A convergence rate for the ADI method for infinite-dimensional Lyapunov equations

Mark Opmeer

doi:10.1016/j.ifacol.2021.06.154

A convergence rate for the ADI method for infinite-dimensional Lyapunov equations

Research output: Contribution to journal › Conference article › peer-review

Abstract

We show that the ADI method for a class of infinite-dimensional Lyapunov equations with appropriately chosen shift parameters converges exponentially in the square root. The main assumption on the class of Lyapunov equations is that the main operator generates an analytic semigroup. Rather than directly analyzing the ADI algorithm, we instead use that the ADI error is bounded by the error made by applying quadrature to the inverse Laplace transform integral of the output map and we analyze the error made by this quadrature approximation.

Original language	English
Pages (from-to)	373-377
Number of pages	5
Journal	IFAC-PapersOnLine
Volume	54
Issue number	9
Early online date	16 Jul 2021
DOIs	https://doi.org/10.1016/j.ifacol.2021.06.154
Publication status	Published - 16 Jul 2021

Bibliographical note

Publisher Copyright:
Copyright © 2021 The Authors. This is an open access article under the CC BY-NC-ND license.

Keywords

Adi method
Convergence of numerical methods
Distributed-parameter systems
Laplace transforms
Lyapunov equation

ASJC Scopus subject areas

Control and Systems Engineering

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10.1016/j.ifacol.2021.06.154Licence: CC BY-NC-ND

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abstract = "We show that the ADI method for a class of infinite-dimensional Lyapunov equations with appropriately chosen shift parameters converges exponentially in the square root. The main assumption on the class of Lyapunov equations is that the main operator generates an analytic semigroup. Rather than directly analyzing the ADI algorithm, we instead use that the ADI error is bounded by the error made by applying quadrature to the inverse Laplace transform integral of the output map and we analyze the error made by this quadrature approximation.",

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AB - We show that the ADI method for a class of infinite-dimensional Lyapunov equations with appropriately chosen shift parameters converges exponentially in the square root. The main assumption on the class of Lyapunov equations is that the main operator generates an analytic semigroup. Rather than directly analyzing the ADI algorithm, we instead use that the ADI error is bounded by the error made by applying quadrature to the inverse Laplace transform integral of the output map and we analyze the error made by this quadrature approximation.

KW - Adi method

KW - Convergence of numerical methods

KW - Distributed-parameter systems

KW - Laplace transforms

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A convergence rate for the ADI method for infinite-dimensional Lyapunov equations

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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