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  • 1. Arjmand, Doghonay
    et al.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    A time dependent approach for removing the cell boundary error in elliptic homogenization problems2016Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 314, s. 206-227Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper concerns the cell-boundary error present in multiscale algorithms for elliptic homogenization problems. Typical multiscale methods have two essential components: a macro 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(epsilon/eta) error in the computation, where epsilon 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 epsilon/eta in the periodic setting. Additionally, we present numerical evidence showing that the method improves the O(epsilon/eta) error to O(epsilon) in general non-periodic media.

  • 2.
    Arjmand, Doghonay
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Analysis of heterogeneous multiscale methods for long time wave propagation problems2014Ingår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 12, nr 3, s. 1135-1166Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we analyze a multiscale method developed under the heterogeneous multiscale method (HMM) framework for numerical approximation of multiscale wave propagation problems in periodic media. In particular, we are interested in the long time O(epsilon(-2)) wave propagation, where e represents the size of the microscopic variations in the media. In large time scales, the solutions of multiscale wave equations exhibit O(1) dispersive effects which are not observed in short time scales. A typical HMM has two main components: a macromodel and a micromodel. The macromodel is incomplete and lacks a set of local data. In the setting of multiscale PDEs, one has to solve for the full oscillatory problem over local microscopic domains of size eta = O(epsilon) to upscale the parameter values which are missing in the macroscopic model. In this paper, we prove that if the microproblems are consistent with the macroscopic solutions, the HMM approximates the unknown parameter values in the macromodel up to any desired order of accuracy in terms of epsilon/eta..

  • 3.
    Arjmand, Doghonay
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Analysis of HMM for Long Time Multiscale Wave Propagation Problems in Locally-Periodic MediaManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    Multiscale wave propagation problems are difficult to solve numerically due to the interaction of different scales inherent in the problem. Extracting information about the average behaviour of the system requires resolving small scales in the problem. This leads to a tremendous computational burden if the size of microscopic variations are much smaller than the size of scales of interest. Heterogeneous multiscale methods (HMM) is a tool to avoid resolving the small scales everywhere. Nevertheless, it approximates the average part of the solution by upscaling the microscopic information on a small part of the domain. This leads to a substantial improvement in the computational cost. In this article, we analyze an HMM-based numerical method which approximates the long time behaviour of multiscale wave equations. In particular, we consider theoretically challenging case of locally-periodic media where fast and slow variations are allowed at the same time. We are interested in the long time regime (T=O(e^{-1})), where e represents the wavelength of the fast variations in themedia. We first use asymptotic expansions to derive effective equations describing the long time effects of the multiscale waves in multi-dimensional locally-periodic media. We then show that HMM captures these non-trivial long time eects. All the theoretical statements are general in terms of dimension. Two dimensional numericale xamples are considered to support our theoretical arguments

  • 4. Arjmand, Doghonay
    et al.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Estimates for the upscaling error in heterogeneous multiscale methods for wave propagation problems in locally periodic media2017Ingår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 15, nr 2, s. 948-976Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper concerns the analysis of a multiscale method for wave propagation problems in microscopically nonhomogeneous media. A direct numerical approximation of such problems is prohibitively expensive as it requires resolving the microscopic variations over a much larger physical domain of interest. The heterogeneous multiscale method (HMM) is an efficient framework to approximate the solutions of multiscale problems. In the HMM, one assumes an incomplete macroscopic model which is coupled to a known but expensive microscopic model. The micromodel is solved only locally to upscale the parameter values which are missing in the macro model. The resulting macroscopic model can then be solved at a cost independent of the small scales in the problem. In general, the accuracy of the HMM is related to how good the upscaling step approximates the right macroscopic quantities. The analysis of the method that we consider here was previously addressed only in purely periodic media, although the method itself is numerically shown to be applicable to more general settings. In the present study, we consider a more realistic setting by assuming a locally periodic medium where slow and fast variations are allowed at the same time. We then prove that the HMM captures the right macroscopic effects. The generality of the tools and ideas in the analysis allows us to establish convergence rates in a multidimensional setting. The theoretical findings here imply an improved convergence rate in one dimension, which also justifies the numerical observations from our earlier study.

  • 5. Benamou, J. D.
    et al.
    Collino, F.
    Runborg, Olof
    KTH, Tidigare Institutioner                               , Numerisk analys och datalogi, NADA.
    Numerical microlocal analysis of harmonic wavefields2004Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 199, nr 2, s. 717-741Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present and test a numerical method which, given an analytical or numerical solution of the Helmholtz equation in a neighborhood of a fixed observation point and assuming that the geometrical optics approximation is relevant, determines at this point the number of crossing rays and computes their directions and associated complex amplitudes.

