A nodal discontinuous Galerkin method for reverse-time migration on GPU clusters

Modave, Axel; St-Cyr, Amik; Mulder, W.A.; Warburton, Tim

doi:10.1093/gji/ggv380

Cited by 28 publications

(36 citation statements)

References 64 publications

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“…Additionally, the computational structure of DG methods has been shown to be well-suited to many-core and accelerator architectures such as Graphics Processing Units (GPU). DG implementations on a single GPU have demonstrated significant speedups over conventional architectures [12,13], while implementations using multiple GPUs still demonstrate high scalability [14,15].…”

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confidence: 99%

Weight-Adjusted Discontinuous Galerkin Methods: Wave Propagation in Heterogeneous Media

Chan¹,

Hewett²,

Warburton³

2017

SIAM J. Sci. Comput.

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Abstract. Time-domain discontinuous Galerkin (DG) methods for wave propagation require accounting for the inversion of dense elemental mass matrices, where each mass matrix is computed with respect to a parameter-weighted L 2 inner product. In applications where the wavespeed varies spatially at a sub-element scale, these matrices are distinct over each element, necessitating additional storage. In this work, we propose a weight-adjusted DG (WADG) method which reduces storage costs by replacing the weighted L 2 inner product with a weight-adjusted inner product. This equivalent inner product results in an energy stable method, but does not increase storage costs for locally varying weights. A priori error estimates are derived, and numerical examples are given illustrating the application of this method to the acoustic wave equation with heterogeneous wavespeed.1. Introduction. Accurate numerical simulations of wave propagation through complex media are becoming increasingly important in seismology, especially as modern computational resources make the use of high fidelity subsurface models feasible for seismic imaging and full waveform inversion. A host of different numerical methods are currently in use, the most popular of which are high order finite difference methods [1]. While finite difference methods tend to perform excellently for simple geometries and smoothly varying data, their accuracy is degraded for heterogeneous media with interfaces or sharp gradients [2].In order to address these issues, high order finite element methods for wave propagation have been considered as alternatives to finite difference methods. A drawback of using continuous finite elements for time-domain simulations using explicit timestepping is the inversion of a global mass matrix system at each timestep. Spectral Element Methods (SEM) [3] address this issue by diagonalizing this mass matrix system through the use of mass-lumping, which co-locates interpolation nodes for Lagrange basis functions and Gauss-Legendre-Lobatto quadrature points. Since SEM is limited to unstructured hexahedral meshes, which are less geometrically flexible than tetrahedral meshes, triangular and tetrahedral mass-lumped spectral element methods have been investigated as alternatives [4,5,6]. However, due to a mismatch in the number of natural quadrature nodes and the dimension of polynomial approximation spaces on simplices, these methods necessitate adding additional nodes in the interior of the element to construct sufficiently accurate nodal points suitable for mass-lumping. Additionally, mass-lumpable nodal points on tetrahedra have only been determined for polynomial bases of degree four or less [4].High order discontinuous Galerkin (DG) methods have been considered as an alternative to Spectral Element Methods for seismic wave propagation [7,8,9,10]. Instead of using mass-lumping to arrive at a diagonal mass matrix, DG methods naturally induce a block diagonal mass matrix through the use of arbitrary-order approximation spaces which are disconti...

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confidence: 99%

Weight-Adjusted Discontinuous Galerkin Methods: Wave Propagation in Heterogeneous Media

Chan¹,

Hewett²,

Warburton³

2017

SIAM J. Sci. Comput.

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“…These methods can provide accurate solutions to realistic transient wave‐like problems thanks to heterogeneous, nonconforming and curvilinear meshes, high‐order discontinuous basis functions, and stable formulations for complicated physical models (see, eg, these studies). In addition, the discrete structure of the numerical schemes is well suited for efficient massively parallel computing on distributed memory architectures and modern many‐core accelerators …”

Section: Introductionmentioning

confidence: 99%

“…In addition, the discrete structure of the numerical schemes is well suited for efficient massively parallel computing on distributed memory architectures and modern many-core accelerators. [11][12][13][14][15][16] A critical issue for the simulation of wave phenomena is to correctly account for radiation of waves at artificial boundaries of the computational domain. Nonreflective boundary treatments must be incorporated into the discontinuous Galerkin formulations to simulate the outward propagation of signals and perturbations generated from within the computational domain, even if they are not a priori known.…”

Section: Introductionmentioning

confidence: 99%

“…We use elaborate implementation techniques to improve the computational performance, while keeping the compatibility of the final implementation with implementations in the literature. 11,14 This paper is organized as follows. In Section 2, the HABCs are presented and edge/corner compatibility conditions are derived for both the wave equation and the pressure-velocity system.…”

