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  • 1. Abdulle, A.
    et al.
    Engquist, Björn
    Mathematics Department, University of Texas at Austin.
    Finite element heterogeneous multiscale methods with near optimal computational complexity2007In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 6, no 4, 1059-1084 p.Article in journal (Refereed)
    Abstract [en]

    This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the heterogeneous multiscale methods (HMM), we propose a micro-macro approach which combines the finite element method (FEM) for the macroscopic solver and the pseudospectral method for the microsolver. Unlike the micro-macromethods based on the standard FEM proposed so far, in the HMM we obtain, for periodic homogenization problems, a method that (slow) variable.

  • 2. Andersson, U
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Ledfelt, G
    Runborg, O
    A contribution to wavelet-based subgrid modeling1999In: Applied and Computational Harmonic Analysis, ISSN 1063-5203, E-ISSN 1096-603X, Vol. 7, no 2, 151-164 p.Article in journal (Refereed)
    Abstract [en]

    A systematic technique for the derivation of subgrid scale models in the numerical solution of partial differential equations is described. The technique is based on Haar wavelet projections of the discrete operator followed by a sparse approximation. As numerical testing suggests, the resulting numerical method will accurately represent subgrid scale phenomena on a coarse grid. Applications to numerical homogenization and wave propagation in materials with subgrid inhomogeneities are presented.

  • 3. Ariel, G.
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30). University of Texas at Austin, United States .
    Kreiss, H. -O
    Tsai, R.
    Multiscale computations for highly oscillatory problems2009In: Multiscale Modeling and Simulation in Science, Springer Berlin/Heidelberg, 2009, 237-287 p.Conference paper (Refereed)
    Abstract [en]

    We review a selection of essential techniques for constructing computational multiscale methods for highly oscillatory ODEs. Contrary to the typical approaches that attempt to enlarge the stability region for specialized problems, these lecture notes emphasize how multiscale properties of highly oscillatory systems can be characterized and approximated in a truly multiscale fashion similar to the settings of averaging and homogenization. Essential concepts such as resonance, fast-slow scale interactions, averaging, and techniques for transformations to non-stiff forms are discussed in an elementary manner so that the materials can be easily accessible to beginning graduate students in applied mathematics or computational sciences.

  • 4. Ariel, G.
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Kreiss, H.-O.
    Tsai, R.
    Multiscale computations for highly oscillatory problems, Multiscale Modeling and Simulation in Science2009In: IEEE Computational Science & Engineering, ISSN 1070-9924, E-ISSN 1558-190XArticle in journal (Refereed)
  • 5.
    Ariel, Gil
    et al.
    Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
    Engquist, Björn
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Tanushev, Nicolay M.
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Tsai, Richard
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Gaussian Beam Decomposition of High Frequency Wave Fields Using Expectation-Maximization2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 6, 2303-2321 p.Article in journal (Refereed)
    Abstract [en]

    A new numerical method for approximating highly oscillatory wave fields as a superposition of Gaussian beams is presented. The method estimates the number of beams and their parameters automatically. This is achieved by an expectation–maximization algorithm that fits real, positive Gaussians to the energy of the highly oscillatory wave fields and its Fourier transform. Beam parameters are further refined by an optimization procedure that minimizes the difference between the Gaussian beam superposition and the highly oscillatory wave field in the energy norm.

  • 6.
    Ariel, Gil
    et al.
    Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
    Engquist, Björn
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Tsai, Richard
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    A multiscale method for stiff ordinary differential equations with resonance2009In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 78, no 266, 929-956 p.Article in journal (Refereed)
    Abstract [en]

    A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam.

  • 7.
    Ariel, Gil
    et al.
    Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
    Engquist, Björn
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Tsai, Richard
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    A reversible multiscale integration method2009In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 7, no 3, 595-610 p.Article in journal (Refereed)
    Abstract [en]

    A multiscale, time reversible method for computing the effective slow behavior of systems of highly oscillatory ordinary differential equations is presented. The proposed method relies on correctly tracking a set of slow variables that is sufficient to approximate any variable and functional that are slow under the dynamics of the system. The algorithm follows the framework of the heterogeneous multiscale method. The notion of time reversibility in the multiple time-scale setting is discussed. The algorithm requires nontrivial matching between the microscopic state variables and the macroscopic slow ones. Numerical examples show the efficiency of the multiscale method and the advantages of time reversibility.