  • 6. Castella, F.
    et al.
    Perthame, B.
    Runborg, Olof
    High frequency limit of the Helmholtz equation. II. Source on a general smooth manifold2002Ingår i: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 27, nr 04-mar, s. 607-651Artikel i tidskrift (Refereegranskat)
  • 7. Daubechies, I.
    et al.
    Runborg, Olof
    KTH, Tidigare Institutioner, Numerisk analys och datalogi, NADA.
    Sweldens, W.
    Normal multiresolution approximation of curves2004Ingår i: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 20, nr 3, s. 399-463Artikel i tidskrift (Refereegranskat)
    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 study properties such as regularity, convergence, and stability of a normal multiresolution analysis. In particular, we show that these properties critically depend on the underlying subdivision scheme and that, in general, the convergence of normal multiresolution approximations equals the convergence of the underlying subdivision scheme.

  • 8. Daubechies, Ingrid
    et al.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Zou, Jing
    A sparse spectral method for homogenization multiscale problems2007Ingår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 6, nr 3, s. 711-740Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We develop a new sparse spectral method, in which the fast Fourier transform (FFT) is replaced by RAlSFA (randomized algorithm of sparse Fourier analysis); this is a sublinear randomized algorithm that takes time O(B log N) to recover a B-term Fourier representation for a signal of length N, where we assume B << N. To illustrate its potential, we consider the parabolic homogenization problem with a characteristic. ne scale size epsilon. For fixed tolerance the sparse method has a computational cost of O(vertical bar log epsilon vertical bar) per time step, whereas standard methods cost at least O(epsilon(-1)). We present a theoretical analysis as well as numerical results; they show the advantage of the new method in speed over the traditional spectral methods when epsilon is very small. We also show some ways to extend the methods to hyperbolic and elliptic problems.

  • 9. Di, Yana
    et al.
    Popovic, Jelena
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    AN ADAPTIVE FAST INTERFACE TRACKING METHOD2015Ingår i: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 33, nr 6, s. 576-586Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    An adaptive numerical scheme is developed for the propagation of an interface in a velocity field based on the fast interface tracking method proposed in [2]. A multiresolution stategy to represent the interface instead of point values, allows local grid refinement while controlling the approximation error on the interface. For time integration, we use an explicit Runge-Kutta scheme of second-order with a multiscale time step, which takes longer time steps for finer spatial scales. The implementation of the algorithm uses a dynamic tree data structure to represent data in the computer memory. We briefly review first the main algorithm, describe the essential data structures, highlight the adaptive scheme, and illustrate the computational efficiency by some numerical examples.

  • 10.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Holst, Henrik
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time2011Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Multiscale problems are computationally costly to solve by direct simulation because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multiscale wave propagation following the framework of the heterogeneous multiscale method. The numerical methods couple simulations on macro- and microscales for problems with rapidly fluctuating material coefficients. The computational complexity of the new method is significantly lower than that of traditional techniques. We focus on HMM approximation applied to long time integration of one-dimensional wave propagation problems in both periodic and non-periodic medium and show that the dispersive effect that appear after long time is fully captured.

  • 11.
    Engquist, Björn
    et al.
    University of Texas Austin.
    Holst, Henrik
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Multiscale Methods for One Dimensional Wave Propagation with High Frequency Initial Data2011Rapport (Övrigt vetenskapligt)
    Abstract [en]

    High frequency wave propagation problems are computationally costly to solve by traditional techniques because the short wavelength must be well represented over a domain determined by the largest scales of the problem. We have developed and analyzed a new numerical method for high frequency wave propagation in the framework of heterogeneous multiscale methods, closely related to the analytical method of geometrical optics. The numerical method couples simulations on macro- and micro-scales for problems with highly oscillatory initial data. The method has a computational complexity essentially independent of the wavelength. We give one numerical example with a sharp but regular jump in velocity on the microscopic scale for which geometrical optics fails but our HMM gives correct results. We briefly discuss how the method can be extended to higher dimensional problems.

