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  • 1.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Kozlov, Vladimir A.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Mpinganzima, Lydie
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. University of Rwanda.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Numerical Solution of the Cauchy Problem for the Helmholtz Equation2014Report (Other academic)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem.

  • 2.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Mpinganzima, L.
    University of Rwanda, Rwanda.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation2017In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 73, no 1, p. 163-172Article in journal (Refereed)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation appears in various applications. The problem is severely ill-posed and regularization is needed to obtain accurate solutions. We start from a formulation of this problem as an operator equation on the boundary of the domain and consider the equation in (H-1/2)* spaces. By introducing an artificial boundary in the interior of the domain we obtain an inner product for this Hilbert space in terms of a quadratic form associated with the Helmholtz equation; perturbed by an integral over the artificial boundary. The perturbation guarantees positivity property of the quadratic form. This inner product allows an efficient evaluation of the adjoint operator in terms of solution of a well-posed boundary value problem for the Helmholtz equation with transmission boundary conditions on the artificial boundary. In an earlier paper we showed how to take advantage of this framework to implement the conjugate gradient method for solving the Cauchy problem. In this work we instead use the Conjugate gradient method for minimizing a Tikhonov functional. The added penalty term regularizes the problem and gives us a regularization parameter that can be used to easily control the stability of the numerical solution with respect to measurement errors in the data. Numerical tests show that the proposed algorithm works well. (C) 2016 Elsevier Ltd. All rights reserved.

  • 3.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Mpinganzima, Lydie
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation2014In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 68, no 1-2, p. 44-60Article in journal (Refereed)
    Abstract [en]

    In this paper we study the Cauchy problem for the Helmholtz equation. This problem appears in various applications and is severely ill–posed. The modified alternating procedure has been proposed by the authors for solving this problem but the convergence has been rather slow. We demonstrate how to instead use conjugate gradient methods for accelerating the convergence. The main idea is to introduce an artificial boundary in the interior of the domain. This addition of the interior boundary allows us to derive an inner product that is natural for the application and that gives us a proper framework for implementing the steps of the conjugate gradient methods. The numerical results performed using the finite difference method show that the conjugate gradient based methods converge considerably faster than the modified alternating iterative procedure studied previously.

  • 4.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Mpinganzima, Lydie
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    An alternating iterative procedure for the Cauchy problem for the Helmholtz equation2014In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 22, no 1, p. 45-62Article in journal (Refereed)
    Abstract [en]

    We present a modification of the alternating iterative method, which was introduced by V.A. Kozlov and V. Maz’ya in for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The method is implemented numerically using the finite difference method.

  • 5.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Mpinganzima, Lydie
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation2014Manuscript (preprint) (Other academic)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers in the Helmholtz equation. We prove here that if we alternate Robin and Dirichlet boundary conditions instead of Neumann and Dirichlet boundary conditions, then the algorithm will converge. We present also another algorithm based on the same idea, which converges for large wavenumbers. Numerical implementations obtained using the finite difference method are presented. Numerical results illustrate that the algorithms suggested in this paper, produce a convergent iterative sequences.

  • 6.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Mpinganzima, Lydie
    Univ Rwanda, Rwanda.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Robin-Dirichlet algorithms for the Cauchy problem for the Helmholtz equation2018In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 26, no 7, p. 1062-1078Article in journal (Refereed)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Mazya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann-Dirichlet iterations by the Robin-Dirichlet ones which repairs the convergence for less than the first Dirichlet-Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin-Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences.

