Rational Krylov Research Articles

Adaptive rational block Arnoldi methods for model reductions in large-scale MIMO dynamical systems

In recent years, a great interest has been shown towards Krylov subspace techniques applied to model order reduction of large-scale dynamical systems. A special interest has been devoted to single-input single-output (SISO) systems by using moment matching techniques based on Arnoldi or Lanczos algorithms. In this paper, we consider multiple-input multiple-output (MIMO) dynamical systems and introduce the rational block Arnoldi process to design low order dynamical systems that are close in some sense to the original MIMO dynamical system. Rational Krylov subspace methods are based on the choice of suitable shifts that are selected a priori or adaptively. In this paper, we propose an adaptive selection of those shifts and show the efficiency of this appr oach in our numerical tests. We also give some new block Arnoldi-like relations that are used to propose an upper bound for the norm of the error on the transfer function. m ∈ R m×p , xm(t), ym(t) ∈ R m , and m≪ n, while maintaining the most relevant properties of

Simulation Algorithms With Exponential Integration for Time-Domain Analysis of Large-Scale Power Delivery Networks

We design an algorithmic framework using matrix exponentials for time-domain simulation of power delivery network (PDN). Our framework can reuse factorized matrices to simulate the large-scale linear PDN system with variable stepsizes. In contrast, current conventional PDN simulation solvers have to use fixed step-size approach in order to reuse factorized matrices generated by the expensive matrix decomposition. Based on the proposed exponential integration framework, we design a PDN solver R-MATEX with the flexible time-stepping capability. The key operation of matrix exponential and vector product (MEVP) is computed by the rational Krylov subspace method. To further improve the runtime, we also propose a distributed computing framework DR-MATEX. DR-MATEX reduces Krylov subspace generations caused by frequent breakpoints from a large number of current sources during simulation. By virtue of the superposition property of linear system and scaling invariance property of Krylov subspace, DR-MATEX can divide the whole simulation task into subtasks based on the alignments of breakpoints among those sources. The subtasks are processed in parallel at different computing nodes without any communication during the computation of transient simulation. The final result is obtained by summing up the partial results among all the computing nodes after they finish the assigned subtasks. Therefore, our computation model belongs to the category known as Embarrassingly Parallel model. Experimental results show R-MATEX and DR-MATEX can achieve up to around 14.4X and 98.0X runtime speedups over traditional trapezoidal integration based solver with fixed timestep approach.

Analysis of the Rational Krylov Subspace Projection Method for Large-Scale Algebraic Riccati Equations

In the numerical solution of the algebraic Riccati equation $A^* X + X A - X BB^* X + C^* C =0$, where $A$ is large, sparse, and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the more established Newton--Kleinman iteration. In spite of convincing numerical experiments, a systematic matrix analysis of this class of methods is still lacking. We derive new relations for the approximate solution, the residual, and the error matrices, giving new insights into the role of the matrix $A-BB^*X$ and of its approximations in the numerical procedure. In the context of linear-quadratic regulator problems, we show that the Riccati approximate solution is related to the optimal value of the reduced cost functional, thus completely justifying the projection method from a model order reduction point of view. Finally, the new results provide theoretical ground for recently proposed modifications of projection methods onto rational Krylov subspaces.

Projection-based model reduction for time-varying descriptor systems: New results

We have presented a Krylov-based projection method for model reduction of linear time-varying descriptor systems in [13] which was based on earlier ideas in the work of J. Philips [17] and others. This contribution continues that work by presenting more details of linear time-varying descriptor systems and new results coming from real fields of application. The idea behind the proposed procedure is based on a multipoint rational approximation of the monodromy matrix of the corresponding differential-algebraic equation. This is realized by orthogonal projection onto a rational Krylov subspace. The algorithmic realization of the method employs recycling techniques for shifted Krylov subspaces and their invariance properties. The proposed method works efficiently for macro-models, such as time varying circuit systems and models arising in network interconnection, on limited frequency ranges. Bode plots and step response are used to illustrate both the performance and accuracy of the reduced-order model.

