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  • 1. Order onlineBuy this publication >>
    Achieng, Pauline
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Analysis of the Robin-Dirichlet iterative procedure for solving the Cauchy problem for elliptic equations with extension to unbounded domains2020Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    In this thesis we study the Cauchy problem for elliptic equations. It arises in many areas of application in science and engineering as a problem of reconstruction of solutions to elliptic equations in a domain from boundary measurements taken on a part of the boundary of this domain. The Cauchy problem for elliptic equations is known to be ill-posed.

    We use an iterative regularization method based on alternatively solving a sequence of well-posed mixed boundary value problems for the same elliptic equation. This method, based on iterations between Dirichlet-Neumann and Neumann-Dirichlet mixed boundary value problems was first proposed by Kozlov and Maz’ya [13] for Laplace equation and Lame’ system but not Helmholtz-type equations. As a result different modifications of this original regularization method have been proposed in literature. We consider the Robin-Dirichlet iterative method proposed by Mpinganzima et.al [3] for the Cauchy problem for the Helmholtz equation in bounded domains.

    We demonstrate that the Robin-Dirichlet iterative procedure is convergent for second order elliptic equations with variable coefficients provided the parameter in the Robin condition is appropriately chosen. We further investigate the convergence of the Robin-Dirichlet iterative procedure for the Cauchy problem for the Helmholtz equation in a an unbounded domain. We derive and analyse the necessary conditions needed for the convergence of the procedure.

    In the numerical experiments, the precise behaviour of the procedure for different values of k2 in the Helmholtz equation is investigated and the results show that the speed of convergence depends on the choice of the Robin parameters, μ0 and μ1. In the unbounded domain case, the numerical experiments demonstrate that the procedure is convergent provided that the domain is truncated appropriately and the Robin parameters, μ0 and μ1 are also chosen appropriately.

    List of papers
    1. Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations
    Open this publication in new window or tab >>Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations
    2021 (English)In: Bulletin of the Iranian Mathematical Society, ISSN 1735-8515, Vol. 47, p. 1681-1699Article in journal (Refereed) Published
    Abstract [en]

    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 k2, 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 k2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

    Place, publisher, year, edition, pages
    Springer, 2021
    Keywords
    Helmholtz equation, Cauchy problem, Inverse problem, Ill-posed problem
    National Category
    Mathematical Analysis
    Identifiers
    urn:nbn:se:liu:diva-170834 (URN)10.1007/s41980-020-00466-7 (DOI)000575739300001 ()2-s2.0-85092146699 (Scopus ID)
    Available from: 2020-10-26 Created: 2020-10-26 Last updated: 2024-02-22Bibliographically approved
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  • 2. Order onlineBuy this publication >>
    Achieng, Pauline
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Reconstruction of solutions of Cauchy problems for elliptic equations in bounded and unbounded domains using iterative regularization methods2023Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Cauchy problems for elliptic equations arise in applications in science and engineering. These problems often involve finding important information about an elliptical system from indirect or incomplete measurements. Cauchy problems for elliptic equations are known to be disadvantaged in the sense that a small pertubation in the input can result in a large error in the output. Regularization methods are usually required in order to be able to find stable solutions. In this thesis we study the Cauchy problem for elliptic equations in both bounded and unbounded domains using iterative regularization methods. In Paper I and II, we focus on an iterative regularization technique which involves solving a sequence of mixed boundary value well-posed problems for the same elliptic equation. The original version of the alternating iterative technique is based on iterations alternating between Dirichlet-Neumann and Neumann-Dirichlet boundary value problems. This iterative method is known to possibly work for Helmholtz equation. Instead we study a modified version based on alternating between Dirichlet-Robin and Robin-Dirichlet boundary value problems. First, we study the Cauchy problem for general elliptic equations of second order with variable coefficients in a limited domain. Then we extend to the case of unbounded domains for the Cauchy problem for Helmholtz equation. For the Cauchy problem, in the case of general elliptic equations, we show that the iterative method, based on Dirichlet-Robin, is convergent provided that parameters in the Robin condition are chosen appropriately. In the case of an unbounded domain, we derive necessary, and sufficient, conditions for convergence of the Robin-Dirichlet iterations based on an analysis of the spectrum of the Laplacian operator, with boundary conditions of Dirichlet and Robin types.

    In the numerical tests, we investigate the precise behaviour of the Dirichlet-Robin iterations, for different values of the wave number in the Helmholtz equation, and the results show that the convergence rate depends on the choice of the Robin parameter in the Robin condition. In the case of unbounded domain, the numerical experiments show that an appropriate truncation of the domain and an appropriate choice of Robin parameter in the Robin condition lead to convergence of the Robin-Dirichlet iterations.

