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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 complexity2007Inngår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 6, nr 4, s. 1059-1084Artikkel i tidsskrift (Fagfellevurdert)
    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.
    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 oscillators2009Inngår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 7, nr 3, s. 1387-1404Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 3.
    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 problems2014Inngår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 12, nr 3, s. 1135-1166Artikkel i tidsskrift (Fagfellevurdert)
    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..

  • 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 media2017Inngår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 15, nr 2, s. 948-976Artikkel i tidsskrift (Fagfellevurdert)
    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. 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 problems2007Inngår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 6, nr 3, s. 711-740Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 6.
    Engblom, Stefan
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    PARALLEL IN TIME SIMULATION OF MULTISCALE STOCHASTIC CHEMICAL KINETICS2009Inngår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 8, nr 1, s. 46-68Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A version of the time-parallel algorithm parareal is analyzed and applied to stochastic models in chemical kinetics. A fast predictor at the macroscopic scale (evaluated in serial) is available in the form of the usual reaction rate equation. A stochastic simulation algorithm is used to obtain an exact realization of the process at the mesoscopic scale (in parallel). The underlying stochastic description is a jump process driven by the Poisson measure. A convergence result in this arguably difficult setting is established, suggesting that a homogenization of the solution is advantageous. We devise a simple but highly general such technique. Three numerical experiments on models representative to the field of computational systems biology illustrate the method. For nonstiff problems, it is shown that the method is able to quickly converge even when stochastic effects are present. For stiff problems, we are instead able to obtain fast convergence to a homogenized solution. Overall, the method builds an attractive bridge between, on the one hand, macroscopic deterministic scales and, on the other hand, mesoscopic stochastic ones. This construction is clearly possible to apply also to stochastic models within other fields.

  • 7.
    Engquist, Björn
    et al.
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Ying, Lexing
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers2011Inngår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 9, nr 2, s. 686-710Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This paper introduces a new sweeping preconditioner for the iterative solution of the variable coefficient Helmholtz equation in two and three dimensions. The algorithms follow the general structure of constructing an approximate LDLt factorization by eliminating the unknowns layer by layer starting from an absorbing layer or boundary condition. The central idea of this paper is to approximate the Schur complement matrices of the factorization using moving perfectly matched layers (PMLs) introduced in the interior of the domain. Applying each Schur complement matrix is equivalent to solving a quasi-1D problem with a banded LU factorization in the 2D case and to solving a quasi-2D problem with a multifrontal method in the 3D case. The resulting preconditioner has linear application cost, and the preconditioned iterative solver converges in a number of iterations that is essentially independent of the number of unknowns or the frequency. Numerical results are presented in both two and three dimensions to demonstrate the efficiency of this new preconditioner.

  • 8.
    Martin, Lindsay
    et al.
    Univ Texas Austin, Dept Math, Austin, TX 78712 USA..
    Tsai, Yen-Hsi R.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. Univ Texas Austin, Dept Math, Austin, TX 78712 USA.;Univ Texas Austin, Oden Inst Computat Engn & Sci, Austin, TX 78712 USA.
    A multiscale domain decomposition algorithm for boundary value problems for eikonal equations2019Inngår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 17, nr 2, s. 620-649Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper, we present a new multiscale domain decomposition algorithm for computing solutions of static Eikonal equations. In our new method, the decomposition of the domain does not depend on the slowness function in the Eikonal equation or the boundary conditions. The novelty of our new method is a coupling of coarse grid and fine grid solvers to propagate information along the characteristics of the equation efficiently. The method involves an iterative parareal-like update scheme in order to stabilize the method and speed up convergence. One can view the new method as a general framework where an effective coarse grid solver is computed "on the fly" from coarse and fine grid solutions that are computed in previous iterations. We study the optimal weights used to define the effective coarse grid solver and the stable update scheme via a model problem. To demonstrate the framework, we develop a specific scheme using Cartesian grids and the fast sweeping method for solving Eikonal equations. Numerical examples are given to show the method's effectiveness on Eikonal equations involving a variety of multiscale slowness functions.

  • 9.
    Sun, Yi
    et al.
    Courant Institute of Mathematical Sciences, New York University, New York, USA.
    Caflisch, Russel
    Department of Mathematics, California NanoSystem Institute and Institute for Pure and Applied.
    Engquist, Björn
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    A multiscale method for epitaxial growth2011Inngår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 9, nr 1, s. 335-354Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we investigate a heterogeneous multiscale method (HMM) for interface tracking and apply the technique to the simulation of epitaxial growth. HMM relies on an efficient coupling between macroscale and microscale models. When the macroscale model is not fully known explicitly or not accurate enough, HMM provides a procedure for supplementing the missing data from a microscale model. Here we design a multiscale method that couples kinetic Monte-Carlo (KMC) simulations on the microscale with the island dynamics model based on the level set method and a diffusion equation. We perform the numerical simulations for submonolayer island growth and step edge evolutions on the macroscale domain while keeping the KMC modeling of the internal boundary conditions. Our goal is to get comparably accurate solutions at potentially lower computational cost than for the full KMC simulations, especially for the step flow problem without nucleation.

  • 10. Sun, Yi
    et al.
    Engquist, Björn
    Heterogeneous multiscale methods for interface tracking of combustion fronts2006Inngår i: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 5, nr 2, s. 532-563Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we investigate the heterogeneous multiscale methods (HMM) for interface tracking and apply the technique to the simulation of combustion fronts. Our goal is to overcome the numerical difficulties, which are caused by different time scales between the transport part and the reactive part in the model equations of some interface tracking problems, such as combustion processes. HMM relies on an efficient coupling between the macroscale and microscale models. When the macroscale model is not fully known explicitly or not valid in localized regions, HMM provides a procedure for supplementing the missing data from a microscale model. Here we design and analyze a multiscale scheme in which a localized microscale model resolves the details in the model and a phase field or a front tracking method defines the interface on the macroscale. This multiscale technique overcomes the difficulty of stiffness of common problems in combustion processes. Numerical results for Majda's model and reactive Euler equations in one and two dimensions show substantially improved efficiency over traditional methods.

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