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  • 51.
    Popovic, Jelena
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Analysis of a fast method for solving the high frequency Helmholtz equation in one dimension2011In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, no 3, p. 721-755Article in journal (Refereed)
    Abstract [en]

    We propose and analyze a fast method for computing the solution of the high frequency Helmholtz equation in a bounded one-dimensional domain with a variable wave speed function. The method is based on wave splitting. The Helmholtz equation is split into one-way wave equations with source functions which are solved iteratively for a given tolerance. The source functions depend on the wave speed function and on the solutions of the one-way wave equations from the previous iteration. The solution of the Helmholtz equation is then approximated by the sum of the one-way solutions at every iteration. To improve the computational cost, the source functions are thresholded and in the domain where they are equal to zero, the one-way wave equations are solved with geometrical optics with a computational cost independent of the frequency. Elsewhere, the equations are fully resolved with a Runge-Kutta method. We have been able to show rigorously in one dimension that the algorithm is convergent and that for fixed accuracy, the computational cost is asymptotically just for a pth order Runge-Kutta method, where omega is the frequency. Numerical experiments indicate that the growth rate of the computational cost is much slower than a direct method and can be close to the asymptotic rate.

  • 52.
    Popovic, Jelena
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Time Upscaling for Hamilton-Jacobi EquationsManuscript (preprint) (Other academic)
    Abstract [en]

     In this paper, we suggest an accurate and computationally efficient numerical method for time-dependent Hamilton-Jacobi equations with convex Hamiltonians. The method is based on a reformulation of the Hamilton-Jacobi equation as a front tracking problem, which is solved with the fast interface tracking methods together with a post-processing step. The complexity of standard numerical methods for such problems is O(dt^(-(d+1))) in d dimensions, where dt is the time step. The complexity of the method that we propose in this paper is reduced to O(dt^(-d)|log dt|) or even to O(dt^(-d)).

  • 53.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Analysis of high order fast interface tracking methods2014In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 128, no 2, p. 339-375Article in journal (Refereed)
    Abstract [en]

    Fast high order methods for the propagation of an interface in a velocity field are constructed and analyzed. The methods are generalizations of the fast interface tracking method proposed in Runborg (Commun Math Sci 7:365-398, 2009). They are based on high order subdivision to make a multiresolution decomposition of the interface. Instead of tracking marker points on the interface the related wavelet vectors are tracked. Like the markers they satisfy ordinary differential equations (ODEs), but fine scale wavelets can be tracked with longer timesteps than coarse scale wavelets. This leads to methods with a computational cost of rather than for markers and reference timestep . These methods are proved to still have the same order of accuracy as the underlying direct ODE solver under a stability condition in terms of the order of the subdivision, the order of the ODE solver and the time step ratio between wavelet levels. In particular it is shown that with a suitable high order subdivision scheme any explicit Runge-Kutta method can be used. Numerical examples supporting the theory are also presented.

  • 54.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    FAST INTERFACE TRACKING VIA A MULTIRESOLUTION REPRESENTATION OF CURVES AND SURFACES2009In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 7, no 2, p. 365-398Article in journal (Refereed)
    Abstract [en]

    We consider the propagation of an interface in a velocity field. The initial interface is described by a normal mesh [Guskov, et al, SIGGRAPH Proc., 259-268, 2000] which gives us a multiresolution decomposition of the interface and the related wavelet vectors. Instead of tracking marker points on the interface we track the wavelet vectors, which like the markers satisfy ordinary differential equations. We show that the finer the spatial scale, the slower the wavelet vectors evolve. By designing a numerical method which takes longer time steps for finer spatial scales weareable to track the interface with the same overall accuracy as when directly tracking the markers, but at a computational cost of O(logN/Delta t) rather than O(N/Delta t) for N markers and timestep Delta t. We prove this rigorously and give numerical examples supporting the theory. We also consider extensions to higher dimensions and co-dimensions.

  • 55.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Introduction to normal multiresolution approximation2005In: Lecture Notes in Computational Science and Engineering, ISSN 1439-7358, Vol. 44, p. 205-224Article in journal (Refereed)
    Abstract [en]

    A multiresolution analysis of a curve is normal if each wavelet detail vector with respect to a certain subdivision scheme lies in the local normal direction. In this paper we give an introduction to the analysis of normal approximations in [3]. We define the normal approximation in its basic form and show simplified proofs of the method's convergence, approximation quality and stability. We also explain how higher order approximations can be constructed using subdivision operators and give a brief summary of the corresponding results for these more general schemes.