The ADI iteration for Lyapunov equations implicitly performs pseudo-optimal model order reduction

ABSTRACTTwo approaches for approximating the solution of large-scale Lyapunov equations are considered: the alternating direction implicit (ADI) iteration and projective methods by Krylov subspaces. We show that they are linked in the way that the ADI iteration can always be identified by a Petrov–Galerkin projection with rational block Krylov subspaces. Therefore, a unique Krylov-projected dynamical system can be associated with the ADI iteration, which is proven to be an pseudo-optimal approximation. This includes the generalisation of previous results on pseudo-optimality to the multivariable case. Additionally, a low-rank formulation of the residual in the Lyapunov equation is presented, which is well-suited for implementation, and which yields a measure of the ‘obliqueness’ that the ADI iteration is associated with.

A new Interpolatory Model Reduction Approach for Quadratic Bilinear Descriptor Systems

AbstractWe propose interpolation based projection techniques for model reduction of quadratic bilinear descriptor systems. The approach identifies projection matrices from the bilinear approximation of the original quadratic bilinear descriptor system and applies Petrov‐Galerkin projection to the original system to construct the required reduced quadratic bilinear descriptor system. Two different techniques of computing the projection matrices from the bilinear approximation are proposed. One is based on linear parametric model reduction and the other is the bilinear rational Krylov method. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An Adaptive Rational Block Lanczos-Type Algorithm for Model Reduction of Large Scale Dynamical Systems

Multipoint moment matching based methods are considered as powerful methods for model-order reduction problems. They are related to rational Krylov subspaces (classical or block ones) and are based on the selection of some interpolation points which is the major problem for these methods. In this work, an adaptive rational block Lanczos-type algorithm is proposed and applied for model order reduction of dynamical multi-input and multi-output linear time independent dynamical systems. We give some algebraic properties of the proposed algorithm and derive an explicit formulation of the error between the original and the reduced transfer functions. An adaptive method for choosing the interpolation points is also introduced. Finally, some numerical experiments are reported to show the effectiveness of the proposed adaptive rational block Lanczos-type process.

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.

An implicitly restarted rational Krylov strategy for Lyapunov inverse iteration

An implicitly restarted rational Krylov strategy for Lyapunov inverse iteration

Near-optimal frequency-weighted interpolatory model reduction

This paper develops an interpolatory framework for weighted-H2 model reduction of MIMO dynamical systems. A new representation of the weighted-H2 inner products in MIMO settings is introduced and used to derive associated first-order necessary conditions satisfied by optimal weighted-H2 reduced-order models. Equivalence of these new interpolatory conditions with earlier Riccati-based conditions given by Halevi is also shown. An examination of realizations for equivalent weighted-H2 systems leads then to an algorithm that remains tractable for large state-space dimension. Several numerical examples illustrate the effectiveness of this approach and its competitiveness with Frequency Weighted Balanced Truncation and an earlier interpolatory approach, the Weighted Iterative Rational Krylov Algorithm.

TRAX: An approach for the time rational analysis of complex dynamic systems

SummaryThis paper introduces a theoretical and algorithmic reduced model approach to efficiently evaluate time responses of complex dynamic systems. The proposed approach combines four main components: analytical expressions of the average of the system's transfer functions in the frequency domain, precise and convergent rational approximations of these exact expressions, exact evaluation of these approximations through model reduction in rational Krylov subspaces and semi‐analytical interpolation at just a few frequency points. The resulting algorithmic principles to evaluate the time response of a particular system are relatively straightforward: one first evaluates the response of the system with slight additional damping at a few frequencies and one then projects or reduces the system in the subspace spanned by these responses. The time response of the reduced model implicitly provides a precise evaluation of that of the original system. The properties of the reduced models and the precision of the proposed approach are studied, and applications on complex matrix systems are presented and discussed. While the theory and numerical algorithms are presented in a matrix context, they are also transposable in a continuous functional context. Copyright © 2015 John Wiley & Sons, Ltd.