  • 8.
    Ariel, Gil
    et al.
    Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
    Engquist, Björn
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Tsai, Richard
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Numerical multiscale methods for coupled oscillators2009In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 7, no 3, 1387-1404 p.Article in journal (Refereed)
    Abstract [en]

    A multiscale method for computing the effective slow behavior of a system of weakly coupled nonlinear planar oscillators is presented. The oscillators may be either in the form of a periodic solution or a stable limit cycle. Furthermore, the oscillators may be in resonance with one another and thereby generate some hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables that is sufficient to approximate any variable and functional that are slow under the dynamics of the ordinary differential equation. The technique is more efficient than existing methods, and its advantages are demonstrated with examples. The algorithm follows the framework of the heterogeneous multiscale method.

  • 9. Atle, Andreas
    et al.
    Engquist, Björn
    On surface radiation conditions for high-frequency wave scattering2007In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 204, no 2, 306-316 p.Article in journal (Refereed)
    Abstract [en]

    A new approximation of the logarithmic derivative of the Hankel function is derived and applied to high-frequency wave scattering. We re-derive the on surface radiation condition (OSRC) approximations that are well suited for a Dirichlet boundary in acoustics. These correspond to the Engquist-Majda absorbing boundary conditions. Inverse OSRC approximations are derived and they are used for Neumann boundary conditions. We obtain an implicit OSRC condition, where we need to solve a tridiagonal system. The OSRC approximations are well suited for moderate wave numbers. The approximation of the logarithmic derivative is also used for deriving a generalized physical optics approximation, both for Dirichlet and Neumann boundary conditions. We have obtained similar approximations in electromagnetics, for a perfect electric conductor. Numerical computations are done for different objects in 2D and 3D and for different wave numbers. The improvement over the standard physical optics is verified.

  • 10. Bamberberg, A
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Halpern, L
    Joly, P
    Construction and analysis of paraxial approximations in heterogeneous media1986Conference paper (Refereed)
    Abstract [en]

    We design a new family of paraxial equations in heterogeneous medium. For these equations, the reflection-transmission on an interface is continuous, which is not the case for the classical ones.

  • 11. Bamberger, A
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Halpern, L
    Joly, P
    Higher order paraxial wave equation approximations in heterogeneous media1988In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 48, 129-154 p.Article in journal (Refereed)
    Abstract [en]

    A new family of paraxial wave equation approximations is derived. These approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computational problems, e.g., in seismology, underwater acoustics and as artificial boundary conditions. The equations are written as systems which simplify computations. The support and singular support are studied; energy estimates are given which prove the well-posedness. The reflection and transmission are shown to be continuously dependent on material interfaces in heterogeneous media

  • 12. Bamberger, A
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Halpern, L
    Joly, P
    Parabolic wave equation approximations in heterogenous media,1988In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 48, 99-128 p.Article in journal (Refereed)
    Abstract [en]

    The properties of different variants of parabolic approximations of scalar wave equations are analyzed. These equations are of general form which includes those used in seismology, underwater acoustics and other applications. A new version of the parabolic approximation is derived for heterogeneous media. It has optimal properties with respect to wave reflection at material interfaces. The amplitud of the reflected and transmitted waves depend continuously on the interface. Existence, uniqueness and energy estimates are proved.

  • 13. Brenan, K.E.
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Backward differentiation approximations of nonlinear differential/algebraic systems1988In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 51, no 84, 659-676 p.Article in journal (Refereed)
    Abstract [en]

    Finite-difference approximations of dynamical systems modeled by nonlinear, semiexplicit, differential/algebraic equations are analyzed. Convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints. The appropriate asymptotic convergence rates and the leading error terms are determined.

  • 14.
    Carstensen, Carsten
    et al.
    Humboldt-Universität zu Berlin, BERLIN, GERMANY.
    Engquist, Björn
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Computational Multiscale Methods2009In: Oberwolfach Reports, ISSN 1660-8933, E-ISSN 1660-8941, Vol. 6, no 2, 1597-1599 p.Article in journal (Refereed)
    Abstract [en]

    Computational Multiscale Methods play an important role in many modern computer simulations in material sciences with different time scales and different scales in space. Besides various computational challenges, the meeting brought together various applications from many disciplines and scientists from various scientific communities.

