Abstract
An efficient and robust restart strategy is important for any Krylov-based method for eigenvalue problems. The tensor infinite Arnoldi method (TIAR) is a Krylov-based method for solving nonlinear eigenvalue problems (NEPs). This method can be interpreted as an Arnoldi method applied to a linear and infinite dimensional eigenvalue problem where the Krylov basis consists of polynomials. We propose new restart techniques for TIAR and analyze efficiency and robustness. More precisely, we consider an extension of TIAR which corresponds to generating the Krylov space using not only polynomials, but also structured functions, which are sums of exponentials and polynomials, while maintaining a memory efficient tensor representation. We propose two restarting strategies, both derived from the specific structure of the infinite dimensional Arnoldi factorization. One restarting strategy, which we call semi-explicit TIAR restart, provides the possibility to carry out locking in a compact way. The other strategy, which we call implicit TIAR restart, is based on the Krylov–Schur restart method for the linear eigenvalue problem and preserves its robustness. Both restarting strategies involve approximations of the tensor structured factorization in order to reduce the complexity and the required memory resources. We bound the error introduced by some of the approximations in the infinite dimensional Arnoldi factorization showing that those approximations do not substantially influence the robustness of the restart approach. We illustrate the effectiveness of the approaches by applying them to solve large scale NEPs that arise from a delay differential equation and a wave propagation problem. The advantages in comparison to other restart methods are also illustrated.
Highlights
We consider the nonlinear eigenvalue problem (NEP) defined as finding (λ, v) ∈ C × Cn\ {0} such that M(λ)v = 0 (1)where λ ∈ Ω ⊆ C, Ω is an open disk centered in the origin and M : Ω → Cn×n is analytic
– Implicit restart (Sect. 5): By only representing polynomials, we show that the tensor infinite Arnoldi method (TIAR) factorization has a particular structure such that it can be accurately approximated
In this work we have derived an extension of the TIAR algorithm and two restarting strategies
Summary
We consider the nonlinear eigenvalue problem (NEP) defined as finding (λ, v) ∈ C × Cn\ {0} such that. We do not assume any particular structure of the NEP except for the analyticity and the computability of certain quantities associated with M(λ) (further described later) This is similar to the infinite Arnoldi method (IAR) [14], which is in the same line of reasoning as our approach. There exist other Arnoldi-like methods combined with a companion linearization that use memory efficient representation of the Krylov basis matrix and that can be restarted. The coefficients that represent the Krylov basis are replaced with their low rank approximations In contrast to those approaches, our specific setting, in particular the infinite dimensional function formulation and the representation of the basis with tensor structure functions, allows us to characterize the impact of the approximations. The matrix Hk denotes the square matrix obtained by removing the last row from the matrix H k ∈ C(k+1)×k
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