  • 12.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Holst, Henrik
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Multiscale methods for the wave equation2007Ingår i: PAMM · Proc. Appl. Math. Mech. 7, 2007, s. 1140903-1140904Konferensbidrag (Övrigt vetenskapligt)
    Abstract [en]

    We consider the wave equation in a medium with a rapidly varying speed of propagation. We construct a multiscale schemebased on the heterogeneous multiscale method, which can compute the correct coarse behavior of wave pulses traveling in themedium, at a computational cost essentially independent of the size of the small scale variations. This is verified by theoreticalresults and numerical examples.

  • 13.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Holst, Henrik
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Multi-scale methods for wave propagation in heterogeneous media2011Ingår i: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 9, nr 1, s. 33-56Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Multi-scale wave propagation problems are computationally costly to solve by traditional techniques because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multi-scale wave propagation in the framework of heterogeneous multi-scale method. The numerical methods couple simulations on macro-and micro-scales for problems with rapidly oscillating coefficients. We show that the complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the micro-scale. A convergence proof is given and numerical results are presented for periodic problems in one, two, and three dimensions. The method is also successfully applied to non-periodic problems and for long time integration where dispersive effects occur.

  • 14.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Holst, Henrik
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Multiscale Methods for Wave Propagation in Heterogeneous Media Over Long Time2012Ingår i: Numerical Analysis of Multiscale Computations / [ed] Björn Engquist, Olof Runborg, Yen-Hsi R. Tsai, Springer Verlag , 2012, s. 167-186Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

    Multiscale wave propagation problems are computationally costly to solve by traditional techniques because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multiscale wave propagation in the framework of the heterogeneous multiscale method (HMM). The numerical methods couple simulations on macro- and microscales for problems with rapidly oscillating coefficients. The complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the smallest scale, when computing solutions at a fixed time and accuracy. We show numerical examples of the HMM applied to long time integration of wave propagation problems in both periodic and non-periodic medium. In both cases our HMM accurately captures the dispersive effects that occur. We also give a stability proof for the HMM, when it is applied to long time wave propagation problems.

  • 15.
    Engquist, Björn
    et al.
    Department of Mathematics and Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA.
    Häggblad, Jon
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Numerical subgrid scale models for the Yee schemeManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    The Yee scheme is a very common and practical algorithm for the simulation of wave propagation on uniform grids.  We develop numerical subgrid scale models in order to incorporate effects of obstacles and holes that are smaller than the grid spacing. The models are based on pre-computing at the microscale, and are thus including the effect of the detailed small scale shape.  Numerical examples in 1D, 2D and 3D are given.

  • 16.
    Engquist, Björn
    et al.
    Univ Texas Austin.
    Häggblad, Jon
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    On Energy Preserving Consistent Boundary Conditions for the Yee Scheme in 2D2012Ingår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 52, nr 3, s. 615-637Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The Yee scheme is one of the most popular methods for electromagnetic wave propagation. A main advantage is the structured staggered grid, making it simple and efficient on modern computer architectures. A downside to this is the difficulty in approximating oblique boundaries, having to resort to staircase approximations. In this paper we present a method to improve the boundary treatment in two dimensions by, starting from a staircase approximation, modifying the coefficients of the update stencil so that we can obtain a consistent approximation while preserving the energy conservation, structure and the optimal CFL-condition of the original Yee scheme. We prove this in L_2 and verify it by numerical experiments.

  • 17.
    Engquist, Björn
    et al.
    Department of Mathematics and Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA.
    Häggblad, Jon
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Tornberg, Anna-Karin
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    On Consistent Boundary Conditions for the Yee Scheme in 3DManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    The standard staircase approximation of curved boundaries in the Yee scheme is inconsistent. Consistency can however be achieved by modifying the algorithm close to the boundary.  We consider a technique to consistently model curved boundaries where the coefficients of the update stencil is modified, thus preserving the Yee structure.  The method has previously been successfully applied to acoustics in two and three dimension, as well as electromagnetics in two dimensions.  In this paper we generalize to electromagnetics in three dimensions.  Unlike in previous cases there is a non-zero divergence growth along the boundary that needs to be projected away.  We study the convergence and provide numerical examples that demonstrates the improved accuracy.

