The inverse geothermal problem consists of estimating the temperature distribution below the earth's surface using measurements on the surface. The problem is important since temperature governs a variety of geologic processes, including the generation of magmas and the deformation style of rocks. Since the thermal properties of rocks depend strongly on temperature the problem is non-linear.

The problem is formulated as an ill-posed operator equation, where the righthand side is the heat-flux at the surface level. Since the problem is ill-posed regularization is needed. In this study we demonstrate that Tikhonov regularization can be implemented efficiently for solving the operator equation. The algorithm is based on having a code for solving a well-posed problem related to the above mentioned operator. The algorithm is designed in such a way that it can deal with both 2D and 3D calculations.

Numerical results, for 2D domains, show that the algorithm works well and the inverse problem can be solved accurately with a realistic noise level in the surface data.

Keywords

Natural Sciences, Mathematics

BIBTEX

@article{diva2:1117637,
author = {Berntsson, Fredrik and Chen, Lin and Xu, Tao and Wokiyi, Dennis},
title = {{An efficient regularization method for a large scale ill-posed geothermal problem}},
journal = {Computers \& Geosciences},
year = {2017},
volume = {105},
pages = {1--9},
}

Fredrik Berntsson, Matts Karlsson, Vladimir Kozlov, Sergey A. Nazarov, "A one-dimensional model of a false aneurysm", International Journal of Research in Engineering and Science (IJRES), 5(6):61-73, 2017.

AbstractKeywordsBiBTeX

Abstract

A false aneurysm is a hematoma, i.e. collection ofblood outside of a blood vessel, that forms due to a hole in the wall of an artery . This represents a serious medical condition that needs to be monitored and, under certain conditions, treatedurgently. In this work a one-dimensional model of a false aneurysm isproposed. The new model is based on a one-dimensional model of anartery previously presented by the authors and it takes into accountthe interaction between the hematoma and the surrounding musclematerial. The model equations are derived using rigorous asymptoticanalysis for the case of a simplified geometry. Even though the model is simple it still supports a realisticbehavior for the system consisting of the vessel and the hematoma. Using numerical simulations we illustrate the behavior ofthe model. We also investigate the effect of changing the size of the hematoma. The simulations show that ourmodel can reproduce realistic solutions. For instance we show thetypical strong pulsation of an aneurysm by blood entering the hematoma during the work phase of the cardiac cycle, and the blood returning tothe vessel during the resting phase. Also we show that the aneurysmgrows if the pulse rate is increased due to, e.g., a higher work load.

Keywords

Natural Sciences

BIBTEX

@article{diva2:1116998,
author = {Berntsson, Fredrik and Karlsson, Matts and Kozlov, Vladimir and Nazarov, Sergey A.},
title = {{A one-dimensional model of a false aneurysm}},
journal = {International Journal of Research in Engineering and Science (IJRES)},
year = {2017},
volume = {5},
number = {6},
pages = {61--73},
}

Emanuel Evarest, Fredrik Berntsson, Martin Singull, Wilson M. Charles, "Regime Switching models on Temperature Dynamics", International Journal of Applied Mathematics and Statistics, 56(2), 2017.

AbstractKeywordsBiBTeX

Abstract

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.

Keywords

Weather derivatives, Regime switching, temperature dynamics, expectationmaximization, temperature indices, Natural Sciences, Computational Mathematics

BIBTEX

@article{diva2:1083775,
author = {Evarest, Emanuel and Berntsson, Fredrik and Singull, Martin and Charles, Wilson M.},
title = {{Regime Switching models on Temperature Dynamics}},
journal = {International Journal of Applied Mathematics and Statistics},
year = {2017},
volume = {56},
number = {2},
}

The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work, we use spline interpolation to construct approximate eigenfunctions of a linear operator using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. To demonstrate that the method gives useful error bounds, we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electronâ€“electron interactions in the potential.

Keywords

Energy levels, error estimation, graphene, linear operator, quantum dot, spectrum, Natural Sciences, Mathematical Analysis

BIBTEX

@article{diva2:1083772,
author = {Berntsson, Fredrik and Orlof, Anna and Thim, Johan},
title = {{Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots}},
journal = {Numerical Functional Analysis and Optimization},
year = {2017},
volume = {38},
number = {3},
pages = {293--305},
}

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.

