Effective matrix-free preconditioning for the augmented immersed interface method

We present effective and efficient matrix-free preconditioning techniques for the augmented immersed interface method (AIIM). AIIM has been developed recently and is shown to be very effective for interface problems and problems on irregular domains. GMRES is often used to solve for the augmented variable(s) associated with a Schur complement A in AIIM that is defined along the interface or the irregular boundary. The efficiency of AIIM relies on how quickly the system for A can be solved. For some applications, there are substantial difficulties involved, such as the slow convergence of GMRES (particularly for free boundary and moving interface problems), and the inconvenience in finding a preconditioner (due to the situation that only the products of A and vectors are available). Here, we propose matrix-free structured preconditioning techniques for AIIM via adaptive randomized sampling, using only the products of A and vectors to construct a hierarchically semiseparable matrix approximation to A. Several improvements over existing schemes are shown so as to enhance the efficiency and also avoid potential instability. The significance of the preconditioners includes: (1) they do not require the entries of A or the multiplication of A T with vectors; (2) constructing the preconditioners needs only O ( log ? N ) matrix-vector products and O ( N ) storage, where N is the size of A; (3) applying the preconditioners needs only O ( N ) flops; (4) they are very flexible and do not require any a priori knowledge of the structure of A. The preconditioners are observed to significantly accelerate the convergence of GMRES, with heuristical justifications of the effectiveness. Comprehensive tests on several important applications are provided, such as Navier-Stokes equations on irregular domains with traction boundary conditions, interface problems in incompressible flows, mixed boundary problems, and free boundary problems. The preconditioning techniques are also useful for several other problems and methods.

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

[2]  Hongkai Zhao,et al.  A semi-implicit augmented IIM for Navier-Stokes equations with open and traction boundary conditions , 2013 .

[3]  Sabine Le Borne,et al.  H-matrix Preconditioners in Convection-Dominated Problems , 2005, SIAM J. Matrix Anal. Appl..

[4]  Zhilin Li,et al.  The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (Frontiers in Applied Mathematics) , 2006 .

[5]  Xiaoye S. Li,et al.  Direction-Preserving and Schur-Monotonic Semiseparable Approximations of Symmetric Positive Definite Matrices , 2009, SIAM J. Matrix Anal. Appl..

[6]  Anne Greenbaum,et al.  Any Nonincreasing Convergence Curve is Possible for GMRES , 1996, SIAM J. Matrix Anal. Appl..

[7]  Ahmed H. Sameh,et al.  A parallel hybrid banded system solver: the SPIKE algorithm , 2006, Parallel Comput..

[8]  Ray Luo,et al.  A semi-implicit augmented IIM for Navier-Stokes equations with open, traction, or free boundary conditions , 2015, J. Comput. Phys..

[9]  Ronald Kriemann,et al.  Parallel black box $$\mathcal {H}$$-LU preconditioning for elliptic boundary value problems , 2008 .

[10]  Ming-Chih Lai,et al.  New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows. , 2011, East Asian journal on applied mathematics.

[11]  Shivkumar Chandrasekaran,et al.  A Fast ULV Decomposition Solver for Hierarchically Semiseparable Representations , 2006, SIAM J. Matrix Anal. Appl..

[12]  Ying Zhang,et al.  A preconditioned conjugate gradient algorithm for GeneRank with application to microarray data mining , 2011, Data Mining and Knowledge Discovery.

[13]  Jacek Gondzio,et al.  Matrix-free interior point method , 2012, Comput. Optim. Appl..

[14]  John K. Hunter,et al.  Autophobic spreading of drops , 1999 .

[15]  Jie Liu,et al.  Open and traction boundary conditions for the incompressible Navier-Stokes equations , 2009, J. Comput. Phys..

[16]  Jianlin Xia,et al.  On the Complexity of Some Hierarchical Structured Matrix Algorithms , 2012, SIAM J. Matrix Anal. Appl..

[17]  Ming Gu,et al.  Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization , 1996, SIAM J. Sci. Comput..