  • 15. Chung, E.
    et al.
    Engquist, Björn
    Princeton University.
    Convergence analysis of fully discrete finite volume methods for Maxwell's equations in nonhomogeneous media2005In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 43, no 1, 303-317 p.Article in journal (Refereed)
    Abstract [en]

    We will consider both explicit and implicit fully discrete finite volume schemes for solving three-dimensional Maxwell's equations with discontinuous physical coefficients on general polyhedral domains. Stability and convergence for both schemes are analyzed. We prove that the schemes are second order accurate in time. Both schemes are proved to be first order accurate in space for the Voronoi-Delaunay grids and second order accurate for nonuniform rectangular grids. We also derive explicit expressions for the dependence on the physical parameters in all estimates.

  • 16.
    Chung, Eric T.
    et al.
    Department of Mathematics, The Chinese University of Hong Kong, Hong Kong.
    Engquist, Björn
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions2009In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 47, no 5, 3820-3848 p.Article in journal (Refereed)
    Abstract [en]

    In this paper, we developed and analyzed a new class of discontinuous Galerkin (DG) methods for the acoustic

    wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these

    schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches

    are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be

    seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally

    and globally. Moreover, it can be seen as a generalized version of the Raviart-Thomas FE method and the finite volume

    method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally

    convergent. Furthermore, in order to apply the new method for unbounded domains, we proposed a new way to handle the

    second order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.

  • 17. Chung, Eric T.
    et al.
    Engquist, Björn
    Optimal discontinuous Galerkin methods for wave propagation2006In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 44, no 5, 2131-2158 p.Article in journal (Refereed)
    Abstract [en]

    We have developed and analyzed a new class of discontinuous Galerkin methods (DG) which can be seen as a compromise between standard DG and the finite element (FE) method in the way that it is explicit like standard DG and energy conserving like FE. In the literature there are many methods that achieve some of the goals of explicit time marching, unstructured grid, energy conservation, and optimal higher order accuracy, but as far as we know only our new algorithms satisfy all the conditions. We propose a new stability requirement for our DG. The stability analysis is based on the careful selection of the two FE spaces which verify the new stability condition. The convergence rate is optimal with respect to the order of the polynomials in the FE spaces. Moreover, the convergence is described by a series of numerical experiments.

  • 18. Clayton, R
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Absorbing boundary conditions for acoustic and elastic wave equations1977In: Bulletin of The Seismological Society of America (BSSA), ISSN 0037-1106, E-ISSN 1943-3573, Vol. 67, no 6, 1529-1540 p.Article in journal (Refereed)
    Abstract [en]

    The two dimensional elastic wave equations are used to model wave propagation in mediums with large or unbounded domains. In order to numerically simulate those problems the equations have to be put on the computer and artificial boundary conditions that allow minimal wave reflection must be introduced. These boundary conditions are well known in the literature as absorbing or radiating boundary conditions. They have been of interest to physicists and meteorologists for some time. In this paper three different methods for deriving radiating boundary conditions for the elastic wave equations are presented. One of these techniques gives exact absorbing boundary conditions for both P (longitudinal) and S (transverse) waves generated from a surface source. From this, approximate absorbing boundary conditions are derived. These conditions are easy to employ in finite difference codes.

  • 19. Clayton, R.W.
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Absorbing boundary conditions for wave-equation migration1980In: Geophysics, ISSN 0016-8033, E-ISSN 1942-2156, Vol. 45, no 5, 895-904 p.Article in journal (Refereed)
    Abstract [en]

    The standard boundary conditions used at the sides of’s seismic section in wave-equation migration generateartificial reflections. These reflections from the edges of the computational grid appear as artifacts in the finalsection. Padding the section with zero traces on either 5ide adds to the cost of’migration and simply delays theine\ itable reflections.We develop stable absorbing boundary condition5 that annihilate almost all of the artificial reflections. Thisih demonstrated analytically and with synthetic examples. The absorbing boundary conditions presented canbe used with any ofthe different type\ of finite-diftcrence wa\e-equation migration, at essentially no extra cost.

  • 20. Deuflhard, P
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Large scale scientific computing: Proceedings of workshop held at the Oberwolfach Mathematical Institute1987 (ed. 7)Book (Refereed)
    Abstract [en]

    The series Lecture Notes in Computer Science (LNCS), including its subseries Lecture Notes in Artificial Intelligence (LNAI) and Lecture Notes in Bioinformatics (LNBI), has established itself as a medium for the publication of new developments in computer science and information technology research and teaching - quickly, informally, and at a high level.