  • 18.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Lotstedt, P.
    Runborg, O.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Multiscale Modeling and Simulation in Science2009Bok (Refereegranskat)
  • 19.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Lotstedt, P.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Multiscale methods in science and engineering2005Bok (Refereegranskat)
  • 20.
    Engquist, Björn
    et al.
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Wavelet based numerical homogenization2009Ingår i: Highly Oscillatory Problems / [ed] Bjorn Engquist, Athanasios Fokas, Ernst Hairer, Arieh Iserles, Cambridge University Press, 2009, s. 98-126Kapitel i bok, del av antologi (Refereegranskat)
    Abstract [en]

    We consider multiscale differential equations in which the operator varies rapidly over fine scales. Direct numerical simulation methods need to resolve the small scales and they therefore become very expensive for such problems when the computational domain is large. Inspired by classical homogenization theory, we describe a numerical procedure for homogenization, which starts from a fine discretization of a multiscale differential equation, and computes a discrete coarse grid operator which incorporates the influence of finer scales. In this procedure the discrete operator is represented in a wavelet space, projected onto a coarser subspace and approximated by a banded or block-banded matrix. This wavelet homogenization applies to a wider class of problems than classical homogenization. The projection procedure is general and we give a presentation of a framework in Hilbert spaces, which also applies to the differential equation directly. We show numerical results when the wavelet based homogenization technique is applied to discretizations of elliptic and hyperbolic equations, using different approximation strategies for the coarse grid operator.

  • 21.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Tidigare Institutioner, Numerisk analys och datalogi, NADA.
    Tornberg, Anna Karin
    High-frequency wave propagation by the segment projection method2002Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 178, nr 2, s. 373-390Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Geometrical optics is a standard technique used for the approximation of high-frequency wave propagation. Computational methods based on partial differential equations instead of the traditional ray tracing have recently been applied to geometrical optics. These new methods have a number of advantages but typically exhibit difficulties with linear superposition of waves. In this paper we introduce a new partial differential technique based on the segment projection method in phase space. The superposition problem is perfectly resolved and so is the problem of computing amplitudes in the neighborhood of caustics. The computational complexity is of the same order as that of ray tracing. The new algorithm is described and a number of computational examples are given. including a simulation of waveguides.

  • 22. Engquist, Björn
    et al.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Tsai, Y. -HR.
    Preface2012Ingår i: Workshop on Numerical Analysis and Multiscale Computations, 2009, Springer Verlag , 2012Konferensbidrag (Refereegranskat)
  • 23. Gosse, L.
    et al.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Finite moment problems and applications to multiphase computations in geometric optics2005Ingår i: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 3, nr 3, s. 373-392Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Recovering a function out of a finite number of moments is generally an ill-posed inverse problem. We focus on two special cases arising from applications to multiphase geometric optics computations where this problem can be carried out in a restricted class of given densities. More precisely, we present a simple algorithm for the inversion of Markov's moment problem which appears in the treatment of Brenier and Corrias' K-multibranch solutions and study Stieltje's algorithm in order to process moment systems arising from a Wigner analysis. Numerical results are provided for moderately intricate wave-fields.

  • 24. Gosse, L.
    et al.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Resolution of the finite Markov moment problem2005Ingår i: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 341, nr 12, s. 775-780Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We expose in full detail a constructive procedure to invert the so-called 'finite Markov moment problem'. The proofs rely on the general theory of Toeplitz matrices together with the classical Newton's relations.

  • 25. Gosse, Laurent
    et al.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Existence, uniqueness, and a constructive solution algorithm for a class of finite Markov moment problems2007Ingår i: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 68, nr 6, s. 1618-1640Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider a class of finite Markov moment problems with an arbitrary number of positive and negative branches. We show criteria for the existence and uniqueness of solutions, and we characterize in detail the nonunique solution families. Moreover, we present a constructive algorithm to solve the moment problems numerically and prove that the algorithm computes the right solution.

  • 26.
    Häggblad, Jon
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Accuracy of staircase approximations in finite-difference methods for wave propagation2014Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 128, nr 4, s. 741-771Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    While a number of increasingly sophisticated numerical methods have been developed for time-dependent problems in electromagnetics, the Yee scheme is still widely used in the applied fields, mainly due to its simplicity and computational efficiency. A fundamental drawback of the method is the use of staircase boundary approximations, giving inconsistent results. Usually experience of numerical experiments provides guidance of the impact of these errors on the final simulation result. In this paper, we derive exact discrete solutions to the Yee scheme close to the staircase approximated boundary, enabling a detailed theoretical study of the amplitude, phase and frequency errors created. Furthermore, we show how evanescent waves of amplitude occur along the boundary. These characterize the inconsistencies observed in electromagnetic simulations and the locality of the waves explain why, in practice, the Yee scheme works as well as it does. The analysis is supported by detailed proofs and numerical examples.

