15 research outputs found
A short note on the nested-sweep polarized traces method for the 2D Helmholtz equation
We present a variant of the solver in Zepeda-N\'u\~nez and Demanet (2014),
for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media.
By changing the domain decomposition from a layered to a grid-like partition,
this variant yields improved asymptotic online and offline runtimes and a lower
memory footprint. The solver has online parallel complexity that scales
\emph{sub linearly} as , where is
the number of volume unknowns, and is the number of processors, provided
that . The variant in Zepeda-N\'u\~nez and Demanet
(2014) only afforded . Algorithmic scalability is a
prime requirement for wave simulation in regimes of interest for geophysical
imaging.Comment: 5 pages, 5 figure
A short note on the nested-sweep polarized traces method for the 2D Helmholtz equation
We present a variant of the solver in Zepeda-Núñez and Demanet (2014), for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media. By changing the domain decomposition from a layered to a grid-like partition, this variant yields improved asymptotic online and offline runtimes and a lower memory footprint. The solver has online parallel complexity that scales sublinearly as θ(N/P), where N is the number of volume unknowns, and P is the number of processors, provided that P = θ(N[superscript 1/5]). The variant in Zepeda-Núñez and Demanet (2014) only afforded P = θ(N[superscript 1/5]). Algorithmic scalability is a prime requirement for wave simulation in regimes of interest for geophysical imaging. Keywords: frequency-domain, finite difference, modeling, wave equation, numericalNational Science Foundation (U.S.)United States. Office of Naval ResearchUnited States. Air Force. Office of Scientific Researc
A short note on a pipelined polarized-trace algorithm for 3D Helmholtz
We present a fast solver for the 3D high-frequency Helmholtz equation in heterogeneous, constant density, acoustic media. The solver is based on the method of polarized traces, coupled with distributed linear algebra libraries and pipelining to obtain a solver with online runtime O(max(1, R/n)N logN) where N = n[superscript 3] is the total number of degrees of freedom and R is the number of right-hand sides.TOTAL (Firm
Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients
We present a robust and scalable preconditioner for the solution of
large-scale linear systems that arise from the discretization of elliptic PDEs
amenable to rank compression. The preconditioner is based on hierarchical
low-rank approximations and the cyclic reduction method. The setup and
application phases of the preconditioner achieve log-linear complexity in
memory footprint and number of operations, and numerical experiments exhibit
good weak and strong scalability at large processor counts in a distributed
memory environment. Numerical experiments with linear systems that feature
symmetry and nonsymmetry, definiteness and indefiniteness, constant and
variable coefficients demonstrate the preconditioner applicability and
robustness. Furthermore, it is possible to control the number of iterations via
the accuracy threshold of the hierarchical matrix approximations and their
arithmetic operations, and the tuning of the admissibility condition parameter.
Together, these parameters allow for optimization of the memory requirements
and performance of the preconditioner.Comment: 24 pages, Elsevier Journal of Computational and Applied Mathematics,
Dec 201
Recommended from our members
Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients
We present a robust and scalable preconditioner for the solution of large-scale linear systems that arise from the discretization of elliptic PDEs amenable to rank compression. The preconditioner is based on hierarchical low-rank approximations and the cyclic reduction method. The setup and application phases of the preconditioner achieve log-linear complexity in memory footprint and number of operations, and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment. Numerical experiments with linear systems that feature symmetry and nonsymmetry, definiteness and indefiniteness, constant and variable coefficients demonstrate the preconditioner applicability and robustness. Furthermore, it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations, and the tuning of the admissibility condition parameter. Together, these parameters allow for optimization of the memory requirements and performance of the preconditioner
A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory
We develop a new dispersion minimizing compact finite difference scheme for
the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly
developed ray theory for difference equations. A discrete Helmholtz operator
and a discrete operator to be applied to the source and the wavefields are
constructed. Their coefficients are piecewise polynomial functions of ,
chosen such that phase and amplitude errors are minimal. The phase errors of
the scheme are very small, approximately as small as those of the 2-D
quasi-stabilized FEM method and substantially smaller than those of
alternatives in 3-D, assuming the same number of gridpoints per wavelength is
used. In numerical experiments, accurate solutions are obtained in constant and
smoothly varying media using meshes with only five to six points per wavelength
and wave propagation over hundreds of wavelengths. When used as a coarse level
discretization in a multigrid method the scheme can even be used with downto
three points per wavelength. Tests on 3-D examples with up to degrees of
freedom show that with a recently developed hybrid solver, the use of coarser
meshes can lead to corresponding savings in computation time, resulting in good
simulation times compared to the literature.Comment: 33 pages, 12 figures, 6 table