Generalized Rational Krylov Decompositions with an Application to Rational Approximation

Generalized rational Krylov decompositions are matrix relations which, under certain conditions, are associated with rational Krylov spaces. We study the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. Transformations on rational Krylov decompositions allow for changing the poles of a rational Krylov space without recomputation, and two algorithms are presented for this task. Using such transformations we develop a rational Krylov method for rational least squares fitting. Numerical experiments indicate that the proposed method converges fast and robustly. A MATLAB toolbox with implementations of the presented algorithms and experiments is provided.

Compact Rational Krylov Methods for Nonlinear Eigenvalue Problems

We propose a new uniform framework of compact rational Krylov (CORK) methods for solving large-scale nonlinear eigenvalue problems $A(\lambda) x = 0$. For many years, linearizations were used for solving polynomial and rational eigenvalue problems. On the other hand, for the general nonlinear case, $A(\lambda)$ can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. However, the major disadvantage of linearization-based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, the CORK family of rational Krylov methods exploits the structure of the linearization pencils by using a generalization of the compact Arnoldi decomposition. In this way, the extra memory and orthogonalization costs due to the linearization of the original eigenvalue problem are negligible for large-scale problems. Furthermore, we prove that each CORK step breaks down into an orthogonalization step of the original problem dimension and a rational Krylov step on small matrices. We also briefly discuss implicit restarting of the CORK method and how to exploit low rank structure. The CORK method is illustrated with two large-scale examples.

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.

Quadrature-based IRKA for optimal H2 model reduction

Iterative Rational Krylov Algorithm (irka) of Gugercin et al. [2008] is an effective tool for optimal H2 rational approximation. Beattie and Gugercin [2012] has recently developed a new formulation of irka that only uses transfer function evaluations, without requiring any particular realization; thus extending irka to H2 approximation of irrational, infinitedimensional dynamical systems. The main computational cost of this new framework is the need for repeated transfer function evaluations. In this paper, based on the reproducing kernel formulation of the underlying Hilbert space, we introduce a quadrature-based formulation of irka where a number of transfer function evaluation are computed only at the beginning (the offline phase) and the irka steps (the online phase) never revisit the original transfer function performing all the computations in the reduced order dimension.

A model reduction approach to numerical inversion for a parabolic partial differential equation

We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The algorithm presented in this paper considers inversion in one and two dimensions. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss–Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated accurate simulations of the time-domain response. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments.

Shift-and-Invert Krylov Methods for Time-Fractional Wave Equations

The article deals with evolution problems involving time derivatives of fractional order α, with 1 < α ≤2. The solutions are expressed in terms of operator Mittag-Leffler functions. The action of such operator functions is approximated by rational Krylov methods whose convergence features are investigated.

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.

Determining bound states in a semiconductor device with contacts using a nonlinear eigenvalue solver

We present a nonlinear eigenvalue solver enabling the calculation of bound solutions of the Schrodinger equation in a system with contacts. We discuss how the imposition of contacts leads to a nonlinear eigenvalue problem and discuss the numerics for a one- and two-dimensional potential. We reformulate the problem so that the eigenvalue problem can be efficiently solved by the recently proposal rational Krylov method for nonlinear eigenvalue problems, known as NLEIGS. In order to improve the convergence of the method, we propose a holomorphic extension such that we can easily deal with the branch points introduced by a square root. We use our method to determine the bound states of the one-dimensional Poschl---Teller potential, a two-dimensional potential describing a particle in a canyon and the multi-band Hamiltonian of a topological insulator.

Model Order Reduction for Systems with Moving Loads

Abstract In this contribution we present two approaches allowing to find a reduced order approximant of a full order model featuring a moving load term. First, we apply the Balanced Truncation (BT) method to a switched linear system (SLS) using the special structure given in the spatially discretized model. The second approach treats the variability as a continuous parameter dependence and uses the iterative rational Krylov algorithm (IRKA) to compute a parameter preserving reduced order model.