@article{diva2:1078811,
author = {Berntsson, Fredrik and Kozlov, Vladimir and Mpinganzima, L. and Turesson, Bengt-Ove},
title = {{Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation}},
journal = {Computers and Mathematics with Applications},
year = {2017},
volume = {73},
number = {1},
pages = {163--172},
}

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.

Keywords

Blood flow; Linear model; Asymptotic analysis; Dimension reduction; Numerical simulation, Natural Sciences, Engineering and Technology

BIBTEX

@article{diva2:899859,
author = {Berntsson, Fredrik and Karlsson, Matts and Kozlov, Vladimir and Nazarov, Sergey A.},
title = {{A one-dimensional model of viscous blood flow in an elastic vessel}},
journal = {Applied Mathematics and Computation},
year = {2016},
volume = {274},
pages = {125--132},
}

In this paper, we introduce the concept of parameter identification problems, which are inverse problems, as a methodology to inpainting. More specifically, as a first study in this new direction, we generalize the method of harmonic inpainting by studying an elliptic equation in divergence form where we assume that the diffusion coefficient is unknown. As a first step, this unknown coefficient is determined from the information obtained by the known data in the image. Next, we fill in the region with missing data by solving an elliptic equation in divergence form using this obtained diffusion coefficient. An error analysis shows that this approach is promising and our numerical experiments produces better results than the harmonic inpainting.

@article{diva2:749467,
author = {Berntsson, Fredrik and Baravdish, George},
title = {{Coefficient identification in PDEs applied to image inpainting}},
journal = {Applied Mathematics and Computation},
year = {2014},
volume = {242},
pages = {227--235},
}

The eastern Tibetan margin bordered by the Longmen Shan range exhibits significant lateral differences in the lithospheric structure and thermal state. To investigate the roles of these differences in mountain building, we construct a thermo-rheological model along a wide-angle seismic profile across the eastern Tibetan margin based on recent seismic and thermal observations. The thermal modeling is constrained by the surface heat flow data and crustal P wave velocity model. The construction of the theological envelopes is based on rock mechanics results, and involves two types of rheology: a weak case where the lower crust is felsic granulite and the lithospheric mantle is wet peridotite, and a strong case where the lower crust is mafic granulite and the lithospheric mantle is dry peridotite. The results demonstrate: (1) one high-temperature anomaly exists within the uppermost mantle beneath eastern Tibet, indicating that the crust in eastern Tibet is remarkably warmer than that in the Sichuan basin, and (2) the rheological strength of the lithospheric mantle beneath eastern Tibet is considerably weaker than that beneath the Sichuan basin. The rheological profiles are in accord with seismicity distribution. By combining these results with the observed crustal/lithospheric architecture, Pn velocity distribution and magmatism in the eastern Tibetan margin, we suggest that the delamination of a thickened lithospheric mantle root beneath eastern Tibet is responsible for the growth of the eastern Tibetan margin.

@article{diva2:739440,
author = {Chen, Lin and Berntsson, Fredrik and Zhang, Zhongjie and Wang, Peng and Wu, Jing and Xu, Tao},
title = {{Seismically constrained thermo-rheological structure of the eastern Tibetan margin:
Implication for lithospheric delamination}},
journal = {Tectonophysics},
year = {2014},
volume = {627},
pages = {122--134},
}

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.

@article{diva2:711804,
author = {Berntsson, Fredrik and Kozlov, Vladimir and Mpinganzima, Lydie and Turesson, Bengt-Ove},
title = {{An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation}},
journal = {Computers and Mathematics with Applications},
year = {2014},
volume = {68},
number = {1-2},
pages = {44--60},
}

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.

