Fredrik Berntsson

Linköping University | LiU · Department of Mathematics (MAI)

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 t...

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 radia...

We consider the Cauchy problem for the Helmholtz equation with a domain in ℝ d {\mathbb{R}^{d}} , d ≥ 2 {d\geq 2} with N cylindrical outlets to infinity with bounded inclusions in ℝ d - 1 . {\mathbb{R}^{d-1}.} 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...

We consider the Cauchy problem for the Helmholtz equation with a domain in R^d, d>2 with N cylindrical outlets to infinity with bounded inclusions in R^{d-1}. 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 Maz'ya proposed an alternating ite...

The Cauchy problem for Helmholtz equation, for moderate wave number $k^{2}$, is considered. In the previous paper of Achieng et al. (2020, Analysis of Dirichlet–Robin iterations for solving the Cauchy problem for elliptic equations. Bull. Iran. Math. Soc.), a proof of convergence for the Dirichlet–Robin alternating algorithm was given for general e...

We consider the Cauchy problem for the Helmholtz equation defined in a rectangular domain. The Cauchy data are prescribed on a part of the boundary and the aim is to find the solution in the entire domain. The problem occurs in applications related to acoustics and is illposed in the sense of Hadamard. In our work we consider regularizing the probl...

In many industrial applications, such as in the steel industry, it is important to monitor the surface temperature, and also heating or cooling rates, during an industrial process. However, often the surface may be inaccessible for direct measurements. Instead we are restricted to using interior measurements to estimate the surface temperature, by...

The Cauchy problem for the heat equation is a model of situation where one seeks to compute the temperature, or heat-flux, at the surface of a body by using interior measurements. The problem is well-known to be ill-posed, in the sense that measurement errors can be magnified and destroy the solution, and thus regularization is needed. In previous...

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 b...

In this work we propose an algorithm for computing a stationary flow in a bifurcation tree. Our idea is to divide the tree into smaller basic blocks, each corresponding to one bifurcation, and solve a sequence of flow problems for the individual blocks. Numerical experiments demonstrate that the algorithm works well. We give a criteria for converge...

We study a non-linear operator equation originating from a Cauchy problem for an elliptic equation. The problem appears in applications where surface measurements are used to calculate the temperature below the earth surface. The Cauchy problem is ill-posed and small perturbations to the used data can result in large changes in the solution. Since...

We present a one-dimensional model describing the blood flow through a moderately curved and elastic blood vessel. We use an existing two dimensional model of the vessel wall along with Navier−Stokes equations to model the flow through the channel while taking factors, namely, surrounding muscle tissue and presence of external forces other than gra...

Abstract In this study, a two-dimensional linear transition inverse heat conduction problem (IHCP) was solved using the Generalized Minimal Residual Method (GMRES) in quenching process by water jets. The inverse solution method was validated by set of artificial data and solution sensitivity analysis was done on data noise level, regularization par...

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 nonzero drift. We develop mathematica...

Subsequent cooling of slabs after continuous casting is an important step to control before further processing such as hot-rolling. In this work the stresses that occur in slabs during the cooling process were investigated by using a finite element model. In the specific situation that we model a stack of slabs are left to cool under a hood, insula...

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 k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence...

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 p...

A false aneurysm is a hematoma, i.e. collection of blood 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, treated urgently. In this work a one-dimensional model of a false aneurysm is proposed. The new model is based on...

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 strong...

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 by using th...

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 reg...

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 spaces. By introducing an artificial boundary in th...

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 vessel's wall and the pressure, and consists of two coupled equations of parabolic and hyperbolic type. Numerical simulations on a straight...

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 k in the Helmholtz equation. Here we present some simple modifications of the algorithm which may restore the convergence....

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 unk...

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 accelera...

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 seism...

We present a modification of the alternating iterative method, which was introduced by Kozlov and Maz’ya, for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The reason for this modification is that the standard alternating iterative algorithm does not always converge for the Cauchy problem for the Helmholtz equation. T...

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 shap...

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 this work we use numerical methods to study the accuracy and dynamic properties of a shielded thermocouple design. Also, we show that...

In this paper we use the convection-diffusion equation to model the transport of pollutant material through the atmosphere. Such models have a wide range of applications such as predicting the environmental impact from new polluting industrial plants. In our study we solve the convection-diffusion equation in a two dimensional setting using the Cra...

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 sur...

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. An experiment was performed in which a steel slab was heated up to 1250 °C in a fuel fired test furnace. The transient surface temperature and heat flux of a steel slab is calculated using a mo...

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°C in an industrial reheating furnace and the temperature data are recorded during the reheating process. The transient local surface temperature,...

A two-dimensional inverse steady state heat conduction problem in the unit square is considered. Cauchy data are given for y = 0, and boundary data are for x = 0 and x = 1. The elliptic operator is self-adjoint with non-constant, smooth coefficients. The solution for y = 1 is sought. This Cauchy problem is ill-posed in an L2-setting. A stability fu...

We consider two dimensional inverse steady state heat conduction problems in complex geometries. The coefficients of the elliptic equation are assumed to be non-constant. Cauchy data are given on one part of the boundary and we want to find the solution in the whole domain. The problem is ill--posed in the sense that the solution does not depend co...

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 h x < 1. Th...

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 1 are assumed to be insulated. From these we wish to determine the temperatu...

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...

Numerical procedures for solving an inverse heat conduction problem 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...

In several industrial applications, it is desirable to determine the temperature on the surface of a body where the surface itself is inaccessible for measurements. Another reason is that locating a measurement device on the surface would disturb the measurements and an incorrect temperature is recorded. In such cases, one is restricted to internal...

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...

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 i...

: In this paper the concern is certain simulation tools for the filling phase of injection moulding of plastics. A mathematical model, called the distance model, of the injection moulding process has been developed by Professor Gunnar Aronsson at the University of Linkoping. From this model one can find the filling pattern (i.e. flow fronts) of the...

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 internal measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data is given along the line x = 1, and a solution is...

In many industrial applications, one wishes to determine the temperature on the surface of a body where the surface itself is inaccessible for measurements. This chapter considers an inverse heat conduction problem, the Sideways Heat Equation, which is a model of a problem, where the temperature on both sides of a wall has to be determined, but whe...

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. We illustrate this in Figure 0.1. Mathematically it can be formulated as a Cauchy problem for the heat equation in...

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...

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 can be formulated as a Cauchy problem for the heat equation in the quarter plane, with data given...

The thermal conductivity properties of a material can be determined experimentally by using temperature measurements taken at specified lo-cations 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)...