    The cornerstone of LNCS's editorial policy is its unwavering commitment to report the latest results from all areas of computer science and information technology research, development, and education. LNCS has always enjoyed close cooperation with the computer science R & D community, with numerous renowned academics, and with prestigious institutes and learned societies. Our mission is to serve this community by providing a most valuable publication service.

    LNCS commenced publication in 1973 and quite rapidly attracted attention, not least because of its thus far unprecedented publication turnaround times. The 1980s and 1990s witnessed a substantial growth in the series, particularly in terms of volumes published. In the late 1990s we developed a systematic approach to providing LNCS in a full-text electronic version, in parallel to the printed books. Another new feature introduced in the late 1990s was the conceptualization of a couple of color-cover sublines. Still, original research results reported in proceedings and postproceedings remain the core of LNCS.

  • 21. Dorobantu, M
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Wavelet-based numerical homogenization1998In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 35, no 2, 540-559 p.Article in journal (Refereed)
    Abstract [en]

    A numerical homogenization procedure for elliptic differential equations is presented. It is based on wavelet decompositions of discrete operators in find and coarse scale components followed by the elemination of the fine scale contributions. If the operator is in divergence form, this is preserved by the homogenization procedure. For periodic problems, results similar to classical effective coefficient theory is proved. The procedure can be applied to problems that are not cell-periodic.

  • 22. Durlofsky, L. J.
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Osher, S
    Triangle based adaptive stencils for the solution of hyperbolic conservation laws1992In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, ISSN 0021-9991, Vol. 98, no 1, 64-73 p.Article in journal (Refereed)
    Abstract [en]

    A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.

  • 23. E., Weinan
    et al.
    Engquist, Björn
    University of Texas, Austin.
    The Heterogeneous Multi-Scale Method for Homogenization Problems2005In: Proceedings of Conference on Multiscale Methods in Science and Engineering, 2005, 71-92 p.Conference paper (Refereed)
  • 24. E., Weinan
    et al.
    Engquist, Björn
    University of Texas, Austin.
    Li, Xiantao
    Ren, Weiqing
    Vanden-Eijnden, Eric
    Heterogeneous multiscale methods: A review2007In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 2, no 3, 367-450 p.Article, review/survey (Refereed)
    Abstract [en]

    This paper gives a systematic introduction to HMM, the heterogeneous multiscale methods, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several application areas, including complex fluids, micro-fluidics, solids, interface problems, stochastic problems, and statistically self-similar problems. Emphasis is given to the technical tools, such as the various constrained molecular dynamics, that have been developed, in order to apply HMM to these problems. Examples of mathematical results on the error analysis of HMM are presented. The review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.

  • 25. Eliasson, P
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    The effect of dissipation and coarse grid resolution for multigrid in flow problem1996Conference paper (Refereed)
    Abstract [en]

    This paper is to investigate the effects of the numerical dissipation and the resolution of the solution on coarser grids for multigrid with the Euler equation approximations. The convergence is accomplished by multi-stage explicit time-stepping to steady state accelerated by FAS multigrid. A theoretical investigation is carried out for linear hyperbolic equations in one and two dimensions. The spectra reveals that for stability and hence robustness of spatial discretizations with a small amount of numerical dissipation the grid transfer operators have to be accurate enough and the smoother of low temporal accuracy. Numerical results give grid independent convergence in one dimension. For twodimensional problems with a small amount of numerical dissipation, however, only a few grid levels contribute to an increased speed of convergence. This is explained by the small numerical dissipation leading to dispersion. Increasing the mesh density and hence making the problem over resolved increases the number of mesh levels contributing to an increased speed of convergence. If the steady state equations are elliptic, all grid levels contribute to the convergence regardless of the mesh density

  • 26.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Far field computational boundary conditions1989In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 7, no 2, 113-120 p.Article in journal (Refereed)
  • 27.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Inverse imaging methods in exploration seismology1980Conference paper (Refereed)
  • 28.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Multi-scale modeling2005In: Perspectives in Analysis: ESSAYS IN HONOR OF LENNART CARLESON'S 75TH BIRTHDAY / [ed] Benedicks, M; Jones, PW; Smirnov, S, BERLIN: SPRINGER-VERLAG BERLIN , 2005, Vol. 27, 51-61 p.Conference paper (Refereed)
    Abstract [en]

    If a mathematical model contains many different scales the computational cost for its numerical solution is very large. The smallest scale must be resolved over the distance of the largest scale. A huge number of unknowns are required and until recently many such problems could not be treated computationally. We will discuss a new set of numerical techniques that couples models for different scales in the same simulation in order to handle many realistic multi-scale problems. In most of this presentation we shall survey existing methods but we shall also give some new observations.

