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Achieng, P., Berntsson, F. & Kozlov, V. (2023). Robin-Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain. Journal of Inverse and Ill-Posed Problems, 31(5)
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
Chepkorir, J., Berntsson, F. & Kozlov, V. (2023). Solving stationary inverse heat conduction in a thin plate. Partial Differential Equations and Applications, 4(6)
Open this publication in new window or tab >>Solving stationary inverse heat conduction in a thin plate
2023 (English)In: Partial Differential Equations and Applications, ISSN 2662-2971, Vol. 4, no 6Article in journal (Refereed) Published
Abstract [en]

We consider a steady state heat conduction problem in a thin plate. In the application, it is used to connect two cylindrical containers and fix their relative positions. At the same time it serves to measure the temperature on the inner cylinder. We derive a two dimensional mathematical model, and use it to approximate the heat conduction in the thin plate. Since the plate has sharp edges on the sides the resulting problem is described by a degenerate elliptic equation. To find the temperature in the interior part from the exterior measurements, we formulate the problem as a Cauchy problem for stationary heat equation. We also reformulate the Cauchy problem as an operator equation, with a compact operator, and apply the Landweber iteration method to solve the equation. The case of the degenerate elliptic equation has not been previously studied in this context. For numerical computation, we consider the case where noisy data is present and analyse the convergence.

Keywords
Cauchy problem, Stationary heat equation, Degenerate elliptic equation, Landweber iterative method
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-201113 (URN)10.1007/s42985-023-00267-7 (DOI)
Available from: 2024-02-22 Created: 2024-02-22 Last updated: 2024-02-22Bibliographically approved
Berntsson, F. & Wikström, P. (2023). Thermal tracking of a ladle during production cycles. International Journal for Computational Methods in Engineering Science and Mechanics, 24(6), 406-416
Open this publication in new window or tab >>Thermal tracking of a ladle during production cycles
2023 (English)In: International Journal for Computational Methods in Engineering Science and Mechanics, ISSN 1550-2287, E-ISSN 1550-2295, Vol. 24, no 6, p. 406-416Article in journal (Refereed) Published
Abstract [en]

Temperature control is important for the steel making process. Knowledge of the amount of thermal energy stored in the ladle allows for better predictions of the steel temperature during the process. This has a potential to improve the quality of the steel. In this work, we present a mathematical model of the heat transfer within a ladle during the production process. The model can be used to compute the current, and also the future, thermal status of the ladle. The model is simple and can be solved efficiently. We also present results from numerical simulations intended to illustrate the model.

Place, publisher, year, edition, pages
TAYLOR & FRANCIS INC, 2023
Keywords
Heat equation; steel making ladle; energy simulation; temperature control; industrial application
National Category
Metallurgy and Metallic Materials
Identifiers
urn:nbn:se:liu:diva-197874 (URN)10.1080/15502287.2023.2253255 (DOI)001058207600001 ()
Available from: 2023-09-19 Created: 2023-09-19 Last updated: 2024-04-09Bibliographically approved
Achieng, P., Berntsson, F., Chepkorir, J. & Kozlov, V. (2021). Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations. Bulletin of the Iranian Mathematical Society, 47, 1681-1699
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
Berntsson, F., Karlsson, M., Kozlov, V. & Nazarov, S. A. (2018). A Modification to the Kirchhoff Conditions at a Bifurcation and Loss Coefficients.
Open this publication in new window or tab >>A Modification to the Kirchhoff Conditions at a Bifurcation and Loss Coefficients
2018 (English)Report (Other academic)
Abstract [en]

One dimensional models for fluid flow in tubes are frequently used tomodel complex systems, such as the arterial tree where a large numberof vessels are linked together at bifurcations. At the junctions transmission conditions are needed. One popular option is the classic Kirchhoffconditions which means conservation of mass at the bifurcation andprescribes a continuous pressure at the joint.

In reality the boundary layer phenomena predicts fast local changesto both velocity and pressure inside the bifurcation. Thus it is not appropriate for a one dimensional model to assume a continuous pressure. In this work we present a modification to the classic Kirchhoff condi-tions, with a symmetric pressure drop matrix, that is more suitable forone dimensional flow models. An asymptotic analysis, that has beencarried out previously shows that the new transmission conditions hasen exponentially small error.

The modified transmission conditions take the geometry of the bifurcation into account and can treat two outlets differently. The conditions can also be written in a form that is suitable for implementationin a finite difference solver. Also, by appropriate choice of the pressuredrop matrix we show that the new transmission conditions can producehead loss coefficients similar to experimentally obtained ones.

