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

Abstract

Abstract: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 approach 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.

Balanced truncation-rational Krylov methods for model reduction in large scale dynamical systems

“…In our numerical tests, we used two strategies: the first one is a priori selection from Lyapack (Mehrmann and Penzl 1998) and the second strategy consists in selecting, at each iteration, a new shift s m+1 which is used to compute a new basis vector. For the second case, we used an adaptive selection from Abidi et al (2016). The rational block Arnoldi algorithm for the pair (A, V ) where V ∈ IR n×r is summarized as follows:…”

“…The matrices T A,m and T D,m could be obtained directly from H A,m and H D,m , respectively; see Abidi et al (2016) as follows:…”

The Block Rational Arnoldi Method

“…We consider the problem of approximating the transfer function H(s) = c T (sE − A) −1 b over a range of frequencies. We compare two different sequences of poles denoted by ξ (1) and ξ (2) . The first sequence ξ (1) is chosen equal to that in [10,Section 5.2], with four poles equidistantly placed on the interval i[0, 40] and cyclically repeated until we have 24 poles in total.…”

“…Block Krylov methods were introduced in the 1970s, starting with the block Lanczos algorithm for linear eigenproblems with repeated eigenvalues [16,27,36]. More recently, block Krylov methods have found applications in model order reduction [2,22,23], for the solution of matrix equations [6,19,31,33], matrix function approximation [24,37,38,40], including multisource electromagnetic modeling [13,15,42,43], and solving linear systems with multiple right-hand sides [12,14,18,21,28,41,48]. While the theory of single-vector rational Krylov spaces is well developed [9,10,44,45,46,47], the block case has only been explored to a limited extent [4,24,29].…”

Residual-based iterations for the generalized Lyapunov equation

“…As noted in [15, just before Section 4.1] the changes when going to blocks are mostly technical. Results needed to generalize [15,Proposition 4.2] are found in, e.g., [1], and the block case is implemented in the code available at Simoncini's webpage 1 .…”