  • 29.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Numerical computation of oscillatory solutions to hyperbolic problems1991Conference paper (Refereed)
  • 30.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Variable velocity: wave extrapolation and reflection1977Conference paper (Refereed)
  • 31.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Wavelet based numerical homogenization1998Conference paper (Refereed)
    Abstract [en]

    Classical homogenization is an analytic technique for approximating multiscale differential equations. The numbers of scales are reduced and the resulting equations are easier to analyze or numerically approximate. The class of problems that classical homogenization applies to is quite restricted. We shall describe a numerical procedure for homogenization, which starts from a discretization of the multiscale differential equation. In this procedure the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The 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 space, which also applies to the differential equation directly. The wavelet based homogenization technique is applied to discretizations of the Helmholtz equation. In one problem from electromagnetic compatibility a subgrid scale geometrical detail is represented on a coarser grid. In another a wave-guide filter is efficiently approximated in a lower dimension. The technique is also applied to the derivation of effective equations for a nonlinear problem and to the derivation of coarse grid operators in multigrid. These multigrid methods work very well for equations with highly oscillatory or discontinuous coefficients.

  • 32.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Well-posedness of one-way wave equations1976Conference paper (Refereed)
  • 33.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Fatemi, E
    Osher, S
    Numerical solution of high frequency expansion for hyperbolic equations1994Conference paper (Refereed)
  • 34.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Fokas, Antonios
    Heirer, E.
    Iserles, A.
    Highly Oscillatory Problems2009Book (Refereed)
  • 35.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Fornberg, B
    Johansson, J
    Studiematerial till Numerisk analys1970Book (Refereed)
  • 36.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Fornberg, B
    Johansson, J
    Studiematerial till Numeriska metoder1970Book (Refereed)
  • 37.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Golub, G.
    From numerical analysis to computational science2000In: Mathematics Unlimited: 2001 and Beyond, Springer, 2000, 433-448 p.Chapter in book (Refereed)
    Abstract [en]

    Introduction The modern development of numerical computing is driven by the rapid increase in computer performance. The present exponential growth approximately follows Moore's law, doubling in capacity every eighteen months. Numerical computing has, of course, been part of mathematics for a very long time. Algorithms by the names of Euclid, Newton and Gauss, originally designed for computation "by hand", are still used today in computer simulations. The electronic computer originated from the intense research and development done during the second world war. In the early applications of these computers the computational techniques that were designed for calculation by pencil and paper or tables and mechanical machines were directly implemented on the new devices. Together with a deeper understanding of the computational processes new algorithms soon emerged. The foundation of modern numerical analysis was built in the period from the late forties to the late fifties. It became justi

  • 38.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Greenbaum, A
    Murphy, W.D.
    Global boundary conditions and fast Helmholtz solvers1989In: IEEE transactions on magnetics, ISSN 0018-9464, E-ISSN 1941-0069, Vol. 25, no 4, 2804-2806 p.Article in journal (Refereed)
    Abstract [en]

    Electromagnetic scattering from a conducting two-dimensional cylinder is modeled by solving Helmholtz's equation with the far-field radiation boundary condition replaced by a global boundary condition allowing. This allows the boundary condition to be applied very near the scatterer. The discrete problem is solved by a biconjugate gradient algorithm. IncompleteLUdecomposition is used as a preconditioning strategy, resulting in very fast convergence. The numerical solution is compared with known solutions and found to converge fasters forka⩽10, wherekis the wave number andais the radius of the cylinder

  • 39.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Gustafsson, B
    Steady state computations for wave propagation problems1987In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 49, 39-64 p.Article in journal (Refereed)
    Abstract [en]

    The behavior of difference approximations of hyperbolic partial differential equations as time t → ∞  is studied. The rate of convergence to steady state is analyzed theoretically and expe imentally for the advection equation and the linearized Euler equations. The choice of difference formulas and boundary conditions strongly influences the rate of convergence in practical steady state calculations. In particular it is shown that upwind difference methods and characteristic boundary conditions have very attractive convergence properties