  • 56.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Mathematical models and numerical methods for high frequency waves2007In: Communications in Computational Physics, ISSN 1815-2406, Vol. 2, no 5, p. 827-880Article, review/survey (Refereed)
    Abstract [en]

    The numerical approximation of high frequency wave propagation is important in many applications. Examples include the simulation of seismic, acoustic, optical waves and microwaves. When the frequency of the waves is high, this is a difficult multiscale problem. The wavelength is short compared to the overall size of the computational domain and direct simulation using the standard wave equations is very expensive. Fortunately, there are computationally much less costly models, that are good approximations of many wave equations precisely for very high frequencies. Even for linear wave equations these models are often nonlinear. The goal of this paper is to review such mathematical models for high frequency waves, and to survey numerical methods used in simulations. We focus on the geometrical optics approximation which describes the infinite frequency limit of wave equations. We will also discuss finite frequency corrections and some other models.

  • 57.
    Runborg, Olof
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Some new results in multiphase geometrical optics2000In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 34, no 6, p. 1203-1231Article in journal (Refereed)
    Abstract [en]

    In order to accommodate solutions with multiple phases, corresponding to crossing rays, we formulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from an associated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based on delta and Heaviside functions and analyze the resulting equations. They form systems of nonlinear conservation laws with source terms. In contrast to the classical eikonal equation, these equations will incorporate a finite superposition principle in the sense that while the maximum number of phases is not exceeded a sum of solutions is also a solution. We present numerical results for a variety of homogeneous and inhomogeneous problems.

  • 58.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Wavelets and wavelet based numerical homogenization2009In: Multiscale Modeling and Simulation in Science, Springer Berlin/Heidelberg, 2009, p. 195-235Conference paper (Refereed)
    Abstract [en]

    Wavelets is a tool for describing functions on different scales or level of detail. In mathematical terms, wavelets are functions that form a basis for with special properties; the basis functions are spatially localized and correspond to different scale levels. Finding the representation of a function in this basis amounts to making a multiresolution decomposition of the function. Such a wavelet representation lends itself naturally to analyzing the fine and coarse scales as well as the localization properties of a function.Wavelets have been used in many applications, from image and signal analysis to numerical methods for partial differential equations (PDEs). In this tutorial we first go through the basic wavelet theory and then show a more specific application where wavelets are used for numerical homogenization.We will mostly give references to the original sources of ideas presented. There are also a large number of books and review articles that cover the topic of wavelets, where the interested reader can find further information, e.g. [25, 51, 48, 7, 39, 26, 23], just to mention a few.

  • 59.
    Runborg, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Arjmand, Doghonay
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A Time Dependent Approach for Removing the Cell Boundary Error in Elliptic Homogenization ProblemsManuscript (preprint) (Other academic)
    Abstract [en]

    This paper concerns the cell-boundary error present in multiscale algorithms for elliptichomogenization problems. Typical multiscale methods have two essential components: amacro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. To solve the micro model, boundary conditions are required on the boundary of the microscopic domain. Imposing a naive boundary condition leads to O(e/eta) error in the computation, where e is the size of the microscopic variations in the media and eta is the size of the micro-domain. The removal of this error in modern multiscale algorithms still remains an important open problem. In this paper, we present a time-dependent approach which is general in terms of dimension. We provide a theorem which shows that we have arbitrarily high order convergence rates in terms of e/eta in theperiodic setting. Additionally, we present numerical evidence showing that the method improves the O(e/eta) error to O(e) in general non-periodic media.

  • 60. Runborg, Olof
    et al.
    Theodoropoulos, C.
    Kevrekidis, I. G.
    Effective bifurcation analysis: a time-stepper-based approach2002In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 15, no 2, p. 491-511Article in journal (Refereed)
    Abstract [en]

    We introduce a numerical approach to perform the effective (coarse-scale) bifurcation analysis of solutions of dissipative evolution equations with spatially varying coefficients. The advantage of this approach is that the 'coarse model' (the averaged, effective equation) need not be explicitly constructed. The method only uses a time-integrator code for the detailed problem and judicious choices of initial data and integration times; the bifurcation computations are based on the so-called recursive projection method.

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