution effects in [18]. Multigrid methods applied to large 3D examples have been implemented by Plessix [56], Riyanti et al. =-=[60]-=- and, more recently, Calandra et al. [13]. A good review of iterative methods for the Helmholtz equation, including complex-shifted Laplace preconditioners, is in [30]. Another review paper that discu... |

14 |
Plane wave approximation of homogeneous Helmholtz solutions.
- Moiola, Hiptmair, et al.
- 2011
(Show Context)
Citation Context ...eak Variational Formulation [5, 15, 16, 17]; • Their work spawned a series of new formulations within the framework of Discontinous Galerkin methods, such as the Trefftz formulation of Perugia et al. =-=[54, 40]-=-, and the discontinuous enrichment method of Farhat et al. [33]; • Hiptmair et al. proposed the multi-trace formulation [42], which involves solving an integral equation posed on the interfaces of a d... |

14 |
The interplay of ranks of submatrices
- Strang, Nguyen
- 2004
(Show Context)
Citation Context ...E↑`,`+1 = E↑`,`+1G`+1(z1, z0), (54) and G`(zn`+1, zn`+1)− hE↓`,`+1 = E↓`,`+1G`(zn` , zn`+1). (55) In one dimension, the proof of Lemma 4 (in Appendix C) is a direct application of the nullity theorem =-=[68]-=- and the cofactor formula, in which E↑j,j+1 is the ratio between two co-linear vectors. In two dimensions, the proof is slightly more complex but follows the same reasoning. Lemma 5. If u↑ is an up-go... |

13 |
A fast direct algorithm for the solution of the Laplace equation on regions with fractal boundaries
- Jones, Ma, et al.
- 1994
(Show Context)
Citation Context ...mally large square submatrices with fixed -rank. We follow [8] in calling this structure partitioned low-rank (PLR). An early reference for PLR matrices is the work of Jones, Ma, and Rokhlin in 1994 =-=[45]-=-. PLR matrices are a special case of H-matrices5. It is known [3], that the blocks of matrices such as M can have low rank, provided they obey an admissibility condition that takes into account the di... |

12 | A fast method for the solution of the Helmholtz equation
- Haber, MacLachlan
(Show Context)
Citation Context ...nction; • Elman and Ernst use a relaxed multigrid method as a preconditioner for an outer Krylov iteration in [27]; • Haber and McLachlan proposed an alternative formulation for the Hemholtz equation =-=[41]-=-, which reduces the problem to solve an eikonal equation and an advection-diffusion-reaction equation, which can be solved efficiently by a Krylov method using a multigrid method as a preconditioner; ... |

12 |
Fast construction of hierarchical matrix representation from matrixvector multiplication
- Lin, Lu, et al.
(Show Context)
Citation Context ...out that the current complexity bottleneck is the extraction of the Green’s functions. It may be possible to reduce the number of solves to a fractional power of n using randomized algorithms such as =-=[47]-=-); or to O(1) solves, if more information of the underlying PDE is available or easy to compute (see [4]). Once the Green’s matrices are computed, we use Alg. 5 to compress them. A simple complexity c... |

12 | Is the Helmholtz equation really sign-indefinite
- Moiola, Spence
- 2014
(Show Context)
Citation Context ...n [30]. Another review paper that discussed the difficulties generated by the high-frequency limit is [32]. Finally, a beautiful mathematical exposition of the Helmholtz equation and a good review is =-=[55]-=-. 1.5 Organization The discrete integral reformulation of the Helmholtz equation is presented in section 2. The polarization trick, which generates the triangular block structure at the expense of dou... |

12 | A rapidly converging domain decomposition method for the Helmholtz equation. ArXiv e-prints
- Stolk
- 2012
(Show Context)
Citation Context ... complexity claim was perhaps made in the work of Engquist and Ying on sweeping preconditioners in 2011 [28, 29] – a special kind of domain decomposition into grid-spacing-thin layers. In 2013, Stolk =-=[65]-=- restored the flexibilty to consider coarser layerings, with a domain decomposition method that realizes interface transmission via an ingenious forcing term, resulting in linear complexity scalings v... |

11 | Stralen, “Exact Constructions of Square-Root Helmholtz Operator Symbols: The Focusing Quadratic Profile
- Fishman, Hoop, et al.
- 2000
(Show Context)
Citation Context ...e slab. With a local G, the conditions (Eq. 1 or 2) are only approximately polarizing. Other approximations of absorbing conditions, such as square-root operators, can be good in certain applications =-=[23, 63, 64, 34, 66]-=-, but are often too coarse to be useful in the context of a 3 fast domain decomposition solver. In particular, it should be noted that square-root operators do not approximate polarizing conditions to... |