  • 27. Kieri, Emil
    et al.
    Kreiss, Gunilla
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrodinger Equations2015Ingår i: Advances in Applied Mathematics and Mechanics, ISSN 2070-0733, E-ISSN 2075-1354, Vol. 7, nr 6, s. 687-714Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the semiclassical regime, solutions to the time-dependent Schrodinger equation for molecular dynamics are highly oscillatory. The number of grid points required for resolving the oscillations may become very large even for simple model problems, making solution on a grid intractable. Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the atomic nuclei are heavy and the potential is smooth. However, when the potential has variations on a small length-scale, quantum phenomena become important. Then asymptotic methods are less accurate. The two classes of methods perform well in different parameter regimes. This opens for hybrid methods, using Gaussian beams where we can and finite differences where we have to. We propose a new method for treating the coupling between the finite difference method and Gaussian beams. The new method reduces the needed amount of overlap regions considerably compared to previous methods, which improves the efficiency.

  • 28. Krishnan, J.
    et al.
    Runborg, Olof
    Kevrekidis, I. G.
    Bifurcation analysis of nonlinear reaction-diffusion problems using wavelet-based reduction techniques2004Ingår i: Computers and Chemical Engineering, ISSN 0098-1354, E-ISSN 1873-4375, Vol. 28, nr 4, s. 557-574Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Using a computational method for numerical homogenization, we perform the coarse-scale bifurcation analysis of nonlinear reaction-diffusion problems in both uniform and spatially varying media. The method is based on wavelet decomposition and projection of the differential equation on coarse scale wavelet spaces. The approach is capable of capturing turning points and pitchfork bifurcations of sharp, front-like solutions at the coarse level.

  • 29. Liu, H.
    et al.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.). KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Tanushev, N. M.
    Sobolev and max norm error estimates for Gaussian beam superpositions2016Ingår i: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 14, nr 7, s. 2037-2072Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This work is concerned with the accuracy of Gaussian beam superpositions, which are asymptotically valid high frequency solutions to linear hyperbolic partial differential equations and the Schrödinger equation. We derive Sobolev and max norms estimates for the difference between an exact solution and the corresponding Gaussian beam approximation, in terms of the short wavelength e. The estimates are performed for the scalar wave equation and the Schrödinger equation. Our result demonstrates that a Gaussian beam superposition with kth order beams converges to the exact solution as O(εk/2-s) in order s Sobolev norms. This result is valid in any number of spatial dimensions and it is unaffected by the presence of caustics in the solution. In max norm, we show that away from caustics the convergence rate is O(ε⌈k/2⌉) and away from the essential support of the solution, the convergence is spectral in ε. However, in the neighborhood of a caustic point we are only able to show the slower, and dimensional dependent, rate O(ε(k-n)/2) in n spatial dimensions.

  • 30. Liu, Hailiang
    et al.
    Ralston, James
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Tanushev, Nicolay M.
    Gaussian beam methods for the helmholtz equation2014Ingår i: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 74, nr 3, s. 771-793Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this work we construct Gaussian beam approximations to solutions of the high frequency Helmholtz equation with a localized source. Under the assumption of nontrapping rays we show error estimates between the exact outgoing solution and Gaussian beams in terms of the wave number k, both for single beams and superposition of beams. The main result is that the relative local L-2 error in the beam approximations decay as k(-N/2) independent of dimension and presence of caustics for Nth order beams.

  • 31. Liu, Hailiang
    et al.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Tanushev, Nicolay M.
    Error estimates for Gaussian beam superpositions2013Ingår i: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 82, nr 282, s. 919-952Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations, concentrated on a single curve through the physical domain. They can also be extended to some dispersive wave equations, such as the Schrodinger equation. Superpositions of Gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve. This work is concerned with the accuracy of Gaussian beam superpositions in terms of the wavelength epsilon. We present a systematic construction of Gaussian beam superpositions for all strictly hyperbolic and Schrodinger equations subject to highly oscillatory initial data of the form Ae(i Phi/) (epsilon). Through a careful estimate of an oscillatory integral operator, we prove that the k-th order Gaussian beam superposition converges to the original wave field at a rate proportional to epsilon(k/2) in the appropriate norm dictated by the well-posedness estimate. In particular, we prove that the Gaussian beam superposition converges at this rate for the acoustic wave equation in the standard, epsilon-scaled, energy norm and for the Schrodinger equation in the L-2 norm. The obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics. We present a numerical study of convergence for the constant coefficient acoustic wave equation in R-2 to analyze the sharpness of the theoretical results.

