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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).
    Kressner, Daniel
    A parallel QZ algorithm for distributed memory HPC systems2014In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 36, no 5, p. C480-C503Article in journal (Refereed)
    Abstract [en]

    Appearing frequently in applications, generalized eigenvalue problems represent one of the core problems in numerical linear algebra. The QZ algorithm of Moler and Stewart is the most widely used algorithm for addressing such problems. Despite its importance, little attention has been paid to the parallelization of the QZ algorithm. The purpose of this work is to fill this gap. We propose a parallelization of the QZ algorithm that incorporates all modern ingredients of dense eigensolvers, such as multishift and aggressive early deflation techniques. To deal with (possibly many) infinite eigenvalues, a new parallel deflation strategy is developed. Numerical experiments for several random and application examples demonstrate the effectiveness of our algorithm on two different distributed memory HPC systems.

  • 4.
    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)
  • 5.
    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.

  • 6. Bosner, Nela
    et al.
    Karlsson, Lars
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Parallel and Heterogeneous $m$-Hessenberg-Triangular-Triangular Reduction2017In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, no 1, p. C29-C47Article in journal (Refereed)
    Abstract [en]

    The m-Hessenberg-triangular-triangular (mHTT) reduction is a simultaneous orthogonal reduction of three matrices to condensed form. It has applications, for example, in solving shifted linear systems arising in various control theory problems. A new heterogeneous CPU/GPU implementation of the mHTT reduction is presented and evaluated against an existing CPU implementation. The algorithm offloads the compute-intensive matrix-matrix multiplications to the GPU and keeps the inner loop, which is memory intensive and has a complicated control flow, on the CPU. Experiments demonstrate that the heterogeneous implementation can be superior to the existing CPU implementation on a system with 2 x 8 CPU cores and one GPU. Future development should focus on improving the scalability of the CPU computations.

  • 7.
    Burman, Erik
    et al.
    University College London.
    Claus, Susanne
    University College London.
    Massing, André
    Simula Res Lab, Fornebu, Norway.
    A Stabilized Cut Finite Element Method for the Three Field Stokes Problem2015In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 37, no 4, p. A1705-A1726Article in journal (Refereed)
    Abstract [en]

    We propose a Nitsche-based fictitious domain method for the three field Stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh. The dependent variables of velocity, pressure, and extra-stress tensor are discretized on the background mesh using linear finite elements. This equal order approximation is stabilized using a continuous interior penalty (CIP) method. On the unfitted domain boundary, Dirichlet boundary conditions are weakly enforced using Nitsche's method. We add CIP-like ghost penalties in the boundary region and prove that our scheme is inf-sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh. Additionally, we demonstrate that the condition number of the system matrix is bounded independently of the boundary location. We corroborate our theoretical findings numerically.

  • 8.
    Carey, V
    et al.
    Departments of Mathematics, Colorado State University, Fort Collins, CO 80523.
    Estep, D
    Department of mathematics and Department of Statistics, Colorado State University, Fort Collins, CO 80523 .
    Johansson, August
    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.
    Tavener, S
    Departments of Mathematics, Colorado State University, Fort Collins, CO 80523.
    Blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions2010In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 32, no 4, p. 2121-2145Article in journal (Refereed)
    Abstract [en]

    We describe and test an adaptive algorithm for evolution problems that employs a sequence of "blocks" consisting of fixed, though non-uniform, space meshes. This approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing, re-meshing, matrix reassembly, and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes. A major issue whith a blockadaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy. We describe several strategies to achieve this goal using adjoint-based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples.

  • 9.
    Cohen, David
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematics, University of Innsbruck, Austria.
    Verdier, Olivier
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Computing, Mathematics and Physics, Bergen University College, Norway.
    MultiSymplectic discretisation of wave map equations2016In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, no 2, p. A953-A972Article in journal (Refereed)
    Abstract [en]

    We present a new multisymplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multisymplectic discretization based on this new formulation is exemplified by means of the Euler box scheme. When applied to the wave map equation, this numerical scheme is explicit, preserves the constraint, and can be seen as a generalization of the \smaller SHAKE algorithm for constrained mechanical systems. Furthermore, numerical experiments show excellent conservation properties of the numerical solutions.