10 |
Notes on Perfectly Matched Layers (PMLs
- Johnson
- 2010
(Show Context)
Citation Context ...mulas. 2.1 Discretization Let Ω = (0, Lx) × (0, Lz) be a rectangular domain. Throughout this paper, we pose Eq. 3 with absorbing boundary conditions on ∂Ω, realized as a perfectly matched layer (PML) =-=[6, 44]-=-. This approach consists of extending the computational domain with an absorbing layer, in which the differential operator is modified to efficiently damp the outgoing waves and reduce the reflections... |

9 |
Electromagnetic integral equations requiring small numbers of Krylov-subspace iterations
- Bruno, Elling, et al.
(Show Context)
Citation Context ...er of sub-domains. In addition, the runtimes scales sub-linearly with respect to the number of volume unknowns. 11Some authors refer to GMRES algorithm as the Generalized conjugate residual algorithm =-=[12]-=-, [61] 32 N ω/2pi [Hz] L = 8 L = 16 L = 32 L = 64 L = 128 195× 870 5.57 (3) 0.094 (3) 0.21 (3) 0.49 (3-4) 1.02 (3-4) 2.11 396× 1720 7.71 (3) 0.14 (3) 0.32 (3) 0.69 (3-4) 1.55 (3-4) 3.47 792× 3420 11.1... |

8 | Communication avoiding rank revealing QR factorization with column pivoting, 2013. Page 60 EXTRA SLIDES EXTRA SLIDES: RRQR Page 61 Rank revealing factorizations • A rank revealing QR (RRQR) factorization is given as • Since , the numerical rank of A is k
- Demmel, Grigori, et al.
(Show Context)
Citation Context ...4) (a more representative figure in Table 4), then the same argument leads to δ = 1/8 and an overall online runtime of O(N7/8). 6.2 Communication cost We use a commonly-used communication cost model (=-=[2, 25, 58, 69]-=-) to perform the cost analysis. The model assumes that each process is only able to send or receive a single message at a time. When the messages has size m, the time to communicate that message is α+... |

8 | Modeling of seismic data in the downward continuation approach.
- Stolk, Hoop
- 2005
(Show Context)
Citation Context |

7 | Fast wave computation via Fourier integral operators
- Demanet, Ying
(Show Context)
Citation Context ...ccount the distance between blocks. In regimes of high frequencies and rough heterogeneous media, this admissibility condition becomes more stringent in ways that are not entirely understood yet. See =-=[14, 24]-=- for partial progress. Neither of the usual non-adaptive admissibility criteria seems adequate in our case. The “nearby interaction” blocks (from one interface to itself) have a singularity along the ... |

7 |
A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media
- Gillman, Barnett, et al.
(Show Context)
Citation Context ...ng et al.’s method [22], which couples multifrontal elimination with H-matrices. Another example is the work of Gillman and Martinsson on computing impedance-to-impedance maps in a multiscale fashion =-=[39]-=-. It is not yet clear whether offline linear complexity scalings can be achieved this way, though good direct methods are often faster in practice than the iterative methods mentioned above. The main ... |

7 |
A Helmholtz Iterative Solver for 3D Seismic-Imaging Problems,” Geophys.
- Plessix
- 2007
(Show Context)
Citation Context .... implemented a 3D solver using the complex-shifted Laplacian with optimal grids to minimize pollution effects in [18]. Multigrid methods applied to large 3D examples have been implemented by Plessix =-=[56]-=-, Riyanti et al. [60] and, more recently, Calandra et al. [13]. A good review of iterative methods for the Helmholtz equation, including complex-shifted Laplace preconditioners, is in [30]. Another re... |

6 |
Parallel solution of block tridiagonal linear systems
- Bevilacqua, Codenotti, et al.
- 1988
(Show Context)
Citation Context ... ordering of the unknowns implies that every block in H correspond to degrees of freedom with fixed depth (fixed z). In order to prove the equivalent of the rank-one property in the 2D case we follow =-=[7]-=- and [53]. Definition 10. Let ∆i and Σi be defined by the following recurrence relations ∆1 = H1, ∆i = Hi −Ci (∆i−1)−1 Cti; (125) Σnz = Hnz , Σi = Hi −Cti+1 (Σi+1)−1 Ci+1. (126) Proposition 3. H is pr... |

