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Multilevel Representations of Random Fields and Sparse Approximations of Solutions to Random PDEs

In: Multiscale, Nonlinear and Adaptive Approximation II

Author

Listed:
  • Markus Bachmayr

    (RWTH Aachen University, Institut für Geometrie und Praktische Mathematik)

  • Albert Cohen

    (Sorbonne Université, Laboratoire Jacques-Louis Lions)

Abstract

In the analysis of random fields as well as in applications, such as in uncertainty quantification, representations of random fields as function series play an important role. While the most usual such expansions are based on Karhunen- Loève decompositions, in many cases of interest, other expansions with independent coefficients exist and may be more relevant. In particular, for certain classes of Gaussian random fields, one can obtain such expansions in terms of function systems withwavelet-type multilevel structure. In this article,we reviewconstructions of such representations that are suitable for numerical methods, as well as their impact on sparse approximations of solutions to random partial differential equations.

Suggested Citation

  • Markus Bachmayr & Albert Cohen, 2024. "Multilevel Representations of Random Fields and Sparse Approximations of Solutions to Random PDEs," Springer Books, in: Ronald DeVore & Angela Kunoth (ed.), Multiscale, Nonlinear and Adaptive Approximation II, pages 25-54, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-75802-7_3
    DOI: 10.1007/978-3-031-75802-7_3
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