  • 32.
    Malenova, Gabriela
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Motamed, M.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Stochastic regularity of a quadratic observable of high-frequency waves2017Ingår i: RESEARCH IN THE MATHEMATICAL SCIENCES, ISSN 2197-9847, Vol. 4, artikel-id UNSP 3Artikel i tidskrift (Refereegranskat)
  • 33.
    Malenova, Gabriela
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Motamed, M.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Tempone, R.
    A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty2016Ingår i: SIAM/ASA Journal on Uncertainty Quantification, ISSN 1560-7526, E-ISSN 2166-2525, Vol. 4, nr 1, s. 1084-1110Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the wave equation with highly oscillatory initial data, where there is uncertainty in the wave speed, initial phase, and/or initial amplitude. To estimate quantities of interest related to the solution and their statistics, we combine a high-frequency method based on Gaussian beams with sparse stochastic collocation. Although the wave solution, u(epsilon), is highly oscillatory in both physical and stochastic spaces, we provide theoretical arguments for simplified problems and numerical evidence that quantities of interest based on local averages of |u(epsilon)|(2) are smooth, with derivatives in the stochastic space uniformly bounded in epsilon, where epsilon denotes the short wavelength. This observable related regularity makes the sparse stochastic collocation approach more efficient than Monte Carlo methods. We present numerical tests that demonstrate this advantage.

  • 34.
    Malenova, Gabriela
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Motamed, Mohammad
    Department of Mathematics and Statistics, The University of New Mexico, Albuquerque, NM 87131, USA.
    Runborg, Olof
    KTH, Centra, SeRC - Swedish e-Science Research Centre. KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Regularity of stochastic observables in Gaussian beam superposition of high-frequency waves2017Ingår i: Research in mathematical sciences, ISSN 2197-9847Artikel i tidskrift (Övrigt vetenskapligt)
    Abstract [en]

    We consider high-frequency waves satisfying the scalar wave equation with highly oscillatory initial data. The wave speed, and the phase and amplitude of the initial data are assumed to be uncertain, described by a finite number of random variables with known probability distributions. We define quantities of interest (QoIs), or observables, as local averages of the squared modulus of the wave solution. We aim to quantify the regularity of these QoIs in terms of the input random parameters, and the wave length, i.e., to estimate the size of their derivatives. The regularity is important for uncertainty quantification methods based on interpolation in the stochastic space. In particular, the size of the derivatives should be bounded independently of the wave length. In this paper, we are able to show that when these QoIs are approximated by Gaussian beam superpositions, they indeed have this property, despite the highly oscillatory character of the waves.

  • 35.
    Malenova, Gabriela
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Fast evaluation of high frequency observables by by Gaussian beams and the numerical steepest descent methodManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We consider observables (quantities of interest) of the wave equation where the initial data has a wavelength which is very small compared to the distance traveled by the wave. Computation by direct methods becomes expensive owing to the short wavelength. Therefore, two fast asymptotic methods are proposed to address the difficulties: the Gaussian beam method to approximate the wave solution, and the numerical steepest descent method to compute the quantity of interest. In this work, we demonstrate the use of the latter method and present the corresponding error estimates. We show that the latter method has a lower computational cost compared to the previously considered methods and present the corresponding error estimates.

  • 36.
    Malenova, Gabriela
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Stochastic regularity of general quadratic observables of high frequency wavesManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We consider solutions to the wave equation with uncertain initial data and medium, whose wavelength is short compared to thedistance traveled by the wave. We are interested in the statistics of the observables, i.e. functionals of the wave solution. Computation by direct methods gets very expensive or outright non-feasible as the wavelength decreases. To address the difficulties, we proposed a method consisting of the Gaussian beam method to treat the high frequencies and the sparse stochastic collocation method to remedy the curse of dimensionality in the stochastic space. For the latter method to converge, we need the observables to satisfy certain stochastic regularity conditions. The main contribution of this work is to show this regularity for a set of quadratic observables obtained by the Gaussian beam approximation of the wave solution.

