Low-rank Right-hand Sides Research Articles

A modified block Hessenberg method for low-rank tensor Sylvester equation

This work focuses on iteratively solving the tensor Sylvester equation with low-rank right-hand sides. To solve such equations, we first introduce a modified version of the block Hessenberg process so that approximation subspaces contain some extra block information obtained by multiplying the initial block by the inverse of each coefficient matrix of the tensor Sylvester equation. Then, we apply a Galerkin-like condition to transform the original tensor Sylvester equation into a low-dimensional tensor form. The reduced problem is then solved using a blocked recursive algorithm based on Schur decomposition. Moreover, we reveal how to stop the iterations without the need to compute the approximate solution by calculating the residual norm or an upper bound. Eventually, some numerical examples are given to assess the efficiency and robustness of the suggested method.

Projection schemes based on Hessenberg process for Sylvester tensor equation with low-rank right-hand side

Projection schemes based on Hessenberg process for Sylvester tensor equation with low-rank right-hand side

On Some Numerical Methods for Solving Large Differential Nonsymmetric Stein Matrix Equations

In this paper, we propose a new numerical method based on the extended block Arnoldi algorithm for solving large-scale differential nonsymmetric Stein matrix equations with low-rank right-hand sides. This algorithm is based on projecting the initial problem on the extended block Krylov subspace to obtain a low-dimensional differential Stein matrix equation. The obtained reduced-order problem is solved by the backward differentiation formula (BDF) method or the Rosenbrock (Ros) method, the obtained solution is used to build the low-rank approximate solution of the original problem. We give some theoretical results and report some numerical experiments.

Tensorized low-rank circulant preconditioners for multilevel Toeplitz linear systems from high-dimensional fractional Riesz equations

The solution of d-dimensional fractional partial differential equations (FPDEs) is challenging in numerical computation. This paper proposes a low-rank preconditioned conjugate gradient (PCG) method for discretizations with a low-rank right-hand side stemming from the d-dimensional fractional Riesz equation. At each iteration step, a combination of low-rank in Tensor Train (TT) format technique and Fast Fourier Transform (FFT) is used to accelerate multilevel Toeplitz matrix-vector multiplications. Furthermore, a low-rank circulant preconditioner is constructed based on the spectrum of the coefficient matrix. The reported numerical results demonstrate the competitiveness of the new method compared with a state-of-the-art matrix-free approach based on FFT.

Krylov subspace projection method for Sylvester tensor equation with low rank right-hand side

Motivated by the effectiveness of Krylov projection methods and the CP decomposition of tensors, which is a low rank decomposition, we propose Arnoldi-based methods (block and global) to solve Sylvester tensor equation with low rank right-hand sides. We apply a standard Krylov subspace method to each coefficient matrix, in order to reduce the main problem to a projected Sylvester tensor equation, which can be solved by a global iterative scheme. We show how to extract approximate solutions via matrix Krylov subspaces basis. Several theoretical results such as expressions of residual and its norm are presented. To show the performance of the proposed approaches, some numerical experiments are given.

Numerical methods for differential linear matrix equations via Krylov subspace methods

In the present paper, we present some numerical methods for computing approximate solutions to some large differential linear matrix equations. In the first part of this work, we deal with differential generalized Sylvester matrix equations with full rank right-hand sides using a global Galerkin and a norm-minimization approaches. In the second part, we consider large differential Lyapunov matrix equations with low rank right-hand sides and use the extended global Arnoldi process to produce low rank approximate solutions. We give some theoretical results and present some numerical examples.

Nonlinear Least-Squares Approach for Large-Scale Algebraic Riccati Equations

We propose a new optimization approach for solving large-scale continuous-time algebraic Riccati equations with a low-rank right-hand side. First, we project the problem onto a Krylov-type low-dime...

Low-rank approximate solutions to large-scale differential matrix Riccati equations

In the present paper, we consider large-scale continuous-time differential matrix Riccati equations having low rank right-hand sides. These equations are generally solved by Backward Differentiation Formula (BDF) or Rosenbrock methods leading to a large scale algebraic Riccati equation which has to be solved for each timestep. We propose a new approach, based on the reduction of the problem dimension prior to integration. We project the initial problem onto an extended block Krylov subspace and obtain a low-dimentional differential algebraic Riccati equation. The latter matrix differential problem is then solved by Backward Differentiation Formula (BDF) method and the obtained solution is used to reconstruct an approximate solution of the original problem. We give some theoretical results and a simple expression of the residual allowing the implementation of a stop test in order to limit the dimension of the projection space. Some numerical experiments will be given.