Keywords

Natural Sciences

BIBTEX

@article{diva2:526240,
author = {Berntsson, Fredrik and Kozlov, Vladimir and Mpinganzima, Lydie and Turesson, Bengt-Ove},
title = {{An alternating iterative procedure for the Cauchy problem for the Helmholtz equation}},
journal = {Inverse Problems in Science and Engineering},
year = {2014},
volume = {22},
number = {1},
pages = {45--62},
}

A shielded thermocouple is a measurement device used for monitoring the temperature in chemically, or mechanically, hostile environments. The sensitive parts of the thermocouple are protected by a shielding layer. In order to improve the accuracy of the measurement device, we study an inverse heat conduction problem where the temperature on the surface of the shielding layer is sought, given measured temperatures in the interior of the thermocouple. The procedure is well suited for real-time applications where newly collected data is continuously used to compute current estimates of the surface temperature. Mathematically we can formulate the problem as a Cauchy problem for the heat equation, in cylindrical coordinates, where data is given along the line r = r_{1} and the solution is sought at r_{1} < r ≤ r_{2}. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. Thus, regularization techniques are needed. The ill–posedness of the problem is analyzed and a numerical method is proposed. Numerical experiments demonstrate that the proposed method works well.

Keywords

Natural Sciences

BIBTEX

@article{diva2:533318,
author = {Berntsson, Fredrik},
title = {{An Inverse Heat Conduction Problem and Improving Shielded Thermocouple Accuracy}},
journal = {Numerical Heat Transfer, Part A Applications},
year = {2012},
volume = {61},
number = {10},
pages = {754--763},
}

@article{diva2:269417,
author = {Wikstrom, P. and Blasiak, W. and Berntsson, Fredrik},
title = {{Estimation of the transient surface temperature and heat flux of a steel slab using an inverse method}},
journal = {Applied Thermal Engineering},
year = {2007},
volume = {27},
number = {14-15},
pages = {2463--2472},
}

In the steel industry it is of great importance to be able to control the surface temperature and heating or cooling rates during heat treatment processes. In this paper, a steel slab is heated up to 1300 degrees C in an industrial reheating furnace and the temperature data are recorded during the reheating process. The transient local surface temperature, heat flux and effective heat transfer coefficient of the steel slab ares calculated using a model for inverse heat conduction. The calculated surface temperatures are compared with the temperatures achieved by using a model of the heating process with the help of the software STEELTEMP (R) 2D. The results obtained show very good agreement and suggest that the inverse method can be applied to similar high temperature applications with very good accuracy.

@article{diva2:268926,
author = {Wikstrom, Patrik and Blasiak, Wlodzimierz and Berntsson, Fredrik},
title = {{Estimation of the transient surface temperature, heat flux and effective heat transfer coefficient of a slab in an industrial reheating furnace by using an inverse method}},
journal = {STEEL RESEARCH INTERNATIONAL},
year = {2007},
volume = {78},
number = {1},
pages = {63--70},
}

@article{diva2:250157,
author = {Elden, Lars and Berntsson, Fredrik},
title = {{A stability estimate for a Cauchy problem for an elliptic partial differential equation}},
journal = {Inverse Problems},
year = {2005},
volume = {21},
number = {5},
pages = {1643--1653},
}

The sideways heat equation is a one-dimensional model of a problem, where one wants to determine the temperature on the surface of a body using interior measurements. More precisely, we consider a heat conduction problem, where temperature and heat-flux data are available along the line x = 1 and the solution is sought in the interval 0 = x < 1. The problem is ill-posed in the sense that the solution does not depend continuously on the data. Stability can be restored by replacing the time derivative in the heat equation with a bounded spectral approximation. The cut off level in the spectral approximation acts as a regularization parameter, that controls the degree of smoothness in the solution. In certain applications one wants to solve the sideways heat equation in real time, i.e. to constantly update the solution as new measurements are recorded. For this case sequential solution methods are required.

Keywords

Ill-posed, Sequential method, Sideways heat equation, Engineering and Technology

BIBTEX

@article{diva2:267579,
author = {Berntsson, Fredrik},
title = {{Sequential solution of the sideways heat equation by windowing of the data}},
journal = {Inverse Problems in Engineering},
year = {2003},
volume = {11},
number = {2},
pages = {91--103},
}

We consider an inverse problem for the two-dimensional steady-state heat equation. More precisely, the heat equation is valid in a domain O, that is a subset of the unit square. Temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and 1 are assumed to be insulated. From these we wish to determine the temperature in the domain O. Furthermore, a part of the boundary ?O is considered to be unknown, and must also be determined. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least-squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equations, which can be solved using standard numerical methods. The inverse problem that we consider in this paper arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an inverse heat conduction problem.