[18]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[19]  Victor Y. Pan,et al.  Additive Preconditioning for Matrix Computations , 2008, CSR.

[20]  Jacek Gondzio,et al.  A Matrix-Free Preconditioner for Sparse Symmetric Positive Definite Systems and Least-Squares Problems , 2013, SIAM J. Sci. Comput..

[21]  Jianlin Xia,et al.  Fast algorithms for hierarchically semiseparable matrices , 2010, Numer. Linear Algebra Appl..

[22]  Stanley Osher,et al.  A Hybrid Method for Moving Interface Problems with Application to the Hele-Shaw Flow , 1997 .

[23]  Raymond H. Chan,et al.  A Fast Randomized Eigensolver with Structured LDL Factorization Update , 2014, SIAM J. Matrix Anal. Appl..

[24]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

[25]  JAMES VOGEL,et al.  Superfast Divide-and-Conquer Method and Perturbation Analysis for Structured Eigenvalue Solutions , 2016, SIAM J. Sci. Comput..

[26]  Huajian Gao,et al.  A Numerical Study of Electro-migration Voiding by Evolving Level Set Functions on a Fixed Cartesian Grid , 1999 .

[27]  J. K. Hunter,et al.  Reactive autophobic spreading of drops , 2002 .

[28]  K. Ito,et al.  An augmented approach for Stokes equations with a discontinuous viscosity and singular forces , 2007 .

[29]  Jianlin Xia,et al.  A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling , 2012, SIAM J. Matrix Anal. Appl..

[30]  Jianlin Xia,et al.  Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices , 2010, SIAM J. Matrix Anal. Appl..

[31]  Zhilin Li.  A Fast Iterative Algorithm for Elliptic Interface Problems , 1998 .

[32]  Wenjun Ying,et al.  A kernel-free boundary integral method for elliptic boundary value problems , 2007, J. Comput. Phys..

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

[34]  Per-Gunnar Martinsson,et al.  Randomized algorithms for the low-rank approximation of matrices , 2007, Proceedings of the National Academy of Sciences.

[35]  Jianlin Xia,et al.  Efficient Structured Multifrontal Factorization for General Large Sparse Matrices , 2013, SIAM J. Sci. Comput..

[36]  J. Cullum,et al.  Matrix-free preconditioning using partial matrix estimation , 2006 .

[37]  P. Colella,et al.  A second-order projection method for the incompressible navier-stokes equations , 1989 .

[38]  Azzam Haidar,et al.  Parallel algebraic hybrid solvers for large 3D convection-diffusion problems , 2008, Numerical Algorithms.

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

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

[41]  V. Rokhlin,et al.  A fast randomized algorithm for the approximation of matrices ✩ , 2007 .

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

[43]  Yousef Saad,et al.  Divide and Conquer Low-Rank Preconditioners for Symmetric Matrices , 2013, SIAM J. Sci. Comput..

[44]  Hongkai Zhao,et al.  An augmented method for free boundary problems with moving contact lines , 2010 .

[45]  Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer , 2015, 1511.07733.

[46]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[47]  Miroslav Tuma,et al.  Preconditioner updates for solving sequences of linear systems in matrix‐free environment , 2010, Numer. Linear Algebra Appl..

[48]  Per-Gunnar Martinsson,et al.  A Fast Randomized Algorithm for Computing a Hierarchically Semiseparable Representation of a Matrix , 2011, SIAM J. Matrix Anal. Appl..

[49]  Jianlin Xia,et al.  Randomized Sparse Direct Solvers , 2013, SIAM J. Matrix Anal. Appl..

[50]  V. Pan,et al.  Randomized Preprocessing of Homogeneous Linear Systems of Equations , 2010 .

[51]  Julien Langou,et al.  Any admissible cycle‐convergence behavior is possible for restarted GMRES at its initial cycles , 2011, Numer. Linear Algebra Appl..

[52]  Jan Vlcek,et al.  Efficient tridiagonal preconditioner for the matrix-free truncated Newton method , 2014, Appl. Math. Comput..