  • 10. Granat, Robert
    et al.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kressner, Daniel
    A novel parallel QR algorithm for hybrid distributed memory HPC systems2010In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 32, no 4, p. 2345-2378Article in journal (Refereed)
    Abstract [en]

    A novel variant of the parallel QR algorithm for solving dense nonsymmetric eigenvalue problems on hybrid distributed high performance computing systems is presented. For this purpose, we introduce the concept of multiwindow bulge chain chasing and parallelize aggressive early deflation. The multiwindow approach ensures that most computations when chasing chains of bulges are performed in level 3 BLAS operations, while the aim of aggressive early deflation is to speed up the convergence of the QR algorithm. Mixed MPI-OpenMP coding techniques are utilized for porting the codes to distributed memory platforms with multithreaded nodes, such as multicore processors. Numerous numerical experiments confirm the superior performance of our parallel QR algorithm in comparison with the existing ScaLAPACK code, leading to an implementation that is one to two orders of magnitude faster for sufficiently large problems, including a number of examples from applications.

  • 11.
    Karlsson, Lars
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Tisseur, Francoise
    Algorithms for Hessenberg-Triangular Reduction of Fiedler Linearization of Matrix Polynomials2015In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 37, no 3, p. C384-C414Article in journal (Refereed)
    Abstract [en]

    Small- to medium-sized polynomial eigenvalue problems can be solved by linearizing the matrix polynomial and solving the resulting generalized eigenvalue problem using the QZ algorithm. The QZ algorithm, in turn, requires an initial reduction of a matrix pair to Hessenberg-triangular (HT) form. In this paper, we discuss the design and evaluation of high-performance parallel algorithms and software for HT reduction of a specific linearization of matrix polynomials of arbitrary degree. The proposed algorithm exploits the sparsity structure of the linearization to reduce the number of operations and improve the cache reuse compared to existing algorithms for unstructured inputs. Experiments on both a workstation and a high-performance computing system demonstrate that our structure-exploiting parallel implementation can outperform both the general LAPACK routine DGGHRD and the prototype implementation DGGHR3 of a general blocked algorithm.

  • 12. Komori, Yoshio
    et al.
    Cohen, David
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Innsbruck university.
    Burrage, Kevin
    Weak second order explicit exponential Runge–Kutta methods for stochastic differential equations2017In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, no 6, p. A2857-A2878Article in journal (Refereed)
    Abstract [en]

    We propose new explicit exponential Runge--Kutta methods for the weak approximation of solutions of stiff Itô stochastic differential equations (SDEs). We also consider the use of exponential Runge--Kutta methods in combination with splitting methods. These methods have weak order 2 for multidimensional, noncommutative SDEs with a semilinear drift term, whereas they are of order 2 or 3 for semilinear ordinary differential equations. These methods are A-stable in the mean square sense for a scalar linear test equation whose drift and diffusion terms have complex coefficients. We carry out numerical experiments to compare the performance of these methods with an existing explicit stabilized method of weak order 2.

  • 13. Massing, Andre
    et al.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Logg, Anders
    Efficient implementation of finite element methods on nonmatching AND overlapping meshes in three dimensions2013In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 35, no 1, p. C23-C47Article in journal (Refereed)
    Abstract [en]

    In recent years, a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on nonmatching or overlapping meshes. Examples of such methods are the fictitious domain method, the extended finite element method, and Nitsche's method. In all these methods, integrals must be computed over cut cells or subsimplices, which is challenging to implement, especially in three space dimensions. In this note, we address the main challenges of such an implementation and demonstrate good performance of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices. As a canonical example of an overlapping mesh method, we consider Nitsche's method, which we apply to Poisson's equation and a linear elastic problem.