6 |
Multiple traces boundary integral formulation for Helmholtz transmission problems.
- Hiptmair, Jerez-Hanckes
- 2012
(Show Context)
Citation Context ...ous Galerkin methods, such as the Trefftz formulation of Perugia et al. [54, 40], and the discontinuous enrichment method of Farhat et al. [33]; • Hiptmair et al. proposed the multi-trace formulation =-=[42]-=-, which involves solving an integral equation posed on the interfaces of a decomposed domain, which is naturally well suited for operator preconditionning in the case of piece-wise constant medium; • ... |

6 | A parallel butterfly algorithm
- Poulson, Demanet, et al.
(Show Context)
Citation Context |

6 | Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem
- Vion, Geuzaine
(Show Context)
Citation Context ...mission via an ingenious forcing term, resulting in linear complexity scalings very similar to those of the sweeping preconditioners. Other authors have since then proposed related methods, including =-=[19, 73]-=-, which we review in section 1.4. Many of these references present isolated instances of what should eventually become a systematic understanding of how to couple absorption/transmission conditions wi... |

5 |
An Improved Two-Grid Preconditioner for the Solution of Three-Dimensional Helmholtz Problems in Heterogeneous Media,” Numer. Linear Algebra Appl.
- Calandra, Gratton, et al.
- 2013
(Show Context)
Citation Context ...with optimal grids to minimize pollution effects in [18]. Multigrid methods applied to large 3D examples have been implemented by Plessix [56], Riyanti et al. [60] and, more recently, Calandra et al. =-=[13]-=-. A good review of iterative methods for the Helmholtz equation, including complex-shifted Laplace preconditioners, is in [30]. Another review paper that discussed the difficulties generated by the hi... |

5 |
Inversion of Jacobi’s tridiagonal matrix
- Usmani
- 1994
(Show Context)
Citation Context ...re H = b1 c1 0 . . . 0 a1 b2 c2 . . . 0 0 . . . . . . . . . 0 0 . . . an−2 bn−1 cn−1 0 . . . 0 an−1 bn , (113) in which a = c by symmetry. We denote by G the inverse of H. We follow =-=[71]-=- in writing a simple recurrence formula based on the minors θ−1 = 0, θ0 = 1, θi = biθi−1 − aici−1θi−2, (114) φn+2 = 0, φn+1 = 1, φi = biφi+1 − ciai+1φi+2. (115) 40 Lemma 8 ( Lemma 2 in [71] ). We have... |

4 | A source transfer domain decomposition method for Helmholtz equations in unbounded domain part II: Extensions
- Chen, Xiang
(Show Context)
Citation Context ...mission via an ingenious forcing term, resulting in linear complexity scalings very similar to those of the sweeping preconditioners. Other authors have since then proposed related methods, including =-=[19, 73]-=-, which we review in section 1.4. Many of these references present isolated instances of what should eventually become a systematic understanding of how to couple absorption/transmission conditions wi... |

4 | Optimal interface conditions for an arbitrary decomposition into subdomains
- Gander, Kwok
- 2011
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Citation Context ...decomposition framework, the use of such exact polarizing conditions would enable solving the Helmholtz equation in two sweeps of the domain – typically a top-down sweep followed by a bottom-up sweep =-=[36]-=-. However, constructing exact polarizing conditions is a difficult task. No matter the ordering, elimination of the exterior unknowns by direct linear algebra is costly and difficult to parallelize. 1... |

4 | A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations
- Poulson, Engquist, et al.
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Citation Context ... algorithm, which can handle transmitted waves in very favorable complexity [48]; • Poulson et al. parallelized the sweeping preconditioners in 3D to deal with very large scale problems in geophysics =-=[59]-=-. Tsuji et al. designed a spectrally accurate sweeping preconditioner for time-harmonic elastic waves [70]. • Finally, an earlier version of the work presented in this paper can be found in [74], in w... |

4 |
Full-wave-equation depth extrapolation for migration
- Sandberg, Beylkin
- 2009
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Citation Context |

4 | Sweeping preconditioners for elastic wave propagation with spectral element methods
- Tsuji, Poulson, et al.
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Citation Context ...lized the sweeping preconditioners in 3D to deal with very large scale problems in geophysics [59]. Tsuji et al. designed a spectrally accurate sweeping preconditioner for time-harmonic elastic waves =-=[70]-=-. • Finally, an earlier version of the work presented in this paper can be found in [74], in which the notion of polarized traces using Green’s representation formula was first introduced, obtaining f... |

3 |
A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation
- Chen, Cheng, et al.
(Show Context)
Citation Context ...ted Laplace preconditioner with multigrid, which shines for its simplicity. Chen et al. implemented a 3D solver using the complex-shifted Laplacian with optimal grids to minimize pollution effects in =-=[18]-=-. Multigrid methods applied to large 3D examples have been implemented by Plessix [56], Riyanti et al. [60] and, more recently, Calandra et al. [13]. A good review of iterative methods for the Helmhol... |