Keywords

Engineering and Technology

BIBTEX

@article{diva2:267746,
author = {Berntsson, Fredrik},
title = {{Boundary identification for an elliptic equation}},
journal = {Inverse Problems},
year = {2002},
volume = {18},
number = {6},
pages = {1579--1592},
}

@article{diva2:416902,
author = {Berntsson, Fredrik},
title = {{A survey of methods for determinig surface temperatures using interior measurements}},
journal = {Trends in Heat, Mass \& Momentum Transfer},
year = {2001},
volume = {7},
number = {pp},
pages = {105--128},
}

We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.

Keywords

Engineering and Technology

BIBTEX

@article{diva2:268192,
author = {Berntsson, Fredrik and Eld\'{e}n, Lars},
title = {{Numerical solution of a Cauchy problem for the Laplace equation}},
journal = {Inverse Problems},
year = {2001},
volume = {17},
number = {4},
pages = {839--853},
}

We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 ≤ x < 1.

The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge–Kutta method.

We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.

Keywords

Engineering and Technology

BIBTEX

@article{diva2:268556,
author = {Elden, Lars and Berntsson, Fredrik and Reginska, Teresa},
title = {{Wavelet and Fourier methods for solving the sideways heat equation}},
journal = {SIAM Journal on Scientific Computing},
year = {2000},
volume = {21},
number = {6},
pages = {2187--2205},
}

We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using interior measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data are given along the line x = 1, and a solution is sought in the interval 0 ≤ x < 1.

The problem is ill-posed, in the sense that the solution does not depend continuously on the data. Continuous dependence of the data is restored by replacing the time derivative in the heat equation with a bounded spectral-based approximation. The cut-off level in the spectral approximation acts as a regularization parameter. Error estimates for the regularized solution are derived and a procedure for selecting an appropriate regularization parameter is given. The discretized problem is an initial value problem for an ordinary differential equation in the space variable, which can be solved using standard numerical methods, for example a Runge-Kutta method. As test problems we take equations with constant and variable coefficients.

Keywords

Natural Sciences

BIBTEX

@article{diva2:416914,
author = {Berntsson, Fredrik},
title = {{A spectral method for solving the sideways heat equation}},
journal = {Inverse Problems},
year = {1999},
volume = {15},
number = {4},
pages = {891--906},
}

We consider two dimensional inverse steady state heat conductionproblems in complex geometries. The coefficients of the ellipticequation are assumed to be non-constant. Cauchy data are given on onepart of the boundary and we want to find the solution in the wholedomain. The problem is ill--posed in the sense that the solution doesnot depend continuously on the data.Using an orthogonal coordinate transformation the domain is mappedonto a rectangle. The Cauchy problem can then be solved by replacingone derivative by a bounded approximation. The resulting well--posed problem can then be solved by a method of lines. A bounded approximation of the derivative can be obtained by differentiating a cubic spline, that approximate the function in theleast squares sense. This particular approximation of the derivativeis computationally efficient and flexible in the sense that its easyto handle different kinds of boundary conditions.This inverse problem arises in iron production, where the walls of amelting furnace are subject to physical and chemical wear. Temperatureand heat--flux data are collected by several thermocouples locatedinside the walls. The shape of the interface between the molten ironand the walls can then be determined by solving an inverse heatconduction problem. In our work we make extensive use of Femlab for creating testproblems. By using Femlab we solve relatively complex model problemsfor the purpose of creating numerical test data used for validatingour methods. For the types of problems we are intressted in numericalartefacts appear, near corners in the domain, in the gradients that Femlab calculates. We demonstrate why this happen and also how we deal with the problem.

@inproceedings{diva2:416937,
author = {Berntsson, Fredrik and Eld\'{e}n, Lars},
title = {{Numerical Solution of an Inverse Steady State Heat Conduction Problem}},
booktitle = {International Symposium on Inverse Problems In Engineering Mechanics, Nagano, Japan, March 2000},
year = {2000},
}

We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem where one wants to determine the temperature on the surface of a body using internal measurements. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We discuss the nature of the ill-posedness as well as methods for restoring stability with respect to measurement errors.