  • 37.
    Marin, Oana
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Corrected trapezoidal rules for a class of singular functions2014Ingår i: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 34, nr 4, s. 1509-1540Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A set of accurate quadrature rules applicable to a class of integrable functions with isolated singularities is designed and analysed theoretically in one and two dimensions. These quadrature rules are based on the trapezoidal rule with corrected quadrature weights for points in the vicinity of the singularity. To compute the correction weights, small-size ill-conditioned systems have to be solved. The convergence of the correction weights is accelerated by the use of compactly supported functions that annihilate boundary errors. Convergence proofs with error estimates for the resulting quadrature rules are given in both one and two dimensions. The tabulated weights are specific for the singularities under consideration, but the methodology extends to a large class of functions with integrable isolated singularities. Furthermore, in one dimension we have obtained a closed form expression based on which the modified weights can be computed directly.

  • 38. Motamed, M.
    et al.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    A wavefront-based Gaussian beam method for computing high frequency wave propagation problems2015Ingår i: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 69, nr 9, s. 949-963Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a novel wavefront method based on Gaussian beams for computing high frequency wave propagation problems. Unlike standard geometrical optics, Gaussian beams compute the correct solution of the wave field also at caustics. The method tracks a front of two canonical beams with two particular initial values for width and curvature. In a fast post-processing step, from the canonical solutions we recreate any other Gaussian beam with arbitrary initial data on the initial front. This provides a simple mechanism to include a variety of optimization processes, including error minimization and beam width minimization, for a posteriori selection of optimal beam initial parameters. The performance of the method is illustrated with two numerical examples.

  • 39.
    Motamed, Mohammad
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    A fast method for the creeping ray contribution to scattering problems2006Ingår i: Mathematical Modeling of Wave Phenomena / [ed] Nilsson, B; Fishman, L, MELVILLE, NY: AMER INST PHYSICS , 2006, Vol. 834, s. 56-64Konferensbidrag (Refereegranskat)
    Abstract [en]

    Creeping rays can give an important contribution to the solution of medium to high frequency scattering problems. They are generated at the shadow lines of the illuminated scatterer by grazing incident rays and propagate along geodesics on the scatterer surface, continuously shedding diffracted rays in their tangential direction. In this paper we show how the ray propagation problem can be formulated as a partial differential equation (PDE) in a three-dimensional phase space. To solve the PDE we use a fast marching method. The PDE solution contains information about all possible creeping rays. This information includes the phase and amplitude of the field, which are extracted by a fast postprocessing. Computationally the cost of solving the PDE is less than tracing all rays individually by solving a system of ordinary differential equations(ODE). We consider an application to monostatic radar cross section problems where creeping rays from all illumination angles must be computed.

  • 40.
    Motamed, Mohammad
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    A fast phase space method for computing creeping rays2006Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 219, nr 1, s. 276-295Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Creeping rays can give an important contribution to the solution of medium to high frequency scattering problems. They are generated at the shadow lines of the illuminated scatterer by grazing incident rays and propagate along geodesics on the scatterer surface, continuously shedding diffracted rays in their tangential direction. In this paper, we show how the ray propagation problem can be formulated as a partial differential equation (PDE) in a three-dimensional phase space. To solve the PDE we use a fast marching method. The PDE solution contains information about all possible creeping rays. This information includes the phase and amplitude of the field, which are extracted by a fast post-processing. Computationally, the cost of solving the PDE is less than tracing all rays individually by solving a system of ordinary differential equations. We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method and a comparison with the results of ray tracing are presented.

  • 41.
    Motamed, Mohammad
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems2007Ingår i: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 5, nr 3, s. 617-648Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a multiple-patch phase space method for computing trajectories on two-dimensional manifolds possibly embedded in a higher-dimensional space. The dynamics of trajectories are given by systems of ordinary differential equations (ODEs). We split the manifold into multiple patches where each patch has a well-defined regular parameterization. The ODEs are formulated as escape equations, which are hyperbolic partial differential equations (PDEs) in a three- dimensional phase space. The escape equations are solved in each patch, individually. The solutions of individual patches are then connected using suitable inter-patch boundary conditions. Properties for particular families of trajectories are obtained through a fast post-processing. We apply the method to two different problems : the creeping ray contribution to mono-static radar cross section computations and the multivalued travel-time of seismic waves in multi-layered media. We present numerical examples to illustrate the accuracy and efficiency of the method.