  • 40.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Gustafsson, B
    Third International Conference on Hyperbolic Problems: Proceedings of the conference dedicated to Professor Heinz-Otto Kreiss on his 60th birthday1991 (ed. 1-2)Book (Refereed)
    Abstract [en]

    These volumes contain papers from the third International Conference on Hyperbolic Problems, which was held on June 11-15, 1990 in Uppsala, Sweden. The conference reflected the current vitality of research in hyperbolic problems and the interaction between theory, numerical methods and applications. Most of the papers deal with non-linear problems. This is particularly true for the applications where fluid mechanics dominates.

  • 41.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Gustafsson, B
    Vreeburg, J
    Numerical solution of a PDE system describing a catalytic converter1978In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 27, no 3, 295-314 p.Article in journal (Refereed)
    Abstract [en]

    Numerical approximations are studied for a large hyperbolic system coupled to a parabolic equation and a system of algebraic equations. The equations, which all are nonlinear, describe nonviscous compressible one-dimensional gas flow in a catalytic converter. Chemical reactions within the gas are included in the model. Well-posedness of the partial differential equations is analyzed together with stability of the numerical models. In particular an investigation is made of the effect of numerical dissipation and different boundary conditions. Numerical results are presented.

  • 42.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Halpern, L
    Far field boundary conditions for computation over long time1988In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 4, no 1, 21-45 p.Article in journal (Refereed)
    Abstract [en]

    A new class of computational far field boundary conditions for hyperbolic partial differential equations is developed. These boundary conditions combine properties of absorbing boundary conditions for transient solutions and properties of far field boundary conditions for steady-state problems. The conditions can be used to limit the computational domain when both traveling waves and evanescent waves are present. Boundary conditions for scalar wave equations are derived and analyzed. Extensions to systems of equations are discussed and results from numerical experiments are presented.

  • 43.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Halpern, L
    Long-time behavior of absorbing boundary conditions1990In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, ISSN 0170-4214, Vol. 13, no 3, 189-203 p.Article in journal (Refereed)
    Abstract [en]

    A new class of computational far-field boundary conditions for hyperbolic partial differential equations was recently introduced by the authors. These boundary conditions combine properties of absorbing conditions for transient solutions and properties of far-field conditions for steady states. This paper analyses the properties of the wave equation coupled with these new boundary conditions: well-posedness, dissipativity and convergence in time.

  • 44.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Harten, A
    Osher, S
    A high order essentially non-oscillatory shock capturing method1987Conference paper (Refereed)
  • 45.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Holst, Henrik
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time2011Article in journal (Refereed)
    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.

  • 46.
    Engquist, Björn
    et al.
    University of Texas Austin.
    Holst, Henrik
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Multiscale Methods for One Dimensional Wave Propagation with High Frequency Initial Data2011Report (Other academic)
    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.

  • 47.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Holst, Henrik
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Multiscale methods for the wave equation2007In: PAMM · Proc. Appl. Math. Mech. 7, 2007, 1140903-1140904 p.Conference paper (Other academic)
    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.

  • 48.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Holst, Henrik
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Multi-scale methods for wave propagation in heterogeneous media2011In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 9, no 1, 33-56 p.Article in journal (Refereed)
    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.

  • 49.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Holst, Henrik
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Multiscale Methods for Wave Propagation in Heterogeneous Media Over Long Time2012In: Numerical Analysis of Multiscale Computations / [ed] Björn Engquist, Olof Runborg, Yen-Hsi R. Tsai, Springer Verlag , 2012, 167-186 p.Chapter in book (Other academic)
    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.

  • 50.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Hou, T
    Particle method approximation of oscillatory solutions to hyperbolic differential equations1989In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, ISSN 0036-1429, Vol. 26, no 2, 289-319 p.Article in journal (Refereed)
    Abstract [en]

    Particle methods approximating hyperbolic partial differential equations with oscillatory solutions are studied. Convergence is proved for approximations for which the continuous solution is not well resolved on the computational grid. Highly oscillatory solutions to the Broadwell and variable coefficients Carleman models are considered. Homogenization results are given and the approximations of more general systems are discussed. Numercial exampels are presented

123 1 - 50 of 138
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