Successful heat treatment requires good control of the temperature and cooling rates during the process. In an experiment a aluminium block, of the alloy AA7010, was cooled rapidly by spraying water on one surface. Thermocouples inside the block recorded the temperature, and we demonstrate that it is possible to find the temperature distribution in the region between the thermocouple and the surface, by solving numerically the sideways heat equation.

Keywords

Natural Sciences

BIBTEX

@inproceedings{diva2:416928,
author = {Berntsson, Fredrik and Eld\'{e}n, Lars},
title = {{An inverse heat conduction problem and an application to heat treatment of aluminium}},
booktitle = {Inverse Problems in Engineering Mechanics II},
year = {2000},
pages = {99--106},
}

@inproceedings{diva2:417797,
author = {Berntsson, Fredrik},
title = {{Simulation Tools for Injection Moulding}},
booktitle = {Nordic Matlab Conference, Stockholm, October 1997},
year = {1997},
}

@inproceedings{diva2:417793,
author = {Berntsson, Fredrik and Eld\'{e}n, Lars and Loyd, Dan and Garcia-Padrón, Ricardo},
title = {{A Comparison of Three Numerical Methods for an Inverse Heat Conduction Problem and an Industrial Application}},
booktitle = {Numerical Methods in Thermal Problems Conference, University of Wales, Swansea},
year = {1997},
}

In many industrial applications one wishes to determine the temperature history on the surface of a body, where the surface itself is inaccessible for measurements. The sideways heat equation is a model of this situation. In a one-dimensional setting this is formulated mathematically as a Cauchy problem for the heat equation, where temperature and heat--flux data are available along the line x=1, and a solution is sought for 0 ≤ x< 1. This problem is ill-posed in the sense that the solution does not depend continuously on the data. Stability can be restored by replacing the time derivative in the heat equation by a bounded approximation. We consider both spectral and wavelet approximations of the derivative. The resulting problem is a system of ordinary differential equations in the space variable, that can be solved using standard methods, e.g. a Runge-Kutta method. The methods are analyzed theoretically, and error estimates are derived, that can be used for selecting the appropriate level of regularization. The numerical implementation of the proposed methods is discussed. Numerical experiments demonstrate that the proposed methods work well, and can be implemented efficiently. Furthermore, the numerical methods can easily be adapted to solve problems with variable coefficients, and also non-linear equations. As test problems we take model equations, with constant and variable coefficients. Also, we solve problems from applications, with actual measured data.

Inverse problems for the stationary heat equation are also discussed. Suppose that the Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are given on the outer boundary, and we wish to compute the steady state temperature on the inner boundary. A standard approach is to discretize the equation by finite differences, and use Tikhonov's method for stabilizing the discrete problem, which leads to a large sparse least squares problem. Alternatively, we propose to use a conformal mapping to transform the domain into an annulus, where the equivalent problem can be solved using separation of variables. The ill-posedness is dealt with by filtering away high frequencies from the solution. Numerical results using both methods are presented. A closely related problem is that of determining the stationary temperature inside a body, from temperature and heat-flux measurements on a part of the boundary. In practical applications it is sometimes the case that the domain, where the differential equation is valid, is partly unknown. In such cases we want to determine not only the temperature, but also the shape of the boundary of the domain. This problem arises, for instance, in iron production, where the walls of a melting furnace is subject to both physical and chemical wear. In order to avoid a situation where molten metal breaks out through the walls the thickness of the walls should be constantly monitored. This is done by solving an inverse problem for the stationary heat equation, where temperature and heat-flux data are available at certain locations inside the walls of the furnace. Numerical results are presented also for this problem.

In this study we discuss the pricing of weather derivatives whose underlying weather variable is temperature. The dynamics of temperature in this study follows a two state regime switching model with a heteroskedastic mean reverting process as the base regime and a shifted regime defined by Brownian motion with mean different from zero. We develop the mathematical formulas for pricing futures contract on heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. We also present the mathematical expressions for pricing the corresponding options on futures contracts for the same temperature indices. The local volatility nature of the model in the base regime captures very well the dynamics of the underlying process, thus leading to a better pricing processes for temperature derivatives contracts written on various index variables. We provide the description of Montecarlo simulation method for pricing weather derivatives under this model and use it to price a few weather derivatives call option contracts.

