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Multiscale Modeling and Simulation in Science pp 195–235Cite as

Wavelets andWavelet Based Numerical Homogenization

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 66))

Wavelets is a tool for describing functions on different scales or level of detail. In mathematical terms, wavelets are functions that form a basis for \(L^2 \left(\mathbb{R}\right)\) with special properties; the basis functions are spatially localized and correspond to different scale levels. Finding the representation of a function in this basis amounts to making a multiresolution decomposition of the function. Such a wavelet representation lends itself naturally to analyzing the fine and coarse scales as well as the localization properties of a function.Wavelets have been used in many applications, from image and signal analysis to numerical methods for partial differential equations (PDEs). In this tutorial we first go through the basic wavelet theory and then show a more specific application where wavelets are used for numerical homogenization.We will mostly give references to the original sources of ideas presented. There are also a large number of books and review articles that cover the topic of wavelets, where the interested reader can find further information, e.g. [25, 51, 48, 7, 39, 26, 23], just to mention a few.

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Authors and Affiliations

    Authors
    1. Olof Runborg
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    Editors and Affiliations

    1. Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX, 78712-0257, USA

      Björn Engquist

    2. Department of Information Technology, Uppsala University, Uppsala, 751 05, Sweden

      Per Lötstedt

    3. Department of Numerical Analysis and Computer Science, Royal Institute of Technology, Stockholm, 100 44, Sweden

      Olof Runborg

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    Runborg, O. (2009). Wavelets andWavelet Based Numerical Homogenization. In: Engquist, B., Lötstedt, P., Runborg, O. (eds) Multiscale Modeling and Simulation in Science. Lecture Notes in Computational Science and Engineering, vol 66. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88857-4_4

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