  • 42.
    Motamed, Mohammad
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    A Wave Front-based Gaussian Beam Method for Computing High Frequency WavesManuskript (Övrigt vetenskapligt)
  • 43.
    Motamed, Mohammad
    et al.
    Department of Mathematics, Simon Fraser University.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Taylor expansion and discretization errors in Gaussian beam superposition2010Ingår i: Wave motion, ISSN 0165-2125, E-ISSN 1878-433X, Vol. 47, nr 7, s. 421-439Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The Gaussian beam superposition method is an asymptotic method for computing high frequency wave fields in smoothly varying inhomogeneous media. In this paper we study the accuracy of the Gaussian beam superposition method and derive error estimates related to the discretization of the superposition integral and the Taylor expansion of the phase and amplitude off the center of the beam. We show that in the case of odd order beams, the error is smaller than a simple analysis would indicate because of error cancellation effects between the beams. Since the cancellation happens only when odd order beams are used, there is no remarkable gain in using even order beams. Moreover, applying the error estimate to the problem with constant speed of propagation, we show that in this case the local beam width is not a good indicator of accuracy, and there is no direct relation between the error and the beam width. We present numerical examples to verify the error estimates.

  • 44.
    Motamed, Mohammad
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Taylor Expansion Errors in Gaussian Beam SummationManuskript (Övrigt vetenskapligt)
  • 45.
    Möller, Joakim
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Kevrekidis, P.G.
    Lust, K.
    Kevrekidis, I.G.
    Equation-free, effective computation for discrete systems: a time stepper based approach2005Ingår i: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, ISSN 0218-1274, Vol. 15, nr 3, s. 975-996Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Pade approximations.

  • 46. Persson, P. O.
    et al.
    Runborg, Olof
    Simulation of a waveguide filter using wavelet-based numerical homogenization2001Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 166, nr 2, s. 361-382Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We apply wavelet-based numerical homogenization to the simulation of an optical waveguide filter. We use the method to derive approximate one-dimensional models and subgrid models of the filter. Numerical examples of the technique are presented, and the computational gains are investigated.

  • 47.
    Popovic, Jelena
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    A Fast Method for Solving the Helmholtz Equation Based on Wave-Splitting2009Ingår i: WAVES 2009 / [ed] Barucq, H.; Bonnet-Bendhia, A.-S.; Cohen, G.; Diaz, J.; Ezziani, A.; Joly, P., 2009, s. 220-221Konferensbidrag (Refereegranskat)
  • 48. Popovic, Jelena
    et al.
    Runborg, Olof
    KTH, Centra, SeRC - Swedish e-Science Research Centre. KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Adaptive fast interface tracking methods2017Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 337, s. 42-61Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we present a fast time adaptive numerical method for interface tracking. The method uses an explicit multiresolution description of the interface, which is represented by wavelet vectors that correspond to the details of the interface on different scale levels. The complexity of standard numerical methods for interface tracking, where the interface is described by N marker points, is 0 (N/Delta t),when a time step At is used. The methods that we propose in this paper have 0 (TOL-1/P log N + N log N) computational cost, at least for uniformly smooth problems, where TOL is some given tolerance and p is the order of the time stepping method that is used for time advection of the interface. The adaptive method is robust in the sense that it can handle problems with both smooth and piecewise smooth interfaces (e.g. interfaces with corners) while keeping a low computational cost. We show numerical examples that verify these properties.

  • 49.
    Popovic, Jelena
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Adaptive Fast Interface Tracking Methods: Part I: Time AdaptivityManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    In this paper, we present a fast adaptive numerical method for interface tracking that uses an explicit multiresolution description of the interface. The interface is represented by wavelet vectors that correspond to the details of the interface on different scale levels.The complexity of standard numerical methods for interface tracking, where the interface is described by marker points, is O(N/dt), where N is the number of points on the interface and dt is the time step. The methods that we propose in this paper have O(tol^(-1/p)log N) computational cost, where tol is some given tolerance and p is the order of the numerical method for ordinary differential equations that is used for time advection of the interface. The adaptivity makes methods robust in the sense  that they can handle problems with both smooth and non-smooth interfaces (i.e. interfaces with corners) while keeping low computational cost.

  • 50.
    Popovic, Jelena
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Adaptive Fast Interface Tracking Methods: Part II: Spatial AdaptivityManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    In this paper, we present a fast space-time adaptive numerical method for interface propagation in a time varying velocity field based on a multiresolution description of the interface. The interface is represented by wavelet vectors that correspond to the details of the interface on different scale levels.The method is an extension of the method proposed in "J. Popovic and O. Runborg, Adaptive fast interface tracking methods: Part I, preprint (2012)", which is only time adaptive and it is thus not suitable for problems with expanding interfaces. The method that we propose in this paper, remedies that disadvantage of the time adaptive method in an efficient way. 

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