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In this paper, we are interested in finding an optimal control support design for controlling vibrations due to pedestrian-bridge interactions. Therefore, we derive the topological derivatives of the proposed functionals using the averaged adjoint approach. A numerical algorithm initialized by these sensitivities is used as a solution strategy. The algorithm is tested numerically for two different cases of initial conditions., Funding Agencies|SIDA bilateral programme; Makerere University; [316]

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.

In this paper, an optimal actuator placement problem with a linear wave equation as the constraint is considered. In particular, this work presents the frameworks for finding the best location of actuators depending upon the given initial conditions, and where the dependence on the initial conditions is averaged out. The problem is motivated by the need to control vibrations induced by pedestrian-bridge interactions. An approach based on shape optimization techniques is used to solve the problem. Specifically, the shape sensitivities involving a cost functional are determined using the averaged adjoint approach. A numerical algorithm based on these sensitivities is used as a solution strategy. Numerical results are consistent with the theoretical results, in the two examples considered., Funding Agencies|SIDA bilateral programme (2015- 2022); Makerere University [316]

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 radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.

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.

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 work it has been found that a method based on approximating the time derivative by a Fourier series works well [Berntsson F. A spectral method for solving the sideways heat equation. Inverse Probl. 1999;15:891-906; Elden L, Berntsson F, Reginska T. Wavelet and Fourier methods for solving the sideways heat equation. SIAM J Sci Comput. 2000;21(6):2187-2205]. However, in our situation it is not resonable to assume that the temperature is periodic which means that additional techniques are needed to reduce the errors introduced by implicitly making the assumption that the solution is periodic in time. Thus, as an alternative approach, we instead approximate the time derivative by using a cubic smoothing spline. This means avoiding a periodicity assumption which leads to slightly smaller errors at the end points of the measurement interval. The spline method is also shown to satisfy similar stability estimates as the Fourier series method. Numerical simulations shows that both methods work well, and provide comparable accuracy, and also that the spline method gives slightly better results at the ends of the measurement interval., Funding Agencies|SIDA bilateral programme; Makerere University [316]

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 elliptic operators of second order, provided that appropriate Robin parameters were used. Also, it has been noted that the rate of convergence for the alternating iterative algorithm is quite slow. Thus, we reformulate the Cauchy problem as an operator equation and implement iterative methods based on Krylov subspaces. The aim is to achieve faster convergence. In particular, we consider the Landweber method, the conjugate gradient method and the generalized minimal residual method. The numerical results show that all the methods work well. In this work, we discuss also how one can approach non-symmetric differential operators by similar operator equations and model problems which are used for symmetric differential operators.

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 convergence that can be verified numerically and also an analytical convergence proof for an important special case. The application we have in mind is the computation of the time dependent blood flow in the arterial tree of the human body. The work presented here is for a simplified case but we discuss the extension of our work to the realistic cases. Also potential applications are discussed. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved., Funding Agencies|SIDA Phase IV Bilateral Program Project at Makerere University [NS-316-2014-A]

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.

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 the problem is non-linear certain assumptions on the coefficients are needed. We reformulate the problem as an non-linear operator equation and show that under suitable assumptions the operator is well-defined. The proof is based on making a change of variables and removing the non-linearity from the leading term of the equation. As a part of the proof we obtain an iterative procedure that is convergent and can be used for evaluating the operator. Numerical results show that the proposed procedure converges faster than a simple fixed point iteration for the equation in the the original variables.

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 gravity into account. Our model is obtained via a dimension reduction procedure based on the assumption of thinness of the vessel relative to its length. Results of numerical simulations are presented to highlight the influence of different factors on the blood flow. (C) 2018 Elsevier Inc. All rights reserved., Funding Agencies|Russian Foundation of Basic Research [18-01-00325]

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.

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 mathematical formulas for pricing futures and option contracts on heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) 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 use the Monte Carlo simulation method for pricing weather derivatives call option contracts.

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., Funding Agencies|Swedish International Development Cooperation Agency (Sida); University of Rwanda (UR) [51160027-02, 51160059-02]

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.

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, insulated by a ceramic material, which makes the cooling slower and the temperature distribution more even during the cooling process. The accuracy of the numerical model was validated by a comparison with temperature measurements, and also by a comparison with hot tensile tests for the specific cooling situation under investigation. The results show that the highest stress levels achieved during cooling did not exceed the mechanical strength of the material at the temperatures investigated.

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 electron–electron interactions in the potential.