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Achieng, PaulineBerntsson, FredrikChepkorir, JenniferKozlov, Vladimir
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Achieng, PaulineBerntsson, FredrikChepkorir, JenniferKozlov, Vladimir
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Mathematics and Applied MathematicsFaculty of Science & EngineeringComputational Mathematics
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Mathematical Analysis
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Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic EquationsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2021 (English)In: Bulletin of the Iranian Mathematical Society, ISSN 1735-8515, Vol. 47, p. 1681-1699Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2021. Vol. 47, p. 1681-1699
##### Keywords [en]

Helmholtz equation, Cauchy problem, Inverse problem, Ill-posed problem
##### National Category

Mathematical Analysis
##### Identifiers

URN: urn:nbn:se:liu:diva-170834DOI: 10.1007/s41980-020-00466-7ISI: 000575739300001Scopus ID: 2-s2.0-85092146699OAI: oai:DiVA.org:liu-170834DiVA, id: diva2:1479111
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt565",{id:"formSmash:j_idt565",widgetVar:"widget_formSmash_j_idt565",multiple:true}); Available from: 2020-10-26 Created: 2020-10-26 Last updated: 2024-02-22Bibliographically approved
##### In thesis

The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers k^{2}, in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of k^{2} in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

1. Analysis of the Robin-Dirichlet iterative procedure for solving the Cauchy problem for elliptic equations with extension to unbounded domains$(function(){PrimeFaces.cw("OverlayPanel","overlay1479114",{id:"formSmash:j_idt883:0:j_idt888",widgetVar:"overlay1479114",target:"formSmash:j_idt883:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Accelerated Dirichlet-Robin Alternating Algorithm for Solving the Cauchy Problem for an Elliptic Equation using Krylov Subspaces$(function(){PrimeFaces.cw("OverlayPanel","overlay1481581",{id:"formSmash:j_idt883:1:j_idt888",widgetVar:"overlay1481581",target:"formSmash:j_idt883:1:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Reconstruction of solutions of Cauchy problems for elliptic equations in bounded and unbounded domains using iterative regularization methods$(function(){PrimeFaces.cw("OverlayPanel","overlay1811372",{id:"formSmash:j_idt883:2:j_idt888",widgetVar:"overlay1811372",target:"formSmash:j_idt883:2:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Regularization methods for solving Cauchy problems for elliptic and degenerate elliptic equations$(function(){PrimeFaces.cw("OverlayPanel","overlay1839988",{id:"formSmash:j_idt883:3:j_idt888",widgetVar:"overlay1839988",target:"formSmash:j_idt883:3:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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