  • 7.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Radosavljevic, Sonja
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Wennergren, Uno
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Biology. Linköping University, Faculty of Science & Engineering.
    Estimating effective boundaries of population growth in a variable environment2016In: Boundary Value Problems, ISSN 1687-2762, E-ISSN 1687-2770, p. 1-28, article id 2016:172Article in journal (Refereed)
    Abstract [en]

    We study the impact of age-structure and temporal environmental variability on the persistence of populations. We use a linear age-structured model with time-dependent vital rates. It is the same as the one presented by Chipot in (Arch. Ration. Mech. Anal. 82(1):13-25, 1983), but the assumptions on the vital rates are slightly different. Our main interest is in describing the large-time behavior of a population provided that we know its initial distribution and transient vital rates. Using upper and lower solutions for the characteristic equation, we define time-dependent upper and lower boundaries for a solution in a constant environment. Moreover, we estimate solutions for the general time-dependent case and also for a special case when the environment is changing periodically.

  • 8.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Thim, Johan
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    A Fixed Point Theorem in Locally Convex Spaces2010In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 61, no 2, p. 223-239Article in journal (Other academic)
    Abstract [en]

    For a locally convex space , where the topology is given by a familyof seminorms, we study the existence and uniqueness of fixed points for a mapping defined on some set . We require that there exists a linear and positive operator , acting on functions defined on the index set , such that for every

    Under some additional assumptions, one of which is the existence of a fixed point for the operator, we prove that there exists a fixed point of . For a class of elements satisfying as , we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points.We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudo-differential equations with nonlinear terms.

  • 9.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Thim, Johan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Single layer potentials on surfaces with small Lipschitz constants2014In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 418, no 2, p. 676-712Article in journal (Refereed)
    Abstract [en]

    This paper considers to the equation integral(S) U(Q)/vertical bar P - Q vertical bar(N-1) dS(Q) = F(P), P is an element of S, where the surface S is the graph of a Lipschitz function phi on R-N, which has a small Lipschitz constant. The integral on the left-hand side is the single layer potential corresponding to the Laplacian in RN+1. Let Lambda(r) be the Lipschitz constant of phi on the ball centered at the origin with radius 2r. Our analysis is carried out in local L-p-spaces and local Sobolev spaces, where 1 less than p less than infinity, and results are presented in terms of Lambda. Estimates of solutions to the equation are provided, which can be used to obtain knowledge about the behavior of the solutions near a point on the surface. These estimates are given in terms of seminorms. Solutions are also shown to be unique if they are subject to certain growth conditions. Local estimates are provided and some applications are supplied.

  • 10.
    Thim, Johan
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Riesz Potential Equations in Local Lp-spaces.2009In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 54, no 2, p. 125-151Article in journal (Refereed)
    Abstract [en]

    We consider the following equation for the Riesz potential of order one:

    Uniqueness of solutions is proved in the class of solutions for which the integral is absolutely convergent for almost every x. We also prove anexistence result and derive an asymptotic formula for solutions near the origin.Our analysis is carried out in local Lp-spaces and Sobolev spaces, which allows us to obtain optimal results concerning the class of right-hand sides and solutions. We also apply our results to weighted Lp-spaces and homogenous Sobolev spaces.

  • 11.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Nonlinear potential theory and weighted Sobolev spaces2000Book (Refereed)
    Abstract [en]

    The book systematically develops nonlinear potential theory and the Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincare inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities

  • 12.
    Larsson, Torbjörn (Editor)
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Ranjbar, Zohreh (Editor)
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, Faculty of Science & Engineering.
    Skoglund, Ingegerd (Editor)
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Turesson, Bengt-Ove (Editor)
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Lürén, Oskar (Photographer)
    Fotograf Oskar, Linköping.
    Verksamhetsberättelse 2010: Matematiska institutionen2010Report (Other academic)
  • 13.
    Skoglund, Ingegerd (Editor)
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Turesson, Bengt-Ove (Editor)
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Lürén, Oskar (Photographer)
    Fotograf Oskar, Linköping.
    Verksamhetsberättelse 2011: Matematiska institutionen2011Report (Other academic)
  • 14.
    Skoglund, Ingegerd (Editor)
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Turesson, Bengt-Ove (Editor)
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Holgersson, Peter (Photographer)
    Verksamhetsberättelse 2012: Matematiska institutionen2012Report (Other academic)
1 - 14 of 14
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