Fast Singular Value Decay for Lyapunov Solutions with Nonnormal Coefficients

Lyapunov equations with low-rank right-hand sides often have solutions whose singular values decay rapidly, enabling iterative methods that produce low-rank approximate solutions. All previously known bounds on this decay involve quantities that depend quadratically on the departure of the coefficient matrix from normality: these bounds suggest that the larger the departure from normality, the slower the singular values will decay. We show this is only true up to a threshold, beyond which a larger departure from normality can actually correspond to faster decay of singular values: if the singular values decay slowly, the numerical range cannot extend far into the right half-plane.

Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems

We discuss the numerical solution of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

An Alternating Direction Implicit Method for Solving Projected Generalized Continuous-Time Sylvester Equations

We present the generalized low-rank alternating direction implicit method and the low-rank cyclic Smith method to solve projected generalized continuous-time Sylvester equations with low-rank right-hand sides. Such equations arise in control theory including the computation of inner products andℍ2norms and the model reduction based on balanced truncation for descriptor systems. The requirements of these methods are moderate with respect to both computational cost and memory. Numerical experiments presented in this paper show the effectiveness of the proposed methods.

A direct method for solving projected generalized continuous-time Sylvester equations

This article is devoted to the numerical solution of a projected generalized Sylvester equation with relatively small size. Such an equation arises in stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. The algebraic formula of the solution of the projected generalized continuous-time Sylvester equation is presented. A direct method based on the generalized Schur factorization is proposed. Moreover, its low-rank version for problems with low-rank right-hand sides is also proposed. The computational cost of the direct method is estimated. Numerical simulation show that this direct method has high accuracy.

A Parameter Free Iterative Method for Solving Projected Generalized Lyapunov Equations

This paper is devoted to the numerical solution of projected generalized continuous-time Lyapunov equations with low-rank right-hand sides. Such equations arise in stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. A parameter free iterative method is proposed. This method is based upon a combination of an approximate power method and a generalized ADI method. Numerical experiments presented in this paper show the effectiveness of the proposed method.

Analysis of the solution of the Sylvester equation using low-rank ADI with exact shifts

The solution to a general Sylvester equation A X − X B = G F ∗ with a low-rank right-hand side is analyzed quantitatively through the Low-rank Alternating-Directional-Implicit method (LR-ADI) with exact shifts. New bounds and perturbation bounds on X are obtained. A distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of A and B and the right-hand side factors G and F . Numerical examples suggest that because of this inclusion of details, new perturbation bounds are much sharper than the existing ones.

Inverse Iteration for Purely Imaginary Eigenvalues with Application to the Detection of Hopf Bifurcations in Large-Scale Problems

The detection of a Hopf bifurcation in a large-scale dynamical system that depends on a physical parameter often consists of computing the right-most eigenvalues of a sequence of large sparse eigenvalue problems. Guckenheimer, Gueron, and Harris-Warrick [SIAM J. Numer. Anal., 34 (1997), pp. 1–21] proposed a method that computes a value of the parameter that corresponds to a Hopf point without actually computing right-most eigenvalues. This method utilizes a certain sum of Kronecker products and involves the solution of matrices of squared dimension, which is impractical for large-scale applications. However, if good starting guesses are available for the parameter and the purely imaginary eigenvalue at the Hopf point, then efficient algorithms are available. In this paper, we propose a method for obtaining such good starting guesses, based on finding purely imaginary eigenvalues of a two-parameter eigenvalue problem (possibly arising after a linearization process). The problem is formulated as an inexact inverse iteration method that requires the solution of a sequence of Lyapunov equations with low rank right-hand sides. It is this last fact that makes the method feasible for large systems. The power of our method is tested on four numerical examples.

Bounds on the trace of a solution to the Lyapunov equation with a general stable matrix

Some new estimates for the eigenvalue decay rate of the Lyapunov equation AX + XA T = B with a low rank right-hand side B are derived. The new bounds show that the right-hand side B can greatly influence the eigenvalue decay rate of the solution. This suggests a new choice of the ADI-parameters for the iterative solution. The advantage of these new parameters is illustrated on second order damped systems with a low rank damping matrix.