Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction

We present a method for the fast computation of the eigenpairs of a bijective positive symmetric linear operator $\mathcal{L}$. The method is based on a combination of operator adapted wavelets (gamblets) with hierarchical subspace correction.First, gamblets provide a raw but fast approximation of the eigensubspaces of $\mathcal{L}$ by block-diagonalizing $\mathcal{L}$ into sparse and well-conditioned blocks. Next, the hierarchical subspace correction method, computes the eigenpairs associated with the Galerkin restriction of $\mathcal{L}$ to a coarse (low dimensional) gamblet subspace, and then, corrects those eigenpairs by solving a hierarchy of linear problems in the finer gamblet subspaces (from coarse to fine, using multigrid iteration). The proposed algorithm is robust for the presence of multiple (a continuum of) scales and is shown to be of near-linear complexity when $\mathcal{L}$ is an (arbitrary local, e.g.~differential) operator mapping $\mathcal{H}^s_0(\Omega)$ to $\mathcal{H}^{-s}(\Omega)$ (e.g.~an elliptic PDE with rough coefficients).

[1]  Fei Xue,et al.  Preconditioned Eigensolvers for Large-Scale Nonlinear Hermitian Eigenproblems with Variational Characterizations. II. Interior Eigenvalues , 2015, SIAM J. Sci. Comput..

[2]  W. Hackbusch,et al.  A fast iterative method for solving poisson’s equation in a general region , 1978 .

[3]  Marcel Filoche,et al.  Effective Confining Potential of Quantum States in Disordered Media. , 2015, Physical review letters.

[4]  G. Beylkin,et al.  A Multiresolution Strategy for Numerical Homogenization , 1995 .

[5]  Jinchao Xu,et al.  A two-grid discretization scheme for eigenvalue problems , 2001, Math. Comput..

[6]  Hehu Xie,et al.  A type of multilevel method for the Steklov eigenvalue problem , 2014 .

[7]  Y. Saad,et al.  Numerical Methods for Large Eigenvalue Problems , 2011 .

[8]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[9]  A. Brandt Multi-level adaptive technique (MLAT) for fast numerical solution to boundary value problems , 1973 .

[10]  Ziyun Zhang,et al.  A Fast Hierarchically Preconditioned Eigensolver Based On Multiresolution Matrix Decomposition , 2019, Multiscale Model. Simul..

[11]  E. D'yakonov,et al.  Minimization of the computational labor in determining the first eigenvalues of differential operators , 1980 .

[12]  Houman Owhadi,et al.  Multigrid with Rough Coefficients and Multiresolution Operator Decomposition from Hierarchical Information Games , 2015, SIAM Rev..

[13]  Hehu Xie,et al.  A multi-level correction scheme for eigenvalue problems , 2011, Math. Comput..

[14]  Hehu Xie,et al.  A full multigrid method for eigenvalue problems , 2016, J. Comput. Phys..

[15]  B. Engquist,et al.  Wavelet-Based Numerical Homogenization , 1998 .

[16]  Andrew V. Knyazev,et al.  Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method , 2001, SIAM J. Sci. Comput..

[17]  Walter Kohn,et al.  Analytic Properties of Bloch Waves and Wannier Functions , 1959 .

[18]  Daniel Kressner,et al.  An indefinite variant of LOBPCG for definite matrix pencils , 2014, Numerical Algorithms.

[19]  Houman Owhadi,et al.  De-noising by thresholding operator adapted wavelets , 2018, Statistics and Computing.

[20]  Maxim A. Olshanskii,et al.  Iterative Methods for Linear Systems - Theory and Applications , 2014 .

[21]  Charles A. Micchelli,et al.  A Survey of Optimal Recovery , 1977 .

[22]  Daniel Peterseim,et al.  Localization of elliptic multiscale problems , 2011, Math. Comput..

[23]  I. Babuska,et al.  Finite element-galerkin approximation of the eigenvalues and Eigenvectors of selfadjoint problems , 1989 .

[24]  Daniel Peterseim,et al.  Computation of eigenvalues by numerical upscaling , 2012, Numerische Mathematik.

[25]  A. Aspect,et al.  Direct observation of Anderson localization of matter waves in a controlled disorder , 2008, Nature.

[26]  H. Owhadi,et al.  Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization , 2012, 1212.0812.

[27]  H. Owhadi,et al.  Metric‐based upscaling , 2007 .

[28]  S. C. Brenner,et al.  C 0 IPG Method for Biharmonic Eigenvalue Problems , 2014 .

