Preconditioning the Helmholtz Equation via Row-projections

3D frequency-domain full waveform inversion relies on being able to efficiently solve the 3D Helmholtz equation. Iterative methods require sophisticated preconditioners because the Helmholtz matrix is typically indefinite. We review a preconditioning technique that is based on row-projections. Notable advantages of this preconditioner over existing ones are that it has low algorithmic complexity, is easily parallelizable and extendable to time-harmonic vector equations.

[1]  Å. Björck,et al.  Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations , 1979 .

[2]  A. Bayliss,et al.  An Iterative method for the Helmholtz equation , 1983 .

[3]  Cornelis Vuik,et al.  On a Class of Preconditioners for Solving the Helmholtz Equation , 2003 .

[4]  Dan Gordon,et al.  Component-Averaged Row Projections: A Robust, Block-Parallel Scheme for Sparse Linear Systems , 2005, SIAM J. Sci. Comput..

[5]  Cornelis Vuik,et al.  A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation , 2007, J. Comput. Phys..

[6]  Dan Gordon,et al.  Solution Methods for Nonsymmetric Linear Systems with Large off-Diagonal Elements and Discontinuous Coefficients , 2009 .

[7]  B. Engquist,et al.  Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation , 2010, 1007.4290.

[8]  Dan Gordon,et al.  CARP-CG: A robust and efficient parallel solver for linear systems, applied to strongly convection dominated PDEs , 2010, Parallel Comput..

[9]  Y. Saad,et al.  Preconditioning Helmholtz linear systems , 2010 .

[10]  C. Stolk.  On the generalization of wavelet diagonal preconditioning to the Helmholtz equation , 2010, 1010.4764.

[11]  Scott P. MacLachlan,et al.  A fast method for the solution of the Helmholtz equation , 2011, J. Comput. Phys..

[12]  Azzam Haidar,et al.  Three‐dimensional parallel frequency‐domain visco‐acoustic wave modelling based on a hybrid direct/iterative solver , 2011 .

[13]  Jianlin Xia,et al.  On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver , 2011 .

[14]  Dan Gordon,et al.  Robust and highly scalable parallel solution of the Helmholtz equation with large wave numbers , 2013, J. Comput. Appl. Math..