Publisher
p. 11
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2018:5
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-147718 (URN)LiTH-MAT-R--2018/05--SE (ISRN)
Available from: 2018-05-07 Created: 2018-05-07 Last updated: 2018-05-07Bibliographically approved
Berntsson, F. & Ohlson, M. (2017). More on Estimation of Banded and Banded Toeplitz Covariance Matrices. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>More on Estimation of Banded and Banded Toeplitz Covariance Matrices
2017 (English)Report (Other academic)
Abstract [en]

In this paper we consider two different linear covariance structures, e.g., banded and bended Toeplitz, and how to estimate them using different methods, e.g., by minimizing different norms.

One way to estimate the parameters in a linear covariance structure is to use tapering, which has been shown to be the solution to a universal least squares problem. We know that tapering not always guarantee the positive definite constraints on the estimated covariance matrix and may not be a suitable method. We propose some new methods which preserves the positive definiteness and still give the correct structure.

More specific we consider the problem of estimating parameters of a multivariate normal p–dimensional random vector for (i) a banded covariance structure reflecting m–dependence, and (ii) a banded Toeplitz covariance structure.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. p. 12
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:12
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-141051 (URN)LiTH-MAT-R--2017/12--SE (ISRN)
Available from: 2017-09-25 Created: 2017-09-25 Last updated: 2017-10-06Bibliographically approved
Berntsson, F., Karlsson, M., Kozlov, V. & Nazarov, S. A. (2016). A one-dimensional model of viscous blood flow in an elastic vessel. Applied Mathematics and Computation, 274, 125-132
Open this publication in new window or tab >>A one-dimensional model of viscous blood flow in an elastic vessel
2016 (English)In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 274, p. 125-132Article in journal (Refereed) Published
Abstract [en]

In this paper we present a one-dimensional model of blood flow in a vessel segment with an elastic wall consisting of several anisotropic layers. The model involves two variables: the radial displacement of the vessels wall and the pressure, and consists of two coupled equations of parabolic and hyperbolic type. Numerical simulations on a straight segment of a blood vessel demonstrate that the model can produce realistic flow fields that may appear under normal conditions in healthy blood vessels; as well as flow that could appear during abnormal conditions. In particular we show that weakening of the elastic properties of the wall may provoke a reverse blood flow in the vessel. (C) 2015 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
ELSEVIER SCIENCE INC, 2016
Keywords
Blood flow; Linear model; Asymptotic analysis; Dimension reduction; Numerical simulation
National Category
Mathematics Mechanical Engineering
Identifiers
urn:nbn:se:liu:diva-124453 (URN)10.1016/j.amc.2015.10.077 (DOI)000367521900013 ()
Available from: 2016-02-02 Created: 2016-02-01 Last updated: 2017-11-30
Evarest, E., Berntsson, F., Singull, M. & Charles, W. (2016). Regime Switching models on Temperature Dynamics. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Regime Switching models on Temperature Dynamics
2016 (English)Report (Other academic)
Abstract [en]

Two regime switching models for predicting temperature dynamics are presented in this study for the purpose to be used for weather derivatives pricing. One is an existing model in the literature (Elias model) and the other is presented in this paper. The new model we propose in this study has a mean reverting heteroskedastic process in the base regime and a Brownian motion in the shifted regime. The parameter estimation of the two models is done by the use expectation-maximization (EM) method using historical temperature data. The performance of the two models on prediction of temperature dynamics is compared using historical daily average temperature data from five weather stations across Sweden. The comparison is based on the heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. The expected HDDs, CDDs and CAT of the models are compared to the true indices from the real data. Results from the expected HDDs, CDDs and CAT together with their corresponding daily average plots demonstrate that, our model captures temperature dynamics relatively better than Elias model.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. p. 24
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2016:12
Keywords
Weather derivatives, Regime switching, temperature dynamics, expectation-maximization, temperature indices
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-130586 (URN)LiTH-MAT-R--2016/12--SE (ISRN)
Available from: 2016-08-17 Created: 2016-08-17 Last updated: 2016-08-17Bibliographically approved
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.

Place, publisher, year, edition, pages
Elsevier, 2014
Keywords
Cauchy problem; alternating iterative method; conjugate gradient methods; inverse problem; ill–posed problem
National Category
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
National Category
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
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-2681-8965

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