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  • 1.
    Abdulle, Assyr
    et al.
    ANMC, EPFL.
    Cohen, David
    Institut für Angewandte und Numerische Mathematik, KIT.
    Vilmart, Gilles
    ANMC, EPFL.
    Konstantinos, Zygalakis
    ANMC, EPFL.
    High weak order methods for stochastic differential equations based on modified equations2012In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 34, no 3, p. A1800-A1823Article in journal (Refereed)
    Abstract [en]

    Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (mean-square stable) stochastic problems, and implicit integrators that exactly conserve all quadratic firstintegrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.

  • 2.
    Adlerborn, Björn
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Karlsson, Lars
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Distributed one-stage Hessenberg-triangular reduction with wavefront scheduling2018In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 40, no 2, p. C157-C180Article in journal (Refereed)
    Abstract [en]

    A novel parallel formulation of Hessenberg-triangular reduction of a regular matrix pair on distributed memory computers is presented. The formulation is based on a sequential cacheblocked algorithm by K degrees agstrom et al. [BIT, 48 (2008), pp. 563 584]. A static scheduling algorithm is proposed that addresses the problem of underutilized processes caused by two-sided updates of matrix pairs based on sequences of rotations. Experiments using up to 961 processes demonstrate that the new formulation is an improvement of the state of the art and also identify factors that limit its scalability.

  • 3.
    Adlerborn, Björn
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Karlsson, Lars
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Distributed one-stage Hessenberg-triangular reduction with wavefront scheduling2016Report (Other academic)
    Abstract [en]

    A novel parallel formulation of Hessenberg-triangular reduction of a regular matrix pair on distributed memory computers is presented. The formulation is based on a sequential cache-blocked algorithm by Kågstrom, Kressner, E.S. Quintana-Ortí, and G. Quintana-Ortí (2008). A static scheduling algorithm is proposed that addresses the problem of underutilized processes caused by two-sided updates of matrix pairs based on sequences of rotations. Experiments using up to 961 processes demonstrate that the new algorithm is an improvement of the state of the art but also identifies factors that currently limit its scalability.

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  • 4.
    Adlerborn, Björn
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kressner, Daniel
    SB–MATHICSE–ANCHP, EPF Lausanne.
    PDHGEQZ user guide2015Report (Other academic)
    Abstract [en]

    Given a general matrix pair (A,B) with real entries, we provide software routines for computing a generalized Schur decomposition (S, T). The real and complex conjugate pairs of eigenvalues appear as 1×1 and 2×2 blocks, respectively, along the diagonals of (S, T) and can be reordered in any order. Typically, this functionality is used to compute orthogonal bases for a pair of deflating subspaces corresponding to a selected set of eigenvalues. The routines are written in Fortran 90 and targets distributed memory machines.

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  • 5.
    Agvik, Simon
    Umeå University, Faculty of Science and Technology, Department of Physics.
    A deformable terrain model in multi-domain dynamics using elastoplastic constraints: An adaptive approach2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Achieving realistic simulations of terrain vehicles in their work environment does not only require a careful model of the vehicle itself but the vehicle's interactions with the surroundings are equally important. For off-road ground vehicles the terrain will heavily affect the behaviour of the vehicle and thus puts great demands on the terrain model.

    The purpose of this project has been to develop and evaluate a deformable terrain model, meant to be used in real-time simulations with multi-body dynamics. The proposed approach is a modification of an existing elastoplastic model based on linear elasticity theory and a capped Drucker-Prager model, using it in an adaptive way. The original model can be seen as a system of rigid bodies connected by elastoplastic constraints, representing the terrain. This project investigates if it is possible to create dynamic bodies just when it is absolutely necessary, and store information about possible deformations in a grid.

    Two methods used for transferring information between the dynamic bodies and the grid have been evaluated; an interpolating approach and a discrete approach. The test results indicate that the interpolating approach is preferable, with better stability to an equal performance cost. However, stability problems still exist that have to be solved if the model should be useful in a commercial product.

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  • 6.
    Ahlkrona, Josefin
    et al.
    Department of Mathematics, Stockholm University, Stockholm, Sweden; Swedish e-Science Research Centre (SeRC), Stockholm, Sweden.
    Elfverson, Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A cut finite element method for non-Newtonian free surface flows in 2D: application to glacier modelling2021In: Journal of Computational Physics: X, E-ISSN 2590-0552, Vol. 11, article id 100090Article in journal (Refereed)
    Abstract [en]

    In ice sheet and glacier modelling, the Finite Element Method is rapidly gaining popularity. However, constructing and updating meshes for ice sheets and glaciers is a non-trivial and computationally demanding task due to their thin, irregular, and time dependent geometry. In this paper we introduce a novel approach to ice dynamics computations based on the unfitted Finite Element Method CutFEM, which lets the domain boundary cut through elements. By employing CutFEM, complex meshing and remeshing is avoided as the glacier can be immersed in a simple background mesh without loss of accuracy. The ice is modelled as a non-Newtonian, shear-thinning fluid obeying the p-Stokes (full Stokes) equations with the ice atmosphere interface as a moving free surface. A Navier slip boundary condition applies at the glacier base allowing both bedrock and subglacial lakes to be represented. Within the CutFEM framework we develop a strategy for handling non-linear viscosities and thin domains and show how glacier deformation can be modelled using a level set function. In numerical experiments we show that the expected order of accuracy is achieved and that the method is robust with respect to penalty parameters. As an application we compute the velocity field of the Swiss mountain glacier Haut Glacier d'Arolla in 2D with and without an underlying subglacial lake, and simulate the glacier deformation from year 1930 to 1932, with and without surface accumulation and basal melt.

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  • 7.
    Andersson, Emil
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Online Metallurgical Mass Balance and Reconciliation2021Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    In mineral processing, one of the most important and versatile separation methods is flotation. Flotation utilizes the different surface properties of the valuable minerals in the ore to separate them from the less valuable gangue material in the ore. Crushed and ground ore is mixed with water and fed into flotation tanks. In the flotation tanks, the particles of valuable mineral are made hydrophobic. That way, they can be floated by attaching to air bubbles and gather on top of the flotation tank as froth. This froth, containing higher concentrations of valuable mineral, is recovered and then processed further.

    The flotation circuit is controlled and maintained using measurements on the mass flows and grades of different materials. Due to economical, practical, and technological limitations, these measurements are performed at a chosen number of points in the circuit and at discrete points in time. Poor measurement data can have devastating consequences if the operators are left with limited information and errors in the circuit remain undetected.

    The accuracy of the acquired measurements is improved by performing mass balance and reconciliation. Traditionally, mass balance uses the sum of the total mass flows and the average grades over long times to avoid including the internal mass of the circuit in the calculations. It is desirable to perform mass balance directly to allow for faster intervention if any failures occur in the circuit during the on-line process.

    This report describes an on-line dynamic approach towards mass balancing and reconciliation of the mass flows and grades in a flotation circuit. Here, physical models of the flotation circuit are used to construct mass balance constraints using interpolation and test functions and the mass balance problem is posed as an optimization problem. The optimization problem is to adjust the assay such that the residual, the difference between the measured and the adjusted assay, is minimized while maintaining mass balance.

    An implementation in MATLAB and tests on synthetic data show that the dynamic formulation of mass balance does adjust 'erroneous' measurements such that mass balance is fulfilled. Given this result, there are still important aspects of the implementation that have to be addressed. The model uses the unknown and cell specific parameters flotation rate and recovery. Thus, these must be found or properly modeled. This report proposes a possible model for flotation rate as well as a strategy to find the recovery. The requirements of accuracy and speed of the implementation are also discussed.

    Possible next steps of this project is to further confirm a time effective implementation using synthetic data. Consequently, the implementation can be adapted for natural data in order to verify correctness of models.

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  • 8.
    Andrén, Daniel
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hellström, Lars
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On the complexity of matrix reduction over finite fields2007In: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 39, no 4, p. 428-452Article in journal (Refereed)
    Abstract [en]

    We study matrix elimination over finite fields, and present an algorithm which is asymptotically faster than the traditional Gauss--Jordan elimination. We also bound the average and worst-case complexity for the problem, proving that our algorithm is close to being optimal, and show related concentration results for random matrices.

    Next we present the results of a large computational study of the complexities for small matrices and fields. Here we determine the exact distribution of the complexity for matrices from $\mathrm{GL}_{n}(\mathbb{F}_{q})$, with $n$ an $q$ small

    Finally we consider an extension of the problems studied for finite fields to finite semifields. We give a conjecture on the behaviour of a natural analogue of $\mathrm{GL}_{n}$ for semifields and prove this for a certain class of semifields.

  • 9.
    Anton, Rikard
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cohen, David
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematics, University of Innsbruck, A–6020 Innsbruck, Austria.
    Exponential integrators for stochastic Schrödinger equations driven by Itô noise2018In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 36, no 2, p. 276-309Article in journal (Refereed)
    Abstract [en]

    We study an explicit exponential scheme for the time discretisation of stochastic Schr¨odinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schr¨odinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.

  • 10.
    Anton, Rikard
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cohen, David
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Quer-Sardanyons, Lluis
    Department of Mathematics, Universitat Autònoma de Barcelona, Catalonia, Spain.
    A fully discrete approximation of the one-dimensional stochastic heat equation2020In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 40, no 1, p. 247-284Article in journal (Refereed)
    Abstract [en]

    A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space–time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz this explicit time integrator allows for error bounds in Lq(Ω), for all q ≥ 2, improving some existing results in the literature. On top of this we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.

  • 11.
    Araujo-Cabarcas, Juan Carlos
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Reliable hp finite element computations of scattering resonances in nano optics2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Eigenfrequencies are commonly studied in wave propagation problems, as they are important in the analysis of closed cavities such as a microwave oven. For open systems, energy leaks into infinity and therefore scattering resonances are used instead of eigenfrequencies. An interesting application where resonances take an important place is in whispering gallery mode resonators.

    The objective of the thesis is the reliable and accurate approximation of scattering resonances using high order finite element methods. The discussion focuses on the electromagnetic scattering resonances in metal-dielectric nano-structures using a Drude-Lorentz model for the description of the material properties. A scattering resonance pair satisfies a reduced wave equationand an outgoing wave condition. In this thesis, the outgoing wave condition is replaced by a Dirichlet-to-Neumann map, or a Perfectly Matched Layer. For electromagnetic waves and for acoustic waves, the reduced wave equation is discretized with finite elements. As a result, the scattering resonance problem is transformed into a nonlinear eigenvalue problem.

    In addition to the correct approximation of the true resonances, a large number of numerical solutions that are unrelated to the physical problem are also computed in the solution process. A new method based on a volume integral equation is developed to remove these false solutions.

    The main results of the thesis are a novel method for removing false solutions of the physical problem, efficient solutions of non-linear eigenvalue problems, and a new a-priori based refinement strategy for high order finite element methods. The overall material in the thesis translates into a reliable and accurate method to compute scattering resonances in physics and engineering.

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  • 12.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Campos, Carmen
    Engström, Christian
    Roman, Jose E.
    Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures2020In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 407, article id 109220Article in journal (Refereed)
    Abstract [en]

    In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS. Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform h and p strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The hp-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.

  • 13.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Engström, Christian
    On spurious solutions encountered in Helmholtz scatteringresonance computations in Rd with applications tonano-photonics and acoustics2021In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 429, article id 110024Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider a sorting scheme for potentially spurious scattering resonant pairs in one- and two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a Lippmann-Schwinger type of volume integral equation and can, therefore, be applied to structures with graded materials as well as to configurations including piece-wise constant material properties. For TM/TE polarized electromagnetic waves and for acoustic waves, we compute first approximations of scattering resonances with finite elements. Then, we apply the novel sorting scheme to the computed eigenpairs and use it to mark potentially spurious solutions in electromagnetic and acoustic scattering resonances computations at a low computational cost. Several test cases with Drude-Lorentz dielectric resonators as well as with graded material properties are considered.

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  • 14.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Engström, Christian
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On spurious solutions in finite element approximations of resonances in open systems2017In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 74, no 10, p. 2385-2402Article in journal (Refereed)
    Abstract [en]

    In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann–Schwinger equation and we use computations of the pseudospectrum to show that this is a suitable choice. Numerical simulations indicate that the presented test can distinguish between spurious eigenvalues and true eigenvalues also in difficult cases.

  • 15.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Engström, Christian
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Jarlebring, Elias
    Royal Institute of Technology.
    Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map2018In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, p. 177-192Article in journal (Refereed)
    Abstract [en]

    We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method (TIAR). Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.

  • 16.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Science and Mathematics Education. Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Engström, Christian
    Department of Mathematics, Linnaeus University, Sweden.
    Wadbro, Eddie
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Department of Mathematics and Computer Science, Karlstad University, Sweden.
    Shape optimization for the strong routing of light in periodic diffraction gratings2023In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 472, article id 111684Article in journal (Refereed)
    Abstract [en]

    In the quest for the development of faster and more reliable technologies, the ability to control the propagation, confinement, and emission of light has become crucial. The design of guide mode resonators and perfect absorbers has proven to be of fundamental importance. In this project, we consider the shape optimization of a periodic dielectric slab aiming at efficient directional routing of light to reproduce similar features of a guide mode resonator. For this, the design objective is to maximize the routing efficiency of an incoming wave. That is, the goal is to promote wave propagation along the periodic slab. A Helmholtz problem with a piecewise constant and periodic refractive index medium models the wave propagation, and an accurate Robin-to-Robin map models an exterior domain. We propose an optimal design strategy that consists of representing the dielectric interface by a finite Fourier formula and using its coefficients as the design variables. Moreover, we use a high order finite element (FE) discretization combined with a bilinear Transfinite Interpolation formula. This setting admits explicit differentiation with respect to the design variables, from where an exact discrete adjoint method computes the sensitivities. We show in detail how the sensitivities are obtained in the quasi-periodic discrete setting. The design strategy employs gradient-based numerical optimization, which consists of a BFGS quasi-Newton method with backtracking line search. As a test case example, we present results for the optimization of a so-called single port perfect absorber. We test our strategy for a variety of incoming wave angles and different polarizations. In all cases, we efficiently reach designs featuring high routing efficiencies that satisfy the required criteria.

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  • 17.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Wadbro, Eddie
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden.
    Shape optimization for the strong directional scattering of dielectric nanorods2021In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 122, no 15, p. 3683-3704Article in journal (Refereed)
    Abstract [en]

    In this project, we consider the shape optimization of a dielectric scatterer aiming at efficient directional routing of light. In the studied setting, light interacts with a penetrable scatterer with dimension comparable to the wavelength of an incoming planar wave. The design objective is to maximize the scattering efficiency inside a target angle window. For this, a Helmholtz problem with a piecewise constant refractive index medium models the wave propagation, and an accurate Dirichlet-to-Neumann map models an exterior domain. The strategy consists of using a high-order finite element (FE) discretization combined with gradient-based numerical optimization. The latter consists of a quasi-Newton (BFGS) with backtracking line search. A discrete adjoint method is used to compute the sensitivities with respect to the design variables. Particularly, for the FE representation of the curved shape, we use a bilinear transfinite interpolation formula, which admits explicit differentiation with respect to the design variables. We exploit this fact and show in detail how sensitivities are obtained in the discrete setting. We test our strategy for a variety of target angles, different wave frequencies, and refractive indexes. In all cases, we efficiently reach designs featuring high scattering efficiencies that satisfy the required criteria.

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  • 18.
    Arndt, Daniel
    et al.
    Scalable Algorithms and Coupled Physics Group, Computational Sciences and Engineering Division, Oak Ridge National Laboratory, TN, United States.
    Bangerth, Wolfgang
    Department of Mathematics, Colorado State University, CO, Fort Collins, United States; Department of Geosciences, Colorado State University, CO, Fort Collins, United States.
    Feder, Marco
    SISSA, International School for Advanced Studies, Trieste, Italy.
    Fehling, Marc
    Department of Mathematics, Colorado State University, CO, Fort Collins, United States.
    Gassmöller, Rene
    Department of Geological Sciences, University of Florida, FL, Gainesville, United States.
    Heister, Timo
    School of Mathematical and Statistical Sciences, Clemson University, SC, Clemson, United States.
    Heltai, Luca
    SISSA, International School for Advanced Studies, Trieste, Italy.
    Kronbichler, Martin
    Department of Information Technology, Uppsala University, Uppsala, Sweden; Institute of Mathematics, University of Augsburg, Augsburg, Germany.
    Maier, Matthias
    Department of Mathematics, Texas A&M University, 3368 TAMU, TX, College Station, United States.
    Munch, Peter
    Institute of Mathematics, University of Augsburg, Universitätsstr. 12a, Augsburg, Germany; Institute of Material Systems Modeling, Helmholtz-Zentrum Hereon, Geesthacht, Germany.
    Pelteret, Jean-Paul
    Independent Researcher., Geesthacht, Germany.
    Sticko, Simon
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Turcksin, Bruno
    Scalable Algorithms and Coupled Physics Group, Computational Sciences and Engineering Division, Oak Ridge National Laboratory, TN, United States.
    Wells, David
    Department of Mathematics, University of North Carolina, NC, Chapel Hill, United States.
    The deal.II Library, Version 9.42022In: Journal of Numerical Mathematics, ISSN 1570-2820, E-ISSN 1569-3953, Vol. 30, no 3, p. 231-246Article in journal (Refereed)
    Abstract [en]

    This paper provides an overview of the new features of the finite element library deal.II, version 9.4.

  • 19. Backman, Anders
    et al.
    Bodin, Kenneth
    Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Lacoursière, Claude
    Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Servin, Martin
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Democratizing CAE with Interactive Multiphysics Simulation and Simulators2012Conference paper (Other academic)
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  • 20.
    Baek, Seung Ki
    et al.
    Umeå University, Faculty of Science and Technology, Physics.
    Bernhardsson, Sebastian
    Umeå University, Faculty of Science and Technology, Physics.
    Equilibrium solution to the lowest unique positive integer game2010In: Fluctuation and Noise Letters, ISSN 0219-4775, E-ISSN 1793-6780, Vol. 9, no 1, p. 61-68Article in journal (Refereed)
    Abstract [en]

    We address the equilibrium concept of a reverse auction game so that no one can enhance the individual payoff by a unilateral change when all the others follow a certain strategy. In this approach the combinatorial possibilities to consider become very much involved even for a small number of players, which has hindered a precise analysis in previous works. We here present a systematic way to reach the solution for a general number of players, and show that this game is an example of conflict between the group and the individual interests.

  • 21. Balmus, Maximilian
    et al.
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway.
    Hoffman, Johan
    Razavi, Reza
    Nordsletten, David A.
    A partition of unity approach to fluid mechanics and fluid-structure interaction2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 362, article id 112842Article in journal (Refereed)
    Abstract [en]

    For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains a computationally expensive endeavour which continues to drive interest in the development of novel approaches. Overlapping domain techniques have been introduced as a way to combine the fluid-solid mesh conformity, seen in moving-mesh methods, without the need for mesh smoothing or re-meshing, which is a core characteristic of fixed mesh approaches. In this work, we introduce a novel overlapping domain method based on a partition of unity approach. Unified function spaces are defined as a weighted sum of fields given on two overlapping meshes. The method is shown to achieve optimal convergence rates and to be stable for steady-state Stokes, Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for FSI in the case of 2D flow past an elastic beam simulation. These initial results point to the potential applicability of the method to a wide range of FSI applications, enabling boundary layer refinement and large deformations without the need for re-meshing or user-defined stabilization.

  • 22.
    Bartlett, Roscoe A.
    et al.
    Sandia National Laboratories.
    Van Bloemen Waanders, Bart
    Sandia National Laboratories.
    Berggren, Martin
    Department of Information Technology, Uppsala University.
    Hybrid Differentiation Strategies for Simulation and Analysis of Applications in C++2008In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 35, no 1Article in journal (Refereed)
  • 23.
    Berg, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Numerical analysis and simulation of stochastic partial differential equations with white noise dispersion2023Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. 

    The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. 

    The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE.

    Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.

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  • 24.
    Berg, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Numerical simulations of stochastic generalized Benjamin-Bona-Mahony equationsManuscript (preprint) (Other academic)
  • 25.
    Berg, André
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cohen, David
    Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 41296 Gothenburg, Sweden.
    Dujardin, Guillaume
    Univ. Lille, Inria, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France.
    Approximated exponential integrators for the stochastic Manakov equation2023In: Journal of Computational Dynamics, ISSN 2158-2491, Vol. 10, no 2, p. 323-344Article in journal (Refereed)
    Abstract [en]

    This article presents and analyzes an approximated exponential integrator for the (inhomogeneous) stochastic Manakov system. This system of SPDE occurs in the study of pulse propagation in randomly birefringent optical fibers. For a globally Lipschitz-continuous nonlinearity, we prove that the strong order of the time integrator is 1/2. This is then used to prove that the approximated exponential integrator has convergence order 1/2 in probability and almost sure order 1/2−, in the case of the cubic nonlinear coupling which is relevant in optical fibers. Finally, we present several numerical experiments in order to support our theoretical findings and to illustrate the efficiency of the approximated exponential integrator as well as a modified version of it.

  • 26.
    Berg, André
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cohen, David
    Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.
    Dujardin, Guillaume
    Univ. Lille, Inria, CNRS, UMR 8524 - Laboratoire Paul Painlevé, Villeneuve-d’Ascq, France.
    Lie–Trotter Splitting for the Nonlinear Stochastic Manakov System2021In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 88, no 1, article id 6Article in journal (Refereed)
    Abstract [en]

    This article analyses the convergence of the Lie–Trotter splitting scheme for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. First, we prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz. Then, we show that the splitting scheme has convergence order 1/2 in probability and almost sure order 12- in the case of a cubic nonlinearity. We provide several numerical experiments illustrating the aforementioned results and the efficiency of the Lie–Trotter splitting scheme. Finally, we numerically investigate the possible blowup of solutions for some power-law nonlinearities.

  • 27.
    Berg, André
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cohen, David
    Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.
    Dujardin, Guillaume
    Inria Lille Nord-Europe and Laboratoire Paul Painlevé UMR CNRS 8524, Villeneuve d’Asq Cedex, France.
    Numerical study of nonlinear Schrödinger equations with white noise dispersionManuscript (preprint) (Other academic)
  • 28.
    Berggren, Martin
    Department of Information Technology, Uppsala University.
    A vertex-centered dual discontinuous Galerkin method2006In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 192, no 1, p. 175-181Article in journal (Refereed)
  • 29.
    Berggren, Martin
    Department of Information Technology, Uppsala University.
    Approximations of very weak solutions to boundary-value problems2004In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 42, no 2, p. 860-877Article in journal (Refereed)
  • 30.
    Berggren, Martin
    Department of Computational and Applied Methematics, Rice University, Houston.
    Numerical solution of a flow-control problem: vorticity reduction by dynamic boundary action1998In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 19, no 3, p. 829-860Article in journal (Refereed)
  • 31.
    Berggren, Martin
    Department of Computational and Applied Mathematics, Rice University, Houston, TX.
    Solving an advection-diffusion problem on the Connection Machine1994In: Concurrency: Practice and Experience, Vol. 6, no 1, p. 55-68Article in journal (Refereed)
  • 32.
    Berggren, Martin
    FFA, the Aeronautical Research Institute of Sweden.
    The volume discharge approach to geometric conservation1999In: Computational Methods for Fluid-Structure Interaction / [ed] T. Kvamsdal, 1999Conference paper (Refereed)
  • 33.
    Berggren, Martin
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Bernland, Anders
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Noreland, Daniel
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Acoustic boundary layers as boundary conditions2018In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 371, p. 633-650Article in journal (Refereed)
    Abstract [en]

    The linearized, compressible Navier-Stokes equations can be used to model acoustic wave propagation in the presence of viscous and thermal boundary layers. However, acoustic boundary layers are notorious for invoking prohibitively high resolution requirements on numerical solutions of the equations. We derive and present a strategy for how viscous and thermal boundary-layer effects can be represented as a boundary condition on the standard Helmholtz equation for the acoustic pressure. This boundary condition constitutes an O (delta) perturbation, where delta is the boundary-layer thickness, of the vanishing Neumann condition for the acoustic pressure associated with a lossless sound-hard wall. The approximate model is valid when the wavelength and the minimum radius of curvature of the wall is much larger than the boundary layer thickness. In the special case of sound propagation in a cylindrical duct, the model collapses to the classical Kirchhoff solution. We assess the model in the case of sound propagation through a compression driver, a kind of transducer that is commonly used to feed horn loudspeakers. Due to the presence of shallow chambers and thin slits in the device, it is crucial to include modeling of visco-thermal losses in the acoustic analysis. The transmitted power spectrum through the device calculated numerically using our model agrees well with computations using a hybrid model, where the full linearized, compressible Navier-Stokes equations are solved in the narrow regions of the device and the inviscid Helmholtz equations elsewhere. However, our model needs about two orders of magnitude less memory and computational time than the more complete model. 

  • 34.
    Berggren, Martin
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Ekström, Sven-Erik
    Department of Information Technology, Uppsala University.
    Nordström, Jan
    Department of Information Technology, Uppsala University.
    A discontinuous Galerkin extension of the vertex-centered edge-based finite volume method2009In: Communications in Computational Physics, ISSN 1815-2406, Vol. 5, no 2-4, p. 456-468Article in journal (Refereed)
    Abstract [en]

    The finite volume (FV) method is the dominating discretization technique for computational fluid dynamics (CFD), particularly in the case of compressible fluids. The discontinuous Galerkin (DG) method has emerged as a promising high-accuracy alternative. The standard DG method reduces to a cell-centered FV method at lowest order. However, many of today's CFD codes use a vertex-centered FV method in which the data structures are edge based. We develop a new DG method that reduces to the vertex-centered FV method at lowest order, and examine here the new scheme for scalar hyperbolic problems. Numerically, the method shows optimal-order accuracy for a smooth linear problem. By applying a basic hp-adaption strategy, the method successfully handles shocks. We also discuss how to extend the FV edge-based data structure to support the new scheme. In this way, it will in principle be possible to extend an existing code employing the vertex-centered and edge-based FV discretization to encompass higher accuracy through the new DG method.

  • 35.
    Berggren, Martin
    et al.
    Department of Computational and Applied Mathematics, Rice University, Houston TX.
    Glowinski, Roland
    Department of Mathematics, University of Houston, Houston, , TX .
    A spectral preconditioner for control problems associated with linear evolution equations1995In: East-West Journal on Numerical Mathematics, Vol. 3, no 2, p. 81-109Article in journal (Refereed)
  • 36.
    Berggren, Martin
    et al.
    FFA, the Aeronautical Research Institute of Sweden, Bromma.
    Glowinski, Roland
    Department of Mathematics, University of Houston.
    Lions, Jacques-Louis
    College de France.
    A Computational Approach to Controllability Issues for Flow-Related Models. (I): Pointwise Control of the Viscous Burgers Equation1996In: International journal of computational fluid dynamics (Print), ISSN 1061-8562, E-ISSN 1029-0257, Vol. 7, no 3, p. 237-252Article in journal (Refereed)
  • 37.
    Berggren, Martin
    et al.
    FFA, the Aeronautical Research Institute of Sweden, Bromma, Sweden.
    Glowinski, Roland
    Department of Mathematics, University of Houston, Houston.
    Lions, Jacques-Louis
    College de France, Rue d'Ulm, Paris, France .
    A computational approach to controllability issues for flow-related models. (II): Control of two-dimensional, linear advection-diffusion and Stokes models.1996In: International journal of computational fluid dynamics (Print), ISSN 1061-8562, E-ISSN 1029-0257, Vol. 6, no 4, p. 253-274Article in journal (Refereed)
  • 38.
    Berggren, Martin
    et al.
    FFA, the Aeronautical Research Institute of Sweden.
    Heinkenschloss, Matthias
    Department of Computational and Applied Mathematics, Rice University, Houston, TX.
    Parallel Solution of Optimal-Control Problems by Time-Domain Decomposition1997In: Computational Science for the 21st Century / [ed] M.O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux, and M. F. Wheeler, 1997Conference paper (Refereed)
  • 39.
    Berglund, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Numerical Simulations of Linear Stochastic Oscillators: driven by Wiener and Poisson processes2017Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The main component of this essay is the numerical analysis of stochastic differential equations driven by Wiener and Poisson processes. In order to do this, we focus on two model problems, the geometric Brownian motion and the linear stochastic oscillator, studied in the literature for stochastic differential equations only driven by a Wiener process. This essay covers theoretical as well as numerical investigations of jump - or more specifically, Poisson - processes and how they influence the above model problems.

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  • 40.
    Bernland, Anders
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Wadbro, Eddie
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Berggren, Martin
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Acoustic shape optimization using cut finite elements2018In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 113, no 3, p. 432-449Article in journal (Refereed)
    Abstract [en]

    Fictitious domain methods are attractive for shape optimization applications, since they do not require deformed or regenerated meshes. A recently developed such method is the CutFEM approach, which allows crisp boundary representations and for which uniformly well-conditioned system matrices can be guaranteed. Here, we investigate the use of the CutFEM approach for acoustic shape optimization, using as test problem the design of an acoustic horn for favorable impedance-matching properties. The CutFEM approach is used to solve the Helmholtz equation, and the geometry of the horn is implicitly described by a level-set function. To promote smooth algorithmic updates of the geometry, we propose to use the nodal values of the Laplacian of the level-set function as design variables. This strategy also improves the algorithm's convergence rate, counteracts mesh dependence, and, in combination with Tikhonov regularization, controls small details in the optimized designs. An advantage with the proposed method is that the exact derivatives of the discrete objective function can be expressed as boundary integrals, as opposed to when using a traditional method that uses mesh deformations. The resulting horns possess excellent impedance-matching properties and exhibit surprising subwavelength structures, not previously seen, which are possible to capture due to the fixed mesh approach.

  • 41.
    Bernland, Anders
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Wadbro, Eddie
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Berggren, Martin
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    SHAPE OPTIMIZATION OF A COMPRESSION DRIVER PHASE PLUG2019In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 41, no 1, p. B181-B204Article in journal (Refereed)
    Abstract [en]

    A compression driver is an electro-acoustic transducer with considerably higher efficiency than direct radiating loudspeakers, thanks to the increased radiation resistance caused by a large vibrating diaphragm placed in a compression chamber with small openings. The transition section between compression chamber and output waveguide, the phase plug, must be carefully designed to avoid irregularities in the output sound pressure level (SPL) as a function of frequency. Here we present a shape optimization method based on an implicit level-set description and adjoint sensitivity analysis, which enables a large number of design parameters and vast design freedom. The CutFEM approach, a fictitious domain finite element method, removes the need for mesh updates and makes the method robust and computationally inexpensive. Numerical experiments for a generic annular diaphragm compression driver are presented, with optimized designs showing only minor frequency irregularities. Two different objective functions are considered: one for maximum SPL and one where the SPL is fitted to that of a hypothetical ideal design; the latter approach is found to be more effective in reducing irregularities. Visco-thermal boundary-layer losses are included in a post-processing step, and, though the influence of losses is clearly noticeable, the overall performance is similar and the optimized designs still outperform the original design.

  • 42.
    Bertgren, Amanda
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Approximating the element matrices in an unfitted finite element method using neural networks2022Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    This study investigates the possibility of combining an unfitted finite element method, CutFEM, with neural networks, in an attempt to reduce the computational time on evolving domains. Finite element methods are used to solve partial differential equations by fitting a spatial and temporal discretisation to the momentous domain. CutFEM was proposed to overcome a repeated discretisation of evolving domains by introducing a static background mesh allowed to cut the domain boundary. The drawback is an increased complexity in quadrature of the cut contributions to the element matrices, which quickly becomes time consuming for higher order methods.

    As machine learning methods have been successful in a variety of areas recently, this study investigates the possibility of replacing the element matrix quadrature with neural network regression. A classification network is also proposed for a quadrature-free method of identification of the cut elements. The study has been performed by implementing the networks in alternative CutFEM algorithms. Different methods of implementation, pre- and post-processing of the data, as well as different optimisation strategies of the training phase have been investigated in comparison to CutFEM benchmarks.

    The time consumption showed a significant decrease for the modified CutFEM in comparison to the conventional method. The classification was shown to be successful for two elementary domains, yet some difficulties occurred for a non-trivial level set representation. The approximation of cut elements with sufficient domain intersection showed decent results, although elements with minor intersection with the domain contributed with some difficulties and the error tends to propagate towards nearby elements. This was deduced to stem from the nodal contributions furthest from the domain intersection. Therefore, this study proposes a few additions to the current method in the event that the study is extended. These suggestions are based around the introduction or modification of error weights in the neural network training.

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  • 43.
    Billing, Erik
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Servin, Martin
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Composer: A prototype multilingual model composition tool2013In: MODPROD2013: 7th MODPROD Workshop on Model-Based Product Development / [ed] Peter Fritzson, Umeå: Umeå universitet , 2013Conference paper (Other academic)
    Abstract [en]

    Composer - A prototype multilingual model composition tool

    Erik Billing and Martin Servin

    November 10, 2012

    Facing the task to design, simulate or optimize a complex system itis common to find models and data for the system expressed in differentformats, implemented in different simulation software tools. When a newmodel is developed, a target platform is chosen and existing componentsimplemented with different tools have to be converted. This results inunnecessary work duplication and lead times. The Modelica languageinitiative [2] partially solves this by allowing developers to move modelsbetween different tools following the Modelica standard. Another possi-bility is to exchange models using the Functional Mockup Interface (FMI)standard that allows computer models to be used as components in othersimulations, possibly implemented using other programming languages[1]. With the Modelica and FMI standards entering development, there isneed for an easy-to-use tool that supports design, editing and simulationof such multilingual systems, as well as for retracting system informationfor formulating and solving optimization problems.A prototype solution for a graphical block diagram tool for design, edit-ing, simulation and optimization of multilingual systems has been createdand evaluated for a specific system. The tool is named Composer [3].The block diagram representation should be generic, independent ofmodel implementations, have a standardized format and yet support effi-cient handling of complex data. It is natural to look for solutions amongmodern web technologies, specifically HTML5. The format for represent-ing two dimensional vector graphics in HTML5 is Scalable Vector Graphics(SVG). We combine the SVG format with the FMI standard. In a firststage, we take the XML-based model description of FMI as a form for de-scribing the interface for each component, in a language independent way.Simulation parameters can also be expressed on this form, and integratedas metadata into the SVG image. 

    The prototype, using SVG in conjunction with FMI, is implementedin JavaScript and allow creation and modification of block diagrams directly in the web browser. Generated SVG images are sent to the serverwhere they are translated to program code, allowing the simulation ofthe dynamical system to be executed using selected implementations. Analternative mode is to generate optimization problem from the systemdefinition and model parameters. The simulation/optimization result is 

    returned to the web browser where it is plotted or processed using otherstandard libraries.The fiber production process at SCA Packaging Obbola [4] is used asan example system and modeled using Composer. The system consists oftwo fiber production lines that produce fiber going to a storage tank [5].The paper machine is taking fiber from the tank as needed for production.A lot of power is required during fiber production and the purpose of themodel was to investigate weather electricity costs could be reduced byrescheduling fiber production over the day, in accordance with the electricity spot price. Components are implemented for dynamical simulationusing OpenModelica and for discrete event using Python. The Python implementation supports constraint propagation between components andoptimization over specified variables. Each component is interfaced as aFunctional Mock-up Unit (FMU), allowing components to be connectedand properties specified in language independent way. From the SVGcontaining the high-level system information, both Modelica and Pythoncode is generated and executed on the web server, potentially hosted ina high performance data center. More implementations could be addedwithout modifying the SVG system description.We have shown that it is possible to separate system descriptions onthe block diagram level from implementations and interface between thetwo levels using FMI. In a continuation of this project, we aim to integratethe FMI standard also for co-simulation, such that components implemented in different languages could be used together. One open questionis to what extent FMUs of the same component, but implemented withdifferent tools, will have the same model description. For the SVG-basedsystem description to be useful, the FMI model description must remainthe same, or at least contain a large overlap, for a single component implemented in different languages. This will be further investigated in futurework.

    References

    [1] Modelica Association. Functional mock-up interface, http://www.fmi-standard.org, November 2012.

    [2] Modelica Association. Modelica and the modelica association, http://www.modelica.org, November 2012.

    [3] Erik Billing and Martin Servin. Composer, http://imuit.cs.umu.se/composer, November 2012. [4] SCA Packaging. Sca packaging obbola, http://www.scapackaging.com, November 2012.

    [5] Patrik Törmänen and Hussein Jaffal. Reducing electricity cost - case study. Technical report, UMIT Research Lab, Umeå University, 2011.2

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  • 44.
    Björklund, Martin
    Umeå University, Faculty of Science and Technology, Department of Physics.
    A Hierarchical POD Reduction Method of Finite Element Models with Application to Simulated Mechanical Systems2012Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    When simulating mechanical systems the flexibility of the components often has to be taken into account. This is particularly important for simulations when high detailed information is demanded, e.g. to calculate stresses. To this end the Finite Element Method (FEM) is often used. However the models can become very large, containing millions of degrees of freedom. Solving large linear systems are computationally demanding. Therefore ways of reducing the problem is often sought. These reduction does, however, remove much of the details that was to be investigated. In this thesis this problem is addressed by creating a reduction scheme, using Proper Orthogonal Decomposition (POD), that significantly reduces a problem but still captures much of the details. A novel method for enriching regular POD-based model reduction methods with hierarchically determined enrichment POD-modes is developed. The method is proposed and validated in a FEM application towards dynamical simulation. The enriched method is compared against a regular POD reduction technique. An numerical study is made of a model example of linear elasticity in a gearwheel. The numerical study suggests that the error of displacements is around ten times smaller, on average, when using the enriched basis compared to a reference basis of equal dimensionality consisting of only regular POD modes. Also it is shown that local quantities as the von Mises stress in a gearwheel tooth is preserved much better using the enriched basis. An a posteriori error estimate is proposed and proved for the static case, showing that the error is bound.

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  • 45.
    Björklund, Martin
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Error estimates for finite element approximations of viscoelastic dynamics: the generalized Maxwell model2024In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 425, article id 116933Article in journal (Refereed)
    Abstract [en]

    We prove error estimates for a finite element approximation of viscoelastic dynamics based on continuous Galerkin in space and time, both in energy norm and in L2 norm. The proof is based on an error representation formula using a discrete dual problem and a stability estimate involving the kinetic, elastic, and viscoelastic energies. To set up the dual error analysis and to prove the basic stability estimates, it is natural to formulate the problem as a first-order-in-time system involving evolution equations for the viscoelastic stress, the displacements, and the velocities. The equations for the viscoelastic stress can, however, be solved analytically in terms of the deviatoric strain velocity, and therefore, the viscoelastic stress can be eliminated from the system, resulting in a system for displacements and velocities.

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  • 46.
    Björnfot, Fredrik
    Umeå University, Faculty of Science and Technology, Department of Physics.
    GDP Growth Rate Nowcasting and Forecasting2017Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The main purpose of this project was to help Swedbank get a better understandingof how gross domestic product growth rate develops in the future froma data set of macroeconomic variables. Since GDP values are released long aftera quarter has ended Swedbank would like to have a model that could predictupcoming GDP from these data sets. This was solved by a combination ofgrowth rate predictions from a dynamic factor model, a vector autoregressivemodel and two machine learning models. The predictions were combined usinga weighting method called system averaging model where the model predictionwith least historical error receives the largest weight in the nal future prediction.In previous work a simple moving average model has been implementedto achieve this eect however there are several aws in a simple moving averagemodel. Most of these defects could in theory be avoided by using an exponentialweighting scheme instead. This resulted in the use of an exponentialweighting method that is used to calculate weights for future predictions. Themain conclusions from this project were that some predictions could get betterwhen removing bad performing models which had too large of a weight. Puttingtoo high weight on a single well performing model is also not optimal since thepredictions could get very unstable because of varying model performance. Theexponential weighting scheme worked well for some predictions however whenthe parameter , that controls how the weight is distributed between recent andhistorical errors, got too small a problem arose. Too few values were used toform the nal weights for the prediction and the estimate got unsteady results.

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  • 47.
    Blanck, Andreas
    Umeå University, Faculty of Science and Technology, Department of Physics.
    American Option Price Approximation for Real-Time Clearing2018Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    American-style options are contracts traded on financial markets. These are derivatives of some underlying security or securities that in contrast to European-style options allow their holders to exercise at any point before the contracts expire. However, this advantage aggravates the mathematical formulation of an option's value considerably, explaining why essentially no exact closed-formed pricing formulas exist. Numerous price approximation methods are although available, but their possible areas of application as well as performance, measured by speed and accuracy, differ. A clearing house offering real-time solutions are especially dependent on fast pricing methods to calculate portfolio risk, where accuracy is assumed to be an important factor to guarantee low-discrepancy estimations. Conversely, overly biased risk estimates may worsen a clearing house's ability to manage great losses, endangering the stability of a financial market it operates.

    The purpose of this project was to find methods with optimal performance and to investigate if price approximation errors induce biases in option portfolios' risk estimates. Regarding performance, a Quasi-Monte Carlo least squares method was found suitable for at least one type of exotic option. Yet none of the analyzed closed-form approximation methods could be assessed as optimal because of their varying strengths, where although the Binomial Tree model performed most consistently. Moreover, the answer to which method entails the best risk estimates remains inconclusive since only one set of parameters was used due to heavy calculations. A larger study involving a broader range of parameter values must therefore be performed in order to answer this reliably. However, it was revealed that large errors in risk estimates are avoided only if American standard options are priced with any of the analyzed methods and not when a faster European formula is employed. Furthermore, those that were analyzed can yield rather different risk estimates, implying that relatively large errors may arise if an inadequate method is applied.

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  • 48.
    Blöcker, Christopher
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Through the coding-lens: community detection and beyond2022Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    We live in a highly-connected world and find networks wherever we look: social networks, public transport networks, telecommunication networks, financial networks, and more. These networks can be immensely complex, comprising potentially millions or even billions of inter-connected objects. Answering questions such as how to control disease spreading in contact networks, how to optimise public transport networks, or how to diversify investment portfolios requires understanding each network's function and working principles.

    Network scientists analyse the structure of networks in search of communities: groups of objects that form clusters and are more connected to each other than the rest. Communities form the building blocks of networks, corresponding to their sub-systems, and allow us to represent networks with coarse-grained models. Analysing communities and their interactions helps us unravel how networks function.

    In this thesis, we use the so-called map equation framework, an information-theoretic community-detection approach. The map equation follows the minimum description length principle and assumes complete data in networks with one node type. We challenge these assumptions and adapt the map equation for community detection in networks with two node types and incomplete networks where some data is missing. We move beyond detecting communities and derive approaches for how, based on communities, we can identify influential objects in networks, and predict links that do not (yet) exist.

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  • 49.
    Blöcker, Christopher
    et al.
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Nieves, Juan Carlos
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Rosvall, Martin
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Map equation centrality: community-aware centrality based on the map equation2022In: Applied Network Science, E-ISSN 2364-8228, Vol. 7, no 1, article id 56Article in journal (Refereed)
    Abstract [en]

    To measure node importance, network scientists employ centrality scores that typically take a microscopic or macroscopic perspective, relying on node features or global network structure. However, traditional centrality measures such as degree centrality, betweenness centrality, or PageRank neglect the community structure found in real-world networks. To study node importance based on network flows from a mesoscopic perspective, we analytically derive a community-aware information-theoretic centrality score based on network flow and the coding principles behind the map equation: map equation centrality. Map equation centrality measures how much further we can compress the network's modular description by not coding for random walker transitions to the respective node, using an adapted coding scheme and determining node importance from a network flow-based point of view. The information-theoretic centrality measure can be determined from a node's local network context alone because changes to the coding scheme only affect other nodes in the same module. Map equation centrality is agnostic to the chosen network flow model and allows researchers to select the model that best reflects the dynamics of the process under study. Applied to synthetic networks, we highlight how our approach enables a more fine-grained differentiation between nodes than node-local or network-global measures. Predicting influential nodes for two different dynamical processes on real-world networks with traditional and other community-aware centrality measures, we find that activating nodes based on map equation centrality scores tends to create the largest cascades in a linear threshold model.

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  • 50. Boiveau, Thomas
    et al.
    Burman, Erik
    Claus, Susanne
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Fictitious domain method with boundary value correction using penalty-free Nitsche method2018In: Journal of Numerical Mathematics, ISSN 1570-2820, E-ISSN 1569-3953, Vol. 26, no 2, p. 77-95Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider a fictitious domain approach based on a Nitsche type method without penalty. To allow for high order approximation using piecewise affine approximation of the geometry we use a boundary value correction technique based on Taylor expansion from the approximate to the physical boundary. To ensure stability of the method a ghost penalty stabilization is considered in the boundary zone. We prove optimal error estimates in the H1-norm and estimates suboptimal by 𝓞(h1/2) in the L2-norm. The suboptimality is due to the lack of adjoint consistency of our formulation. Numerical results are provided to corroborate the theoretical study.

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