Bi-Lanczos with partial orthogonalization

Veen, H.I. Van der; Vuik, C.

doi:10.1016/0045-7949(94)00565-k

Cited by 6 publications

(3 citation statements)

References 10 publications

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“…Our numerical experience also suggests that the frequency of partial reorthogonalization increases for ill-conditioned problems significantly. On the other hand, the reported frequencies of partial reorthogonalization in [29] and [41] are lower than the ones we have found in this paper.…”

Section: Test Casecontrasting

confidence: 98%

vibro‐Lanczos, a symmetric Lanczos solver for vibro‐acoustic simulations

Tabak

Rixen

2016

Numerical Meth Engineering

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SUMMARYAn efficient symmetric Lanczos method for the solution of vibro-acoustic eigenvalue problems is presented in this paper. Although finite element discretization results in real but nonsymmetric system matrices, we show that an efficient iteration scheme on a symmetric representation can be built up by using a transformation matrix. In order to decrease the numerical costs of the orthogonalizations performed, we propose to use a partial orthogonalization scheme for the symmetric case. The proposed method is tested on two large problems in order to demonstrate its efficiency and accuracy.

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Section: Test Casecontrasting

confidence: 98%

vibro‐Lanczos, a symmetric Lanczos solver for vibro‐acoustic simulations

Tabak

Rixen

2016

Numerical Meth Engineering

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“…There exist nonhermitian extensions of the Lanczos method which produce simultaneous, biorthogonal approximations for the left- and right-hand eigenspaces of nonhermitian matrices and have seen successful applications in both frequency-domain CC applications as well as state selection for TD-EOM-CC . However, the biorthogonalization requirements of these methods can often be numerically unstable, − and as such, we expect the Arnoldi method to yield superior numerical stability in finite precision …”

Section: Theory and Methodsmentioning

confidence: 99%

Approximate Exponential Integrators for Time-Dependent Equation-of-Motion Coupled Cluster Theory

Williams-Young,

Yuwono,

DePrince III

et al. 2023

J. Chem. Theory Comput.

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With a growing demand for time-domain simulations of correlated many-body systems, the development of efficient and stable integration schemes for the time-dependent Schrödinger equation is of keen interest in modern electronic structure theory. In this work, we present two approaches for the formation of the quantum propagator for time-dependent equation-of-motion coupled cluster theory based on the Chebyshev and Arnoldi expansions of the complex, nonhermitian matrix exponential, respectively. The proposed algorithms are compared with the short-iterative Lanczos method of Cooper et al. [ J. Phys. Chem. A 2021 125 , 5438–5447], the fourth-order Runge–Kutta method, and exact dynamics for a set of small but challenging test problems. For each of the cases studied, both of the proposed integration schemes demonstrate superior accuracy and efficiency relative to the reference simulations.

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“…We thank Shinsei Ryu for the discussions on this point 11. For better numerical accuracy, it might be useful to perform partial re-orthogonalization (PRO) to the bi-Lanczos algorithm[51]. A similar PRO was introduced for the Lanczos algorithm in[8].…”

mentioning

confidence: 99%

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Bhattacharya¹,

Nandy²,

Nath³

et al. 2023

Preprint

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Continuing the previous initiatives [1,2], we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces. Unlike the previously studied Arnoldi iteration, this algorithm renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and chaotic systems by computing Krylov complexity at late times. Our study relies on two specific systems, the dissipative transverse-field Ising model (TFIM) and the dissipative interacting XXZ chain. We find that, for the weak coupling, initial Lanczos coefficients can efficiently distinguish integrable and chaotic evolution before the dissipative effect sets in, which results in more fluctuations in higher Lanczos coefficients. This results in the equal saturation of late-time complexity for both integrable and chaotic cases, making the notion of late-time chaos dubious.

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Bi-Lanczos with partial orthogonalization

Cited by 6 publications

References 10 publications

vibro‐Lanczos, a symmetric Lanczos solver for vibro‐acoustic simulations

vibro‐Lanczos, a symmetric Lanczos solver for vibro‐acoustic simulations

Approximate Exponential Integrators for Time-Dependent Equation-of-Motion Coupled Cluster Theory

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

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