54 research outputs found
A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media
We present a fast method for numerically solving the inhomogeneous Helmholtz
equation. Our iterative method is based on the Born series, which we modified
to achieve convergence for scattering media of arbitrary size and scattering
strength. Compared to pseudospectral time-domain simulations, our modified Born
approach is two orders of magnitude faster and nine orders of magnitude more
accurate in benchmark tests in 1-dimensional and 2-dimensional systems
Preconditioning the 2D Helmholtz equation with polarized traces
We present a domain decomposition solver for the 2D Helmholtz equation, with a special choice of integral transmission condition that involves polarizing the waves into oneway components. This refinement of the transmission condition is the key to combining local direct solves into an efficient iterative scheme, which can then be deployed in a highperformance computing environment. The method involves an expensive, but embarrassingly parallel precomputation of local Green's functions, and a fast online computation of layer potentials in partitioned low-rank form. The online part has sequential complexity that scales sublinearly with respect to the number of volume unknowns, even in the high-frequency regime. The favorable complexity scaling continues to hold in the context of low-order finite difference schemes for standard community models such as BP and Marmousi2, where convergence occurs in 5 to 10 GMRES iterations.TOTAL (Firm)United States. Air Force. Office of Scientific ResearchUnited States. Office of Naval ResearchNational Science Foundation (U.S.
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
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation
In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of [1] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements.
The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.
[1] R. Hiptmair. Multigrid method for Maxwell's equations. SIAM J. Numer. Anal., 36(1):204-225, 1999
Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer
This paper analyzes the Krylov convergence rate of a Helmholtz problem
preconditioned with Multigrid. The multigrid method is applied to the Helmholtz
problem formulated on a complex contour and uses GMRES as a smoother substitute
at each level. A one-dimensional model is analyzed both in a continuous and
discrete way. It is shown that the Krylov convergence rate of the continuous
problem is independent of the wave number. The discrete problem, however, can
deviate significantly from this bound due to a pitchfork in the spectrum. It is
further shown in numerical experiments that the convergence rate of the Krylov
method approaches the continuous bound as the grid distance gets small
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