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A GPU‐accelerated nodal discontinuous Galerkin method with high‐order absorbing boundary conditions and corner/edge compatibility

Modave

Atle²,

Chan

et al. 2017

Numerical Meth Engineering

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Discontinuous Galerkin finite element schemes exhibit attractive features for accurate large-scale wave-propagation simulations on modern parallel architectures. For many applications, these schemes must be coupled with non-reflective boundary treatments to limit the size of the computational domain without losing accuracy or computational efficiency, which remains a challenging task. In this paper, we present a combination of a nodal discontinuous Galerkin method with high-order absorbing boundary conditions (HABCs) for cuboidal computational domains. Compatibility conditions are derived for HABCs intersecting at the edges and the corners of a cuboidal domain. We propose a GPU implementation of the computational procedure, which results in a multidimensional solver with equations to be solved on 0D, 1D, 2D and 3D spatial regions. Numerical results demonstrate both the accuracy and the computational efficiency of our approach.

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“…Klöckner et al addressed this nontrivial cost by developing an implementation of DG on Graphics Processing Units (GPUs) in [3], where the structure of time-explicit DG was found to map extremely well to the coarse and fine grained parallelism present in accelerator architectures. Large problem sizes can be accomodated by parallelizing over multiple GPUs, which has been shown to yield scalable and efficient solvers [4,5].…”

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confidence: 99%

Weight-Adjusted Discontinuous Galerkin Methods: Curvilinear Meshes

Chan¹,

Hewett²,

Warburton³

2017

SIAM J. Sci. Comput.

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Traditional time-domain discontinuous Galerkin (DG) methods result in large storage costs at high orders of approximation due to the storage of dense elemental matrices. In this work, we propose a weight-adjusted DG (WADG) methods for curvilinear meshes which reduce storage costs while retaining energy stability. A priori error estimates show that high order accuracy is preserved under sufficient conditions on the mesh, which are illustrated through convergence tests with different sequences of meshes. Numerical and computational experiments verify the accuracy and performance of WADG for a model problem on curved domains. arXiv:1608.03836v1 [math.NA] 12 Aug 2016 topologically (vertex, edges, faces, interior); however, the number of nodes exceeds the cardinality of natural approximation spaces on simplices. Additionally, such nodal sets have only been constructed up to degree 4 for tetrahedra.A similar approach is taken for flux reconstruction schemes on simplices, which are closely related to filtered nodal DG methods [23,24] where nodes are taken to be unisolvent quadrature points [25,26]. Unlike nodal sets for mass-lumped simplices, these quadrature points do not contain nodes which lie on the boundary, necessitating an additional interpolation step in the computation of numerical fluxes. However, numerical evidence indicates that co-locating nodes and quadrature points reduces instabilities resulting from the aliasing of spatially varying Jacobians [27], though an analysis of high order convergence and energy stability for curvilinear simplices are open problems.Krivodonova and Berger introduced an inexpensive treatment of curved boundaries for two-dimensional flow problems by modifying the DG formulation on affine triangles [28]. This was extended to wave propagation problems by Zhang in [8], and by Zhang and Tan for elements with non-boundary curved faces in [7]. A theoretical stability and convergence analysis remains to be shown, though numerical results suggest that each of these approaches preserves stability and high order accuracy on curvilinear meshes under the condition that curved triangles are well-approximated by planar triangles. However, sufficiently large differences between curved and planar triangles still result in unstable schemes [8].An alternative treatment addressing increased storage costs of curvilinear DG was addressed by Warburton using the Low-Storage Curvilinear DG (LSC-DG) method [29,30]. Under LSC-DG, the spatial variation of the Jacobian is incorporated into the physical basis functions over each element, resulting in identical mass matrices over each element. Work in [30] also includes a priori estimates for projection errors under the LSC-DG basis, and gives sufficient conditions under which convergence is guaranteed. Furthermore, the DG variational formulation is constructed to be a priori stable for surface quadratures with positive weights, allowing for stable under-integration of high order integrands present for curvilinear elements.In [31], the weight-adjusted DG (WADG)...

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A nodal discontinuous Galerkin method for reverse-time migration on GPU clusters

Cited by 28 publications

References 64 publications

Weight-Adjusted Discontinuous Galerkin Methods: Wave Propagation in Heterogeneous Media

Weight-Adjusted Discontinuous Galerkin Methods: Wave Propagation in Heterogeneous Media

A GPU‐accelerated nodal discontinuous Galerkin method with high‐order absorbing boundary conditions and corner/edge compatibility

Weight-Adjusted Discontinuous Galerkin Methods: Curvilinear Meshes

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