Compressed Absorbing Boundary Conditions via Matrix Probing

Absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the Helmholtz equation in a heterogeneous medium. It is always possible to reduce an absorbing layer to an operator at the boundary by layer stripping elimination of the exterior unknowns, but the linear algebra involved is costly. We propose bypassing the elimination procedure and directly fitting the surface-to-surface operator in compressed form from a few exterior Helmholtz solves with random Dirichlet data. The result is a concise description of the absorbing boundary condition, with a complexity that grows slowly (often, logarithmically) in the frequency parameter.

[1]  J. Keller,et al.  Exact non-reflecting boundary conditions , 1989 .

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

[3]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .

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

[5]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..

[6]  Laurent Demanet,et al.  Matrix probing: A randomized preconditioner for the wave-equation Hessian , 2011, 1101.3615.

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

[8]  Richard Courant,et al.  Methods of Mathematical Physics II: Partial Di erential Equations , 1963 .

[9]  Julien Diaz,et al.  A time domain analysis of PML models in acoustics , 2006 .

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

[11]  S. Orszag,et al.  Advanced mathematical methods for scientists and engineers I: asymptotic methods and perturbation theory. , 1999 .

[12]  Steven G. Johnson,et al.  The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers. , 2008, Optics express.

[13]  E. Tadmor The exponential accuracy of Fourier and Chebyshev differencing methods , 1986 .

[14]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.

[16]  G. Folland Introduction to Partial Differential Equations , 1976 .

[17]  John K. Reid,et al.  The Multifrontal Solution of Indefinite Sparse Symmetric Linear , 1983, TOMS.

[18]  Ralf Schweizer,et al.  Integral Equation Methods In Scattering Theory , 2016 .

[19]  Stefan A. Sauter,et al.  Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers? , 1997, SIAM Rev..

[20]  I. Graham,et al.  A new frequency‐uniform coercive boundary integral equation for acoustic scattering , 2011 .

[21]  B. Engquist,et al.  Computational high frequency wave propagation , 2003, Acta Numerica.

[22]  Per-Gunnar Martinsson,et al.  A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media , 2013, 1308.5998.

[23]  William E. Schiesser,et al.  Linear and nonlinear waves , 2009, Scholarpedia.

[24]  Wolfgang Hackbusch,et al.  A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices , 1999, Computing.

[25]  D. Ludwig,et al.  An inequality for the reduced wave operator and the justification of geometrical optics , 1968 .

[26]  David Isaacson,et al.  Layer stripping: a direct numerical method for impedance imaging , 1991 .

[27]  D. Zutter,et al.  ON THE COMPLEX SYMMETRY OF THE POINCARÉ-STEKLOV OPERATOR , 2008 .

[28]  Maksim Skorobogatiy,et al.  Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backward-wave structures. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  W. McLean Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[30]  Tim Colonius,et al.  A high-order super-grid-scale absorbing layer and its application to linear hyperbolic systems , 2009, J. Comput. Phys..

[31]  Lexing Ying,et al.  Fast construction of hierarchical matrix representation from matrix-vector multiplication , 2009, J. Comput. Phys..

[32]  G. Pfander Note on sparsity in signal recovery and in matrix identification , 2007 .

[33]  Holger Rauhut,et al.  Edinburgh Research Explorer Identification of Matrices Having a Sparse Representation , 2022 .

[34]  A. Bayliss,et al.  On accuracy conditions for the numerical computation of waves , 1985 .

[35]  Per-Gunnar Martinsson,et al.  A Direct Solver with O(N) Complexity for Variable Coefficient Elliptic PDEs Discretized via a High-Order Composite Spectral Collocation Method , 2013, SIAM J. Sci. Comput..

[36]  Murthy N. Guddati,et al.  On Optimal Finite-Difference Approximation of PML , 2003, SIAM J. Numer. Anal..

[37]  Daniël De Zutter,et al.  On the Complex Symmetry of the Poincar'E-Steklov Operator , 2008 .

[38]  Rosalie B'elanger-Rioux Compressed absorbing boundary conditions for the Helmholtz equation , 2014, 1408.3144.

[39]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[40]  Laurent Demanet,et al.  Scattering in Flatland: Efficient Representations via Wave Atoms , 2010, Found. Comput. Math..

[41]  W. Marsden I and J , 2012 .

[42]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[43]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[44]  Yogi A. Erlangga,et al.  Advances in Iterative Methods and Preconditioners for the Helmholtz Equation , 2008 .

[45]  Thomas Hagstrom,et al.  A new auxiliary variable formulation of high-order local radiation boundary conditions: corner compatibility conditions and extensions to first-order systems , 2004 .

[46]  Robert L. Higdon,et al.  Numerical absorbing boundary conditions for the wave equation , 1987 .

[47]  Steven G. Johnson,et al.  Photonic Crystals: Molding the Flow of Light , 1995 .

[48]  Laurent Demanet,et al.  A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators , 2008, Multiscale Model. Simul..

[49]  Laurent Demanet,et al.  Matrix Probing and its Conditioning , 2012, SIAM J. Numer. Anal..

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