Solving the Three-Dimensional High-frequency Helmholtz Equation Using Contour Integration and Polynomial Preconditioning

We propose an iterative solution method for the three-dimensional high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solut...

[1]  P. Cummings,et al.  SHARP REGULARITY COEFFICIENT ESTIMATES FOR COMPLEX-VALUED ACOUSTIC AND ELASTIC HELMHOLTZ EQUATIONS , 2006 .

[2]  Martin J. Gander,et al.  Optimized Schwarz Methods , 2006, SIAM J. Numer. Anal..

[3]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[4]  Lothar Reichel,et al.  The application of Leja points to Richardson iteration and polynomial preconditioning , 1991 .

[5]  Cornelis Vuik,et al.  Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian , 2007, SIAM J. Sci. Comput..

[6]  Ping Tak Peter Tang,et al.  FEAST As A Subspace Iteration Eigensolver Accelerated By Approximate Spectral Projection , 2013, SIAM J. Matrix Anal. Appl..

[7]  H. E. Wrigley Accelerating the Jacobi Method for Solving Simultaneous Equations by Chebyshev Extrapolation When the Eigenvalues of the Iteration Matrix are Complex , 1963, Computer/law journal.

[8]  R. Freund On conjugate gradient type methods and polynomial preconditioners for a class of complex non-hermitian matrices , 1990 .

[9]  Martin J. Gander,et al.  Optimized Schwarz Methods without Overlap for the Helmholtz Equation , 2002, SIAM J. Sci. Comput..

[10]  Sverre Brandsberg-Dahl,et al.  The Pseudo-analytical Method: Application of Pseudo-Laplacians to Acoustic And Acoustic Anisotropic Wave Propagation , 2009 .

[11]  Martin J. Gander,et al.  How Large a Shift is Needed in the Shifted Helmholtz Preconditioner for its Effective Inversion by Multigrid? , 2017, SIAM J. Sci. Comput..

[12]  Yousef Saad,et al.  Computing Partial Spectra with Least-Squares Rational Filters , 2016, SIAM J. Sci. Comput..

[13]  D. Sorensen Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[14]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[15]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[16]  Yousef Saad,et al.  A Rational Function Preconditioner For Indefinite Sparse Linear Systems , 2017, SIAM J. Sci. Comput..

[17]  Christiaan C. Stolk,et al.  A rapidly converging domain decomposition method for the Helmholtz equation , 2012, J. Comput. Phys..

[18]  Laurent Demanet,et al.  The method of polarized traces for the 2D Helmholtz equation , 2014, J. Comput. Phys..

[19]  Bruno Després,et al.  A Domain Decomposition Method for the Helmholtz equation and related Optimal Control Problems , 1996 .

[20]  I. Bendixson,et al.  Sur les racines d'une équation fondamentale , 1902 .

[21]  Changsoo Shin Sponge boundary condition for frequency-domain modeling , 1995 .

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

[23]  Gerhard Opfer,et al.  Richardson's iteration for nonsymmetric matrices , 1984 .

[24]  Yousef Saad,et al.  ILUT: A dual threshold incomplete LU factorization , 1994, Numer. Linear Algebra Appl..

[25]  Gene H. Golub,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[26]  Lexing Ying,et al.  Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers , 2010, Multiscale Model. Simul..

[27]  Fang Chen,et al.  Modified HSS iteration methods for a class of complex symmetric linear systems , 2010, Computing.

[28]  Cornelis W. Oosterlee,et al.  Algebraic Multigrid Solvers for Complex-Valued Matrices , 2008, SIAM J. Sci. Comput..

[29]  Jie Shen,et al.  Spectral Approximation of the Helmholtz Equation with High Wave Numbers , 2005, SIAM J. Numer. Anal..

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

[31]  Martin B. van Gijzen,et al.  Preconditioned Multishift BiCG for ℋ2-Optimal Model Reduction , 2017, SIAM J. Matrix Anal. Appl..

[32]  Sur les racines d'une équation fondamentale , .

[33]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[34]  A. George Nested Dissection of a Regular Finite Element Mesh , 1973 .

[35]  Richard S. Varga,et al.  Zero-Free Parabolic Regions for Sequences of Polynomials , 1976 .

[36]  Martin H. Gutknecht,et al.  The Chebyshev iteration revisited , 2002, Parallel Comput..

[37]  Martin J. Gander,et al.  Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed? , 2015, Numerische Mathematik.

[38]  JIANLIN XIA,et al.  Parallel Randomized and Matrix-Free Direct Solvers for Large Structured Dense Linear Systems , 2016, SIAM J. Sci. Comput..

[39]  Y. Saad Least squares polynomials in the complex plane and their use for solving nonsymmetric linear systems , 1987 .

[40]  Eugene E. Tyrtyshnikov,et al.  Some Remarks on the Elman Estimate for GMRES , 2005, SIAM J. Matrix Anal. Appl..

[41]  C. Kelley,et al.  Convergence Analysis of Pseudo-Transient Continuation , 1998 .