    In the presence of noise, additional regularization techniques have to implemented for the alternating iterative procedure to converge. Therefore, in Paper III and IV we focus on iterative regularization methods for solving the Cauchy problem for the Helmholtz equation in a semi-infinite strip, assuming that the data contains measurement noise. In addition, we also reconstruct a radiation condition at infinity from the given Cauchy data. For the reconstruction of the radiation condition, we solve a well-posed problem for the Helmholtz equation in a semi-infinite strip. The remaining solution is obtained by solving an ill-posed problem. In Paper III, we consider the ordinary Helmholtz equation and use seperation of variables to analyze the problem. We show that the radiation condition is described by a non-linear well-posed problem that provides a stable oscillatory solution to the Cauchy problem. Furthermore, we show that the ill–posed problem can be regularized using the Landweber’s iterative method and the discrepancy principle. Numerical tests shows that the approach works well.

    Paper IV is an extension of the theory from Paper III to the case of variable coefficients. Theoretical analysis of this Cauchy problem shows that, with suitable bounds on the coefficients, can iterative regularization methods be used to stabilize the ill-posed Cauchy problem.

    List of papers
    1. Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations
    Open this publication in new window or tab >>Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations
    2021 (English)In: Bulletin of the Iranian Mathematical Society, ISSN 1735-8515, Vol. 47, p. 1681-1699Article in journal (Refereed) Published
    Abstract [en]

    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 k2, 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 k2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

    Place, publisher, year, edition, pages
    Springer, 2021
    Keywords
    Helmholtz equation, Cauchy problem, Inverse problem, Ill-posed problem
    National Category
    Mathematical Analysis
    Identifiers
    urn:nbn:se:liu:diva-170834 (URN)10.1007/s41980-020-00466-7 (DOI)000575739300001 ()2-s2.0-85092146699 (Scopus ID)
    Available from: 2020-10-26 Created: 2020-10-26 Last updated: 2024-02-22Bibliographically approved
    2. Robin-Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain
    Open this publication in new window or tab >>Robin-Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain
    2023 (English)In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 31, no 5Article in journal (Refereed) Published
    Abstract [en]

    We consider the Cauchy problem for the Helmholtz equation with a domain in with N cylindrical outlets to infinity with bounded inclusions in . Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Mazya proposed an alternating iterative method for solving Cauchy problems associated with elliptic, selfadjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R-2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters mu(0) and mu(1), the Robin-Dirichlet alternating iterative procedure is convergent.

    Place, publisher, year, edition, pages
    WALTER DE GRUYTER GMBH, 2023
    Keywords
    Helmholtz equation; Cauchy problem; inverse problem ill-posed problem
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-192481 (URN)10.1515/jiip-2020-0133 (DOI)000940871600001 ()
    Available from: 2023-03-21 Created: 2023-03-21 Last updated: 2024-03-18Bibliographically approved
    3. Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment
    Open this publication in new window or tab >>Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment
    2023 (English)In: Computational Methods in Applied Mathematics, ISSN 1609-4840, E-ISSN 1609-9389Article in journal (Refereed) Epub ahead of print
    Abstract [en]

    We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.

    Place, publisher, year, edition, pages
    WALTER DE GRUYTER GMBH, 2023
    Keywords
    Helmholtz Equation; Inverse Problem; Cauchy Problem; Ill-Posed Problem; Well-Posed Problem; Landweber Method
    National Category
    Fluid Mechanics and Acoustics
    Identifiers
    urn:nbn:se:liu:diva-196637 (URN)10.1515/cmam-2022-0244 (DOI)001035412500001 ()
    Available from: 2023-08-17 Created: 2023-08-17 Last updated: 2023-11-13
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  • 3.
    Achieng, Pauline
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Chepkorir, Jennifer
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations2021In: Bulletin of the Iranian Mathematical Society, ISSN 1735-8515, Vol. 47, p. 1681-1699Article in journal (Refereed)
    Abstract [en]

    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 k2, 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 k2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

    Download full text (pdf)
    fulltext
  • 4.
    Achieng, Pauline
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering. Univ Nairobi, Kenya.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Robin-Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain2023In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 31, no 5Article in journal (Refereed)
    Abstract [en]

    We consider the Cauchy problem for the Helmholtz equation with a domain in with N cylindrical outlets to infinity with bounded inclusions in . Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Mazya proposed an alternating iterative method for solving Cauchy problems associated with elliptic, selfadjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R-2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters mu(0) and mu(1), the Robin-Dirichlet alternating iterative procedure is convergent.

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