[29]  Houman Owhadi,et al.  Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization , 2019 .

[30]  I. Oseledets,et al.  Calculating vibrational spectra of molecules using tensor train decomposition. , 2016, The Journal of chemical physics.

[31]  Hehu Xie,et al.  A full multigrid method for nonlinear eigenvalue problems , 2015, 1502.04657.

[32]  Ming Gu,et al.  A Robust and Efficient Implementation of LOBPCG , 2018, SIAM J. Sci. Comput..

[33]  Gregory H. Wannier,et al.  Dynamics of Band Electrons in Electric and Magnetic Fields , 1962 .

[34]  P. Anderson Absence of Diffusion in Certain Random Lattices , 1958 .

[35]  Xia Ji,et al.  A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation , 2018, J. Sci. Comput..

[36]  I. Babuska,et al.  Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods , 1983 .

[37]  Hehu Xie,et al.  A Multilevel Correction Type of Adaptive Finite Element Method for Eigenvalue Problems , 2012, SIAM J. Sci. Comput..

[38]  D. Sorensen IMPLICITLY RESTARTED ARNOLDI/LANCZOS METHODS FOR LARGE SCALE EIGENVALUE CALCULATIONS , 1996 .

[39]  Florian Schäfer,et al.  Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity , 2017, Multiscale Model. Simul..

[40]  Tony F. Chan,et al.  An Energy-minimizing Interpolation for Robust Multigrid Methods , 1999, SIAM J. Sci. Comput..

[41]  Andrew V. Knyazev,et al.  A subspace preconditioning algorithm for eigenvector/eigenvalue computation , 1995, Adv. Comput. Math..

[42]  Zhaojun Bai,et al.  Minimization Principles for the Linear Response Eigenvalue Problem I: Theory , 2012, SIAM J. Matrix Anal. Appl..

[43]  A. Knyazev,et al.  Efficient solution of symmetric eigenvalue problems using multigridpreconditioners in the locally optimal block conjugate gradient method , 2001 .

[44]  N. Marzari,et al.  Maximally-localized Wannier Functions: Theory and Applications , 2011, 1112.5411.

[45]  Q. Lin,et al.  A MULTILEVEL CORRECTION TYPE OF ADAPTIVE FINITE ELEMENT METHOD FOR STEKLOV EIGENVALUE PROBLEMS , 2012 .

[46]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[47]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[48]  Jun-zhi Cui,et al.  Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains , 2004, Numerische Mathematik.

[49]  Daniel Peterseim,et al.  Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials , 2018, Mathematical Models and Methods in Applied Sciences.

[50]  Sushant Sachdeva,et al.  Approximate Gaussian Elimination for Laplacians - Fast, Sparse, and Simple , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[51]  F. Chatelin Spectral approximation of linear operators , 2011 .

[52]  Chao Yang,et al.  A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix , 2014, J. Comput. Phys..

[53]  Houman Owhadi,et al.  Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients , 2016, J. Comput. Phys..

[54]  Andrew Knyazev,et al.  Preconditioned Eigensolvers - an Oxymoron? , 1998 .

[55]  Andrew Knyazev Recent implementations, applications, and extensions of the Locally Optimal Block Preconditioned Conjugate Gradient method (LOBPCG) , 2017, ArXiv.

[56]  Fei Xue,et al.  Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. I. Extreme eigenvalues , 2016, Math. Comput..

[57]  Houman Owhadi,et al.  Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis , 2017, 1703.10761.

[58]  Hehu Xie,et al.  A full multigrid method for eigenvalue problems , 2014, J. Comput. Phys..

[59]  Ralf Kornhuber,et al.  Numerical Homogenization of Elliptic Multiscale Problems by Subspace Decomposition , 2016, Multiscale Model. Simul..

[60]  G. Wannier The Structure of Electronic Excitation Levels in Insulating Crystals , 1937 .

[61]  Xin Wang,et al.  Multiscale finite element algorithm of the eigenvalue problems for the elastic equations in composite materials , 2009 .

[62]  Yalchin Efendiev,et al.  An adaptive GMsFEM for high-contrast flow problems , 2013, J. Comput. Phys..

[63]  H. Owhadi,et al.  Flux Norm Approach to Finite Dimensional Homogenization Approximations with Non-Separated Scales and High Contrast , 2009, 0901.1463.

[64]  Ralf Kornhuber,et al.  An analysis of a class of variational multiscale methods based on subspace decomposition , 2016, Math. Comput..