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Further exploitation of the RB framework Yvon Maday, Laboratoire - PowerPoint PPT Presentation

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Further exploitation of the RB framework Yvon Maday, Laboratoire Jacques-Louis Lions Sorbonne Universit, Paris, Roscoff, France, Institut Universitaire de France Providence February 2020 Mathematics of Reduced Order Models Reduced Basis

  1. Further exploitation of the RB framework Yvon Maday, Laboratoire Jacques-Louis Lions Sorbonne Université, Paris, Roscoff, France, Institut Universitaire de France Providence — February 2020 Mathematics of Reduced Order Models

  2. Reduced Basis Methods is one of the way for Model Reduction

  3. Vague Statements The idea is to use the fact that the « state » we are interested in is described by a quantity u ( x, t ; µ ) that is a function depending on space (and time) and a parameter µ

  4. Parametric model manifold we introduce the set of all solutions to the mathematical model and assume it has a small Kolmogorov n-width

  5. small Kolmogorov n-width means that there exists a small set of functions in or in Span { } such that, any u in is well approximated by a linear combination of these few functions

  6. an example

  7. Kolmogorov n-width Definition Let X be a normed linear space, S be a subset of X and X n be a generic n -dimensional subspace of X . The deviation of S from X n is E ( S ; X n ) = sup v n ∈ X n k u � v n k X . inf u ∈ S The Kolmogorov n -width of S in X is given by d n ( S , X ) = inf X n sup v n ∈ X n k u � v n k X inf u ∈ S The n -width of S thus measures the extent to which S may be approximated by a n -dimensional subspace of X .

  8. How to get the Kolmogorov best space X n ??

  9. How to get the Kolmogorov best space X n ?? X n optimal space is not attainable : an approximation can be given by PCA/SVD … based on some orthogonal decomposition another way is through greedy approach

  10. