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  • 1.
    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.

  • 2.
    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)
  • 3.
    Cohen, David
    et al.
    Mathematisches Institut, Universität Basel.
    Raynaud, Xavier
    CMA, University of Oslo.
    Geometric finite difference schemes for the generalized hyperelastic-rod wave equation2011In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 235, no 8, p. 1925-1940Article in journal (Refereed)
    Abstract [en]

    Geometric integrators are presented for a class of nonlinear dispersive equations which includes the Camassa-Holm equation, the BBM equation and the hyperelastic-rod wave equation. One group of schemes is designed to preserve a global property of the equations: the conservation of energy; while the other one preserves a more local feature of the equations: the multi-symplecticity. (C) 2010 Elsevier B.V. All rights reserved.

  • 4.
    Frentz, Marie
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type2010In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 234, no 1, p. 146-164Article in journal (Refereed)
    Abstract [en]

    Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.

  • 5.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Department of Economics.
    Catastrophe theory applied to the refraction of traffic1988In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 22, no 2-3, p. 315-318Article in journal (Refereed)
  • 6.
    Soni, V.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Roussel, Olivier
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Hadjadj, A.
    On the accuracy and efficiency of point-value multiresolution algorithms for solving scalar wave and Euler equations2017In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 323, p. 159-175Article in journal (Refereed)
    Abstract [en]

    This paper concerns the use of a point-value multiresolution algorithm and its extension to three-dimensional hyperbolic conservation laws. The proposed method is applied to a high-order finite-differences discretization with an explicit time integration. The fluxes are evaluated on the adaptive grid using a fifth-order high-resolution shock capturing scheme based on a WENO solver, while the time is advanced using a third-order Runge-Kutta scheme. The multiresolution prediction operators are presented for one-, two- and three-dimensional problems. To assess the efficiency and the accuracy of the method, a new tolerance-scale diagram is introduced. This diagram enables to properly choose the adequate value of the tolerance in order to maintain an optimal multiresolution quality. Numerical examples based on advection and Euler equations are carried out to show that the proposed method yields accurate results.

  • 7.
    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.
    High contrast microwave tomography using topology optimization techniques2010In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 234, no 6, p. 1773-1780Article in journal (Refereed)
    Abstract [en]

    Microwave tomography for medical applications leads to a difficult reconstruction problem for the dielectric properties of biological tissue due to strongly diffracting waves in combination with large dielectric contrasts. We apply the material distribution technique used for topology optimization of elastic structures in order to solve the nonlinear least-squares problem underlying the reconstruction problem. Using simulated numerical data with an approximate signal-to-noise ratio of 40 dB and geometrical a priori information on the unknown objects, we obtain good estimates of the dielectric properties corresponding to biological objects.

  • 8.
    Wadbro, Eddie
    et al.
    Uppsala University, Department of Information Technology, Division of Scientific Computing.
    Udawalpola, Rajitha
    Uppsala University, Department of Information Technology, Division of Scientific Computing.
    Berggren, Martin
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Shape and topology optimization of an acoustic horn-lens combination2010In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 234, no 6, p. 1781-1787Article in journal (Refereed)
    Abstract [en]

    Using gradient-based optimization combined with numerical solutions of the Helmholtz equation, we design an acoustic device with high transmission efficiency and even directivity throughout a two-octave-wide frequency range. The device consists of a horn, whose flare is subject to boundary shape optimization, together with an area in front of the horn, where solid material arbitrarily can be distributed using topology optimization techniques, effectively creating an acoustic lens.

  • 9.
    Zhou, Zhiyong
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Recovery analysis for weighted mixed ℓ2/ℓp minimization with 0 < p ≤ 12019In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 352, p. 12p. 210-222Article in journal (Refereed)
    Abstract [en]

    We study the recovery conditions of weighted mixed ℓ2/ℓp (0 < p ≤ 1) minimization for block sparse signal reconstruction from compressed measurements when partial block support information is available. We show that the block p-restricted isometry property (RIP) can ensure the robust recovery. Moreover, we present the sufficient and necessary condition for the recovery by using weighted block p-null space property. The relationship between the block p-RIP and the weighted block p-null space property has been established. Finally, we illustrate our results with a series of numerical experiments.

  • 10.
    Önskog, Thomas
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations2010In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 235, p. 563-592Article in journal (Refereed)
    Abstract [en]

    The efficient and accurate calculation of sensitivities of the price of financial derivatives with respect to perturbations of the parameters in the underlying model, the so-called `Greeks', remains a great practical challenge in the derivative industry. This is true regardless of whether methods for partial differential equations or stochastic differential equations (Monte Carlo techniques) are being used. The computation of the `Greeks' is essential to risk management and to the hedging of financial derivatives and typically requires substantially more computing time as compared to simply pricing the derivatives. Any numerical algorithm (Monte Carlo algorithm) for stochastic differential equations produces a time-discretization error and a statistical error in the process of pricing financial derivatives and calculating the associated `Greeks'. In this article we show how a posteriori error estimates and adaptive methods for stochastic differential equations can be used to control both these errors in the context of pricing and hedging of financial derivatives. In particular, we derive expansions, with leading order terms which are computable in a posteriori form, of the time-discretization errors for the price and the associated `Greeks'. These expansions allow the user to simultaneously first control the time-discretization errors in an adaptive fashion, when calculating the price, sensitivities and hedging parameters with respect to a large number of parameters, and then subsequently to ensure that the total errors are, with prescribed probability, within tolerance.

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