Keywords

Weather derivatives, Arbitrage-free pricing, Regime switching, Monte Carlo Simulation, Option Pricing, Natural Sciences

BIBTEX

@techreport{diva2:1082101,
author = {Evarest Sinkwembe, Emanuel and Berntsson, Fredrik and Singull, Martin and Yang, Xiangfeng},
title = {{Weather derivatives pricing using regim switching models}},
institution = {Linköping University, Department of Electrical Engineering},
year = {2017},
type = {Other academic},
number = {LiTH-MAT-R, 2017/04},
address = {Sweden},
}

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.

Keywords

Weather derivatives, Regime switching, temperature dynamics, expectation-maximization, temperature indices, Natural Sciences

BIBTEX

@techreport{diva2:953299,
author = {Evarest, Emanuel and Berntsson, Fredrik and Singull, Martin and Charles, Wilson},
title = {{Regime Switching models on Temperature Dynamics}},
institution = {Linköping University, Department of Electrical Engineering},
year = {2016},
type = {Other academic},
number = {LiTH-MAT-R, 2016:12},
address = {Sweden},
}

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.

@techreport{diva2:709791,
author = {Berntsson, Fredrik and Kozlov, Vladimir A. and Mpinganzima, Lydie and Turesson, Bengt-Ove},
title = {{Numerical Solution of the Cauchy Problem for the Helmholtz Equation}},
institution = {Linköping University, Department of Electrical Engineering},
year = {2014},
type = {Other academic},
number = {LiTH-MAT-R, 2014:04},
address = {Sweden},
}

The thermal conductivity properties of a material can be determined experimentally by using temperature measurements taken at specified locations inside the material. We examine a situation where the properties of a (previously known) material changed locally. Mathematically we aim to find the coefficient k(x) in the stationary heat equation (kTx)x = 0;under the assumption that the function k(x) can be parametrized using only a few degrees of freedom.

The coefficient identification problem is solved using a least squares approach; where the (non-linear) control functional is weighted according to the distribution of the measurement locations. Though we only discuss the 1D case the ideas extend naturally to 2D or 3D. Experimentsdemonstrate that the proposed method works well.

Keywords

Natural Sciences

BIBTEX

@techreport{diva2:412732,
author = {Berntsson, Fredrik and Mpinganzima, Lydie},
title = {{A Data Assimilation Approach to Coefficient Identification}},
institution = {Linköping University, Department of Electrical Engineering},
year = {2011},
type = {Other academic},
number = {LiTH-MAT-R, 2011:9},
address = {Sweden},
}

We consider an inverse problem for the two dimensional steady state heat equation. More precisely, the heat equation is valid in a domain Ω, that is a subset of the unit square, temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and x = 1 are assumed to be insulated. From these we wish to determine the temperature in the domain Ω. Furthermore, a part of the boundry ∂Ω is considered to be unknown, and must also be determined.

The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equation, which can be solved using standard numerical methods.

The inverse problem, that we consider in this paper, arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an invers heat conduction problem.

Keywords

Engineering and Technology

BIBTEX

@techreport{diva2:605787,
author = {Berntsson, Fredrik},
title = {{Boundary identification for an elliptic equation}},
institution = {Linköping University, Department of Electrical Engineering},
year = {2001},
type = {Other academic},
number = {LiTH-MAT-R, 23},
address = {Sweden},
}

Numerical procedures for solving a non-Characteristic Cauchy problem for the heat equation are discussed. More precisely we consider a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted 0 ≤ x <1. The problem is often referred to as the sideways heat equation.

The problem is analyzed, using both Fourier analysis and the singular value decomposition, and is found to be severely ill-posed. The literature is vast, and many authors have proposed numerical methods that regularize the IHCP. In this paper we attempt to give an overview that covers the most popular methods that have been considered.

Numerical examples that illustrate the numerical algorithms are given.

Keywords

Engineering and Technology

BIBTEX

@techreport{diva2:605779,
author = {Berntsson, Fredrik},
title = {{Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation}},
institution = {Linköping University, Department of Electrical Engineering},
year = {2001},
type = {Other academic},
number = {LiTH-MAT-R, 17},
address = {Sweden},
}

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