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Turesson, Bengt-Ove
##### Publications (8 of 8) Show all publications
Berntsson, F., Kozlov, V., Mpinganzima, L. & Turesson, B.-O. (2014). An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation. Computers and Mathematics with Applications, 68(1-2), 44-60
Open this publication in new window or tab >>An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation
2014 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 68, no 1-2, p. 44-60Article in journal (Refereed) Published
##### 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.

Elsevier, 2014
##### Keywords
Cauchy problem; alternating iterative method; conjugate gradient methods; inverse problem; ill–posed problem
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-105877 (URN)10.1016/j.camwa.2014.05.002 (DOI)000338816300004 ()
Available from: 2014-04-11 Created: 2014-04-11 Last updated: 2017-12-05Bibliographically approved
Berntsson, F., Kozlov, V., Mpinganzima, L. & Turesson, B.-O. (2014). An alternating iterative procedure for the Cauchy problem for the Helmholtz equation. Paper presented at 6th International Conference "Inverse Problems: Modeling and Simulation", 21-26 May 2012, Antalya, Turkey. Inverse Problems in Science and Engineering, 22(1), 45-62
Open this publication in new window or tab >>An alternating iterative procedure for the Cauchy problem for the Helmholtz equation
2014 (English)In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 22, no 1, p. 45-62Article in journal (Refereed) Published
##### 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.

##### Place, publisher, year, edition, pages
Taylor & Francis, 2014
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-77298 (URN)10.1080/17415977.2013.827181 (DOI)000328245900005 ()
##### Conference
6th International Conference "Inverse Problems: Modeling and Simulation", 21-26 May 2012, Antalya, Turkey
Available from: 2012-05-11 Created: 2012-05-11 Last updated: 2017-12-07Bibliographically approved
Berntsson, F., Kozlov, V. A., Mpinganzima, L. & Turesson, B.-O. (2014). Numerical Solution of the Cauchy Problem for the Helmholtz Equation. Linköping University Electronic Press
Open this publication in new window or tab >>Numerical Solution of the Cauchy Problem for the Helmholtz Equation
##### 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.

##### Place, publisher, year, edition, pages
Linköping University Electronic Press, 2014. p. 16
##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 2014:04
##### Keywords
Helmholtz equation, Cauchy Problem, Ill-Posed, Regularization, Numerical Methods.
##### National Category
Computational Mathematics Mathematics
##### Identifiers
urn:nbn:se:liu:diva-105707 (URN)LiTH-MAT-R--2014/04--SE (ISRN)
Available from: 2014-04-03 Created: 2014-04-03 Last updated: 2014-04-11Bibliographically approved
Berntsson, F., Kozlov, V., Mpinganzima, L. & Turesson, B.-O. (2014). Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation.
Open this publication in new window or tab >>Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation
##### 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.

Mathematics
##### Identifiers
urn:nbn:se:liu:diva-105876 (URN)
Available from: 2014-04-11 Created: 2014-04-11 Last updated: 2014-04-11Bibliographically approved
Kozlov, V., Thim, J. & Turesson, B.-O. (2014). Single layer potentials on surfaces with small Lipschitz constants. Journal of Mathematical Analysis and Applications, 418(2), 676-712
Open this publication in new window or tab >>Single layer potentials on surfaces with small Lipschitz constants
2014 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 418, no 2, p. 676-712Article in journal (Refereed) Published
##### 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.

Elsevier, 2014
##### Keywords
Single layer potential; Lipschitz surface; Local estimates
Natural Sciences
##### Identifiers
urn:nbn:se:liu:diva-108788 (URN)10.1016/j.jmaa.2014.04.013 (DOI)000336887700007 ()
Available from: 2014-07-07 Created: 2014-07-06 Last updated: 2017-12-05
Kozlov, V., Thim, J. & Turesson, B.-O. (2010). A Fixed Point Theorem in Locally Convex Spaces. Collectanea Mathematica (Universitat de Barcelona), 61(2), 223-239
Open this publication in new window or tab >>A Fixed Point Theorem in Locally Convex Spaces
2010 (English)In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 61, no 2, p. 223-239Article in journal (Other academic) Published
##### Abstract [en]

For a locally convex space $\mathscr{X}$, where the topology is given by a family$\{p( \, \cdot \, ; \alpha )\}_{\alpha \in \Omega}$of seminorms, we study the existence and uniqueness of fixed points for a mapping $\mathscr{K} : \, \mathscr{D_K} \rightarrow \mathscr{D_K}$ defined on some set $\mathscr{D_K} \subset \mathscr{X}$. We require that there exists a linear and positive operator $K$, acting on functions defined on the index set $\Omega$, such that for every $u,v \in \mathscr{D_K}$

$p(\mathscr{K}(u) - \mathscr{K}(v) \, ; \, \alpha ) \leq K(p(u-v \, ; \, \cdot \, )) (\alpha) \text{,} \qquad \qquad \alpha \in \Omega \text{.}$

Under some additional assumptions, one of which is the existence of a fixed point for the operator$K + p( \mathscr{K}(0) \, ; \, \cdot \,)$, we prove that there exists a fixed point of $\mathscr{K}$. For a class of elements satisfying$K^{n}(p(u \, ; \, \cdot \,)) (\alpha) \rightarrow 0$ as $n \rightarrow \infty$, 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.

##### Place, publisher, year, edition, pages
Universitat de Barcelona, 2010
##### Keywords
Fixed point theorem, Locally convex spaces, Ordinary differential equations, Pseudo-differential operators
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:liu:diva-16537 (URN)10.1007/BF03191243 (DOI)000277332400006 ()
Available from: 2009-01-31 Created: 2009-01-30 Last updated: 2017-12-14Bibliographically approved
Thim, J., Kozlov, V. & Turesson, B.-O. (2009). Riesz Potential Equations in Local Lp-spaces.. Complex Variables and Elliptic Equations, 54(2), 125-151
Open this publication in new window or tab >>Riesz Potential Equations in Local Lp-spaces.
2009 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 54, no 2, p. 125-151Article in journal (Refereed) Published
##### Abstract [en]

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

$\int_{\mathbf{R}^N} \frac{u(y) \, dy}{|x-y|^{N-1}} = f(x) \text{,} \qquad x \in \mathbf{R}^N \text{.}$

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.

##### Keywords
Riesz potentials, Singular integral operators
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:liu:diva-16496 (URN)10.1080/17476930802669728 (DOI)
Available from: 2009-01-31 Created: 2009-01-29 Last updated: 2017-12-14Bibliographically approved
Turesson, B.-O. (2000). Nonlinear potential theory and weighted Sobolev spaces. Springer Berlin/Heidelberg, 1736
Open this publication in new window or tab >>Nonlinear potential theory and weighted Sobolev spaces
2000 (English)Book (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

##### Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2000. p. 173
##### Series
Lecture Notes in Mathematics, ISSN 0075-8434 ; 1736
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-49484 (URN)9783540675884 (ISBN)9783540451686 (ISBN)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-14Bibliographically approved

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