  • 14. McLachlan, Robert
    et al.
    Modin, Klas
    Verdier, Olivier
    Wilkins, Matt
    Symplectic Integrators for Index 1 Constraints2013In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 35, no 5, p. A2150-A2162Article in journal (Refereed)
    Abstract [en]

    We show that symplectic Runge–Kutta methods provide effective symplectic integrators for Hamiltonian systems with index 1 constraints. These include the Hamiltonian description of variational problems subject to position and velocity constraints nondegenerate in the velocities, such as those arising in sub-Riemannian geometry and control theory.

  • 15.
    McLachlan, Robert
    et al.
    Institute of Fundamental Sciences, Massey University, New Zealand.
    Modin, Klas
    Department of Mathematical Sciences, Chalmers University of Technology, Göteburg, Sweden.
    Verdier, Olivier
    Department of Mathematical Sciences, NTNU, Trondheim, Norway.
    Wilkins, Matt
    Institute of Fundamental Sciences, Massey University, New Zealand.
    Symplectic integrators for index one constraintsIn: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197Article in journal (Refereed)
    Abstract [en]

    We show that symplectic Runge–Kutta methods provide effective symplectic integratorsfor Hamiltonian systems with index one constraints. These include the Hamiltonian descriptionof variational problems subject to position and velocity constraints nondegenerate in the velocities,such as those arising in sub-Riemannian geometry and control theory.

  • 16.
    Schmidt, Stephan
    et al.
    Institut für Mathematik , Universität Würzburg.
    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.
    Large-Scale Three-Dimensional Acoustic Horn Optimization2016In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, no 6, p. B917-B940Article in journal (Refereed)
    Abstract [en]

    We consider techniques that enable large-scale gradient-based shape optimization of wave-guiding devices in the context of three-dimensional time-domain simulations. The approach relies on a memory efficient boundary representation of the shape gradient together with primal and adjoint solvers semiautomatically generated by the FEniCS framework. The hyperbolic character of the governing linear wave equation, written as a first-order system, is exploited through systematic use of the characteristic decomposition both to define the objective function and to obtain stable numerical fluxes in the discontinuous Galerkin spatial discretization. The methodology is successfully used to optimize the shape of a midrange acoustic horn, described by 1,762 design variables, for maximum transmission efficiency, where the parallel computations involve a total of $3.5\times10^9$ unknowns.

  • 17.
    Wadbro, Eddie
    et al.
    Uppsala University, Department of Information Technology, Division of Scientific Computing.
    Berggren, Martin
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Microwave tomography using topology optimization techniques2008In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 30, no 3, p. 1613-1633Article in journal (Refereed)
    Abstract [en]

    Microwave tomography is a technique in which microwaves illuminate a specimen, and measurements of the scattered electrical field are used to determine and depict the specimen's dielectric and conductive properties. This article presents a new method to perform the reconstruction. The reconstruction method is illustrated by assuming time harmonic scattering in two space dimensions in a setup tailored for medical applications. We prove that the resulting constrained nonlinear least-squares problem admits a solution. The governing Helmholtz equation is discretized by using the finite-element method, and the dielectric properties are allowed to attain different values at each element within a given region. The reconstruction algorithm uses methodologies borrowed from topology optimization of linearly elastic structures. Numerical examples illustrate the reconstruction method in a parameter range typical for human tissue and for the challenging case where the size of the object is in the same order as the wavelength. A reasonable estimate of the dielectric properties is obtained by using one observation per 20 unknowns when the permittivity is allowed to vary continuously within a given interval. Using a priori information that the permittivity attains only certain values results in a good estimate and a sharp image. As opposed to topology optimization for structures, there is no indication of mesh dependency and checkerboarding when forcing the permittivity to attain discrete values.

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