{"title":"Increasing Stability of the First Order Linearized Inverse Schrödinger Potential Problem with Integer Power Type Nonlinearities","authors":"Sen Zou, Shuai Lu, Boxi Xu","doi":"10.1137/22m1542817","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 4, Page 1868-1889, August 2024. <br/> Abstract. We investigate the increasing stability of the inverse Schrödinger potential problem with integer power type nonlinearities at a large wavenumber. By considering the first order linearized problem with respect to the unknown potential function, a combination formula of the first order linearization is proposed, which provides a Lipschitz type stability for the recovery of the Fourier coefficients of the unknown potential function in low frequency mode. These stability results highlight the advantage of nonlinearity in solving this inverse potential problem by explicitly quantifying the dependence on the wavenumber and the nonlinearity index. A reconstruction algorithm for integer power type nonlinearities is also provided. Several numerical examples illuminate the efficiency of our proposed algorithm.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":"28 1","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m1542817","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Applied Mathematics, Volume 84, Issue 4, Page 1868-1889, August 2024. Abstract. We investigate the increasing stability of the inverse Schrödinger potential problem with integer power type nonlinearities at a large wavenumber. By considering the first order linearized problem with respect to the unknown potential function, a combination formula of the first order linearization is proposed, which provides a Lipschitz type stability for the recovery of the Fourier coefficients of the unknown potential function in low frequency mode. These stability results highlight the advantage of nonlinearity in solving this inverse potential problem by explicitly quantifying the dependence on the wavenumber and the nonlinearity index. A reconstruction algorithm for integer power type nonlinearities is also provided. Several numerical examples illuminate the efficiency of our proposed algorithm.
期刊介绍：
SIAM Journal on Applied Mathematics (SIAP) is an interdisciplinary journal containing research articles that treat scientific problems using methods that are of mathematical interest. Appropriate subject areas include the physical, engineering, financial, and life sciences. Examples are problems in fluid mechanics, including reaction-diffusion problems, sedimentation, combustion, and transport theory; solid mechanics; elasticity; electromagnetic theory and optics; materials science; mathematical biology, including population dynamics, biomechanics, and physiology; linear and nonlinear wave propagation, including scattering theory and wave propagation in random media; inverse problems; nonlinear dynamics; and stochastic processes, including queueing theory. Mathematical techniques of interest include asymptotic methods, bifurcation theory, dynamical systems theory, complex network theory, computational methods, and probabilistic and statistical methods.
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