A black-box rational Arnoldi variant for Cauchy-Stieltjes matrix functions

Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when the function to be approximated is of Cauchy-Stieltjes (or Markov) type, such as the matrix square root or the logarithm. The performance of this approach is demonstrated by numerical examples involving symmetric and nonsymmetric matrices. These examples suggest that our black-box method performs at least as well, and typically better, as the standard rational Arnoldi method with parameters being manually optimized for a given matrix

Near-optimal perfectly matched layers for indefinite Helmholtz problems

A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein's interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included.Comment: Accepted for publication in SIAM Review. To appear 201

On application of the Lanczos method to solution of some partial differential equations

AbstractLet A be a square symmetric n × n matrix, φ be a vector from Rn, and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)φ arises very often in mathematical physics.We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos decomposition method (SLDM) solution as um = ∥ φ ∥Qf(H)e1, where Q is the n × m matrix of the m Lanczos vectors and H is the m × m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for ∥ u − um ∥ that are stable in the presence of computer round-off errors when using the simple Lanczos method.We concentrate on computation of exp(− tA)φ, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations

Compressing variable-coefficient exterior Helmholtz problems via RKFIT

The efficient discretization of Helmholtz problems on unbounded domains is a challenging task, in particular, when the wave medium is nonhomogeneous. We present a new numerical approach for compressing finite difference discretizations of such problems, thereby giving rise to efficient perfectly matched layers (PMLs) for nonhomogeneous media. This approach is based on the solution of a nonlinear rational least squares problem using the RKFIT method proposed in [M. Berljafa and S. G���¼ttel, SIAM J. Matrix Anal. Appl., 36(2):894--916, 2015]. We show how the solution of this least squares problem can be converted into an accurate finite difference grid within a rational Krylov framework. Several numerical experiments are included. They indicate that RKFIT computes PMLs more accurate than previous analytic approaches and even works in regimes where the Dirichlet-to-Neumann functions to be approximated are highly irregular. Spectral adaptation effects allow for accurate finite difference grids with point numbers below the Nyquist limit

Near-optimal perfectly matched layers for indefinite Helmholtz problems

A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein's interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included

Model order reduction of layered waveguides via rational Krylov fitting

Network-based data-driven reduced order models have recently emerged as an efficient numerical tool for forward and inverse problems of wave propagation. Currently, this technique is limited to two classes of problems: bounded inhomogeneous domains (with applications in multiscale simulation and imaging) and homogeneous halfspaces (for the solution of exterior forward problems). Here we relax the constant coefficient requirement for the latter by considering reduced order models (ROMs) of unbounded waveguides with layered inclusions, thereby giving rise to efficient discrete perfectly matched layers (PMLs) for nonhomogeneous media. Our approach is based on the solution of a nonlinear rational least squares problem using the RKFIT method [M. Berljafa and S. Güttel, SIAM J. Sci. Comput., 39(5):A2049--A2071, 2017]. We show how the solution of this least squares problem can be converted into an accurate sparse network approximation within a rational Krylov framework. Several numerical experiments are included. They indicate that RKFIT computes PMLs more accurately than previous analytic approaches and even works in regimes where the transfer functions to be approximated are highly irregular due to pronounced scattering resonances. Spectral adaptation effects allow for accurate ROMs with dimensions near or even below the Nyquist limit