Rational Krylov Subspace Method Research Articles

Three-dimensional transient electromagnetic modelling using Rational Krylov methods

A computational method is given for solving the forward modeling problem for transient electromagnetic exploration. Its key features are discretization of the quasi-static Maxwell's equations in space using the first-kind family of curl-conforming Nedelec elements combined with time integration using rational Krylov subspace methods. We show how rational Krylov subspace methods may be used to solve the same problem in the frequency domain followed by a synthesis of the transient solution using the fast Hankel transform, arguing that the pure time-domain is more efficient. We also propose a simple method for selecting the pole parameters of the rational Krylov subspace method which leads to convergence within an a priori determined number of iterations independent of mesh size and conductivity structure. These poles are repeated in a cyclic fashion, which, in combination with direct solvers for the discrete problem, results in significantly faster solution times than previously proposed schemes.

From Low-Rank Approximation to a Rational Krylov Subspace Method for the Lyapunov Equation

We propose a new method for the approximate solution of the Lyapunov equation with rank-1 right-hand side, which is based on an extended rational Krylov subspace approximation with adaptively computed shifts. The shift selection is obtained from the connection between the Lyapunov equation, solution of systems of linear ODEs, and an alternating least squares method for low-rank approximation. We provide numerical comparison of our method with other well-established techniques for the approximation solution of the Lyapunov equation on different problems.

Improvement of Krylov-Subspace-Reduced Models by Iterative Mode-Truncation

Model order reduction techniques are well established in control theory and likewise in structural mechanics, e.g. when dealing with Elastic Multi-Body-Systems. Apart from the reduction method used, the objective lies in finding a minimal reliable model dimension with desired quality. Extensive research has been conducted concerning Krylov-subspace methods. Rational Krylov-subspace methods based on the second-order Arnoldi-algorithm (SOAR) turned out to give promising results. Current investigations are about generating an optimal reduced model by iteratively choosing the expansion points based on a predefined model dimension. Still, the sufficient choice of this initial dimension is uncertain.The novel approach consists of a two-stage strategy and starts with a fairly large reduced model based on a fixed number and distribution of expansion points. By an iterative posttreatment, the dispensable Krylov-modes are truncated. For an efficient and fast calculation, a special reordering scheme for the Krylov-modes is introduced. Numerical experiments at different mechanical models show, that the novel technique is comparatively fast and confidently generates reliable models with a minimal dimension.

Efficient time integration for discontinuous Galerkin approximations of linear wave equations

We consider the combination of discontinuous Galerkin discretizations in space with various time integration methods for linear acoustic, elastic, and electro‐magnetic wave equations. For the discontinuous Galerkin method we derive explicit formulas for the full upwind flux for heterogeneous materials by solving the Riemann problems for the corresponding first‐order systems. In a framework of bounded semigroups we prove convergence of the spatial discretization. For the time integration we discuss advantages and disadvantages of explicit and implicit Runge–Kutta methods compared to polynomial and rational Krylov subspace methods for the approximation of the matrix exponential function. Finally, the efficiency of the different time integrators is illustrated by several examples in 2D and 3D for electro‐magnetic and elastic waves.

Uniform Approximation of $\varphi$-Functions in Exponential Integrators by a Rational Krylov Subspace Method with Simple Poles

We consider the approximation of the matrix $\varphi$-functions that appear in exponential integrators for stiff systems of differential equations. For stiff systems, the field-of-values of the occurring matrices is large and lies somewhere in the left complex half-plane. In order to obtain an efficient method uniformly for all matrices with a field-of-values in the left complex half-plane, we consider the approximation by a rational Krylov subspace method with equidistant poles of order one on the line $\mbox{Re}\,z = \gamma > 0$. We present error bounds that predict a faster convergence rate as for the resolvent Krylov subspace approximation using a single repeated pole at $\gamma > 0$. Poles of order one allow moreover for a parallel implementation of the corresponding rational Krylov subspace decomposition. We analyze the convergence of the proposed rational Krylov subspace method and present numerical experiments that illustrate our results.

Model Order Reduction by Approximate Balanced Truncation: A Unifying Framework / Modellreduktion durch approximatives Balanciertes Abschneiden: eine vereinigende Formulierung

Summary A novel formulation of approximate truncated balanced realization (TBR) is introduced to unify three approaches: two iterative methods for solving the underlying Lyapunov equations - the alternating directions implicit (ADI) iteration and the rational Krylov subspace method (RKSM) - and a two-step procedure that performs a Krylov-based projection and subsequently direct TBR. The framework allows to compare the three methods with respect to solvability, fidelity, numerical effort, stability preservation and global error bounds, which suggests to merge ADI with the two-step procedure

Sensitivity analysis and adaptive multi-point multi-moment model order reduction in MEMS design

We present a model order reduction algorithm for linear time-invariant descriptor systems of arbitrary derivative order that incorporates sensitivity analysis for network parameters in respect to design parameters. It is based on implicit moment matching via rational Krylov subspace methods with adaptive choice of expansion points and number of moments based on an error indicator. Additionally, we demonstrate how parametric reduced order models can be obtained at nearly no extra costs, such that parameter studies are extremely accelerated. The finite element model of a yaw rate sensor MEMS device has been chosen as a numerical example, but our method is also applicable to systems arising in modeling and simulation of electromagnetics, electrical circuits, machine tools, heat conduction and other phenomena.

Resolvent Krylov subspace approximation to operator functions

We consider the approximation of operator functions in resolvent Krylov subspaces. Besides many other applications, such approximations are currently of high interest for the approximation of φ-functions that arise in the numerical solution of evolution equations by exponential integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behaviour if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we analyse a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators whose field of values lies in the left half plane. In contrast to standard Krylov methods, the convergence will be independent of the norm of the discretised operator and thus of the spatial discretisation. We will discuss efficient implementations for finite element discretisations and illustrate our analysis with numerical experiments.

Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation

For large scale problems, an effective approach for solving the algebraic Lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation. Although Krylov subspaces have been used for a long time, only more recent developments have shown that rational Krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit (ADI) recurrence. In this paper we develop a convergence analysis of the rational Krylov subspace method (RKSM) based on the Kronecker product formulation and on potential theory. Moreover, we propose new enlightening relations between this approach and the ADI method. Our results provide solid theoretical ground for recent numerical evidence of the superiority of RKSM over ADI when the involved parameters cannot be computed optimally, as is the case in many practical application problems.

Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments.

The Numerical Solution of Parabolic and Elliptic Differential Equations

The Numerical Solution of Parabolic and Elliptic Differential Equations