Results 1 
4 of
4
SWEEPING PRECONDITIONERS FOR ELASTIC WAVE PROPAGATION WITH SPECTRAL ELEMENT METHODS
, 2013
"... Abstract.We present a parallel preconditioning method for the iterative solution of the timeharmonic elastic wave equation which makes use of higherorder spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently appr ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
Abstract.We present a parallel preconditioning method for the iterative solution of the timeharmonic elastic wave equation which makes use of higherorder spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently approximate the Schur complement matrices of a block LDLT factorization. Both sequential and parallel versions of the algorithm are discussed and results for largescale problems from exploration geophysics are presented.
The method of polarized traces for the 2D Helmholtz equation. ArXiv eprints
, 2014
"... We present a solver for the 2D highfrequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O(NL), where N is the number of volume unknowns, and L is the number of processors, as long as L grows at most like a small fractional power of N ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
We present a solver for the 2D highfrequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O(NL), where N is the number of volume unknowns, and L is the number of processors, as long as L grows at most like a small fractional power of N. The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define “polarized traces”, i.e., up and downgoing waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green’s functions in an embarrassingly parallel way (the offline part), and incomplete Green’s formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive lowrank partitioning of the integral kernels is used to speed up their application to interface data. The method uses secondorder finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of offdiagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations. 1
Recursive Sweeping Preconditioner for the 3D Helmholtz Equation
, 2015
"... This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping preconditioner recursively t ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping preconditioner recursively to the quasi2D auxiliary problems introduced in the 3D sweeping preconditioner. Compared to the nonrecursive 3D sweeping preconditioner, the setup cost of this new approach drops from O(N4/3) to O(N), the application cost per iteration drops from O(N logN) to O(N), and the iteration count only increases mildly when combined with the standard GMRES solver. Several numerical examples are tested and the results are compared with the nonrecursive sweeping preconditioner to demonstrate the efficiency of the new approach.
Additive Sweeping Preconditioner for the Helmholtz Equation
, 2015
"... We introduce a new additive sweeping preconditioner for the Helmholtz equation based on the perfect matched layer (PML). This method divides the domain of interest into thin layers and proposes a new transmission condition between the subdomains where the emphasis is on the boundary values of the in ..."
Abstract
 Add to MetaCart
(Show Context)
We introduce a new additive sweeping preconditioner for the Helmholtz equation based on the perfect matched layer (PML). This method divides the domain of interest into thin layers and proposes a new transmission condition between the subdomains where the emphasis is on the boundary values of the intermediate waves. This approach can be viewed as an effective approximation of an additive decomposition of the solution operator. When combined with the standard GMRES solver, the iteration number is essentially independent of the frequency. Several numerical examples are tested to show the efficiency of this new approach.