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Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems

Zhang, Ziyun (2022) Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems. Communications in Mathematical Sciences, 20 (2). pp. 377-403. ISSN 1539-6746. doi:10.4310/cms.2022.v20.n2.a4.

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We propose to use the Łojasiewicz inequality as a general tool for analyzing the convergence rate of gradient descent on a Hilbert manifold, without resorting to the continuous gradient flow. Using this tool, we show that a Sobolev gradient descent method with adaptive inner product converges exponentially fast to the ground state for the Gross–Pitaevskii eigenproblem. This method can be extended to a class of general high-degree optimizations or nonlinear eigenproblems under certain conditions. We demonstrate this generalization using several examples, in particular a nonlinear Schrödinger eigenproblem with an extra high-order interaction term. Numerical experiments are presented for these problems.

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Additional Information:© 2022 International Press. Received: September 23, 2020; Accepted (in revised form): July 10, 2021. Published 28 January 2022. This research was in part supported by NSF grants DMS-1912654 and DMS-1907977. The author would like to thank Thomas Y. Hou for the helpful comments on earlier versions of this work, and Zhenzhen Li for introducing the Lojasiewicz inequality to the author. The author would also like to acknowledge the warm hospitality of Oberwolfach Research Institute for Mathematics during the seminar Beyond Numerical Homogenization, where the early ideas of this work started.
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Subject Keywords:Sobolev gradient descent, Gross-Pitaevskii eigenproblem, Lojasiewicz inequality, Schr odinger equation, nonlinear eigenproblems
Issue or Number:2
Classification Code:AMS subject classifications. 35P30; 47J10; 65K10; 65N25; 81Q05.
Record Number:CaltechAUTHORS:20220309-676806000
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:113832
Deposited By: George Porter
Deposited On:11 Mar 2022 20:32
Last Modified:11 Mar 2022 20:32

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