3 |
A DDM double sweep preconditioner for the Helmholtz equation with matrix probing of the DtN map
- Vion, Bélanger-Rioux, et al.
- 2013
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Citation Context ...n the case of piece-wise constant medium; • Plessix and Mulder proposed a method similar to AILU in 2003 [57]; • More recently, Geuzaine and Vion explored approximate transmission boundary conditions =-=[73, 72]-=- coupled with a multiplicative Schwartz iteration, to improve on traditional domain decomposition methods; • Chen and Xiang proposed another instance of efficient domain decomposition where the emphas... |

2 |
An O(n log n) fast direct solver for partial hierarchically semiseparable matrices
- Ambikasaran, Darve
- 2013
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Citation Context ... matrix-vector multiplications at the same time, which can be performed as a matrix-matrix multiplication. This kind of sparse factorization is by no means new; we suggest as a reference Section 4 in =-=[1]-=-. 27 The maximum local rank of the compression scheme can be seen as a tuning parameter. If it is too small, it will induce small off-diagonal blocks, hindering compression and deteriorating the compl... |

2 |
The 2004 BP velocity benchmark.: EAGE
- Billette, Brandsberg-Dahl
- 2005
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Citation Context ...sed in this paper deteriorates only marginally as a function of the frequency and number of subdomains. We use the Marmousi2 model [49], and another geophysical community benchmark, the BP 2004 model =-=[9]-=-, depicted in Fig. 11 and Fig. 12 respectively. Tables 7, 8, 9, and 10 were generated by running 200 randomly generated right hand sides 33 0 2000 4000 6000 8000 10000 12000 14000 16000 0 500 1000 150... |

2 |
An elastic upgrade for Marmousi: The Leading Edge, Society for Exploration Geophysics
- Martin, W, et al.
- 2006
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Citation Context ...teriorating the convergence rate. The performance of the method proposed in this paper deteriorates only marginally as a function of the frequency and number of subdomains. We use the Marmousi2 model =-=[49]-=-, and another geophysical community benchmark, the BP 2004 model [9], depicted in Fig. 11 and Fig. 12 respectively. Tables 7, 8, 9, and 10 were generated by running 200 randomly generated right hand s... |

2 |
Preconditioning the 2D Helmholtz equation with polarized traces
- Zepeda-Núñez, Hewett, et al.
- 2014
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Citation Context ...ysics [59]. Tsuji et al. designed a spectrally accurate sweeping preconditioner for time-harmonic elastic waves [70]. • Finally, an earlier version of the work presented in this paper can be found in =-=[74]-=-, in which the notion of polarized traces using Green’s representation formula was first introduced, obtaining favorable scalings; Some progress has also been achieved with multigrid methods, though t... |

1 |
Compressed absorbing boundary conditions via matrix probing. ArXiv e-prints, [math.NA
- Bélanger-Rioux, Demanet
- 1401
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Citation Context ... 1The upper bound on L is typically of the form O(N1/8), with some caveats. This discussion is covered in sections 1.3 and 6. 2 Probing from random vectors has been proposed to alleviate this problem =-=[4]-=-, but requires significant user intervention. Note that elimination of the interior unknowns by Schur complements, or equivalently computation of an interior Dirichlet-to-Neumann map, may be interesti... |

1 |
Fast Huygens sweeping methods for Helmholtz equations in inhomogeneous media in the high frequency regime
- Luo, Qian, et al.
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Citation Context .... proposed a large-subdomain sweeping method, based on an approximate Green’s function by geometric optics and the butterfly algorithm, which can handle transmitted waves in very favorable complexity =-=[48]-=-; • Poulson et al. parallelized the sweeping preconditioners in 3D to deal with very large scale problems in geophysics [59]. Tsuji et al. designed a spectrally accurate sweeping preconditioner for ti... |

1 |
A multigrid method for the Helmholtz equation with optimized coarse grid corrections. ArXiv e-prints, [math.NA] 1304.4103
- Stolk, Ahmed, et al.
- 2013
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Citation Context ...the complexity scalings do not appear to be optimal in two and higher dimensions: • Bhowmik and Stolk recently proposed new rules of optimized coarse grid corrections for a two level multigrid method =-=[67]-=-; • Brandt and Livshits developed the wave-ray method [11], in which the oscillatory error components are eliminated by a ray-cycle, that exploits a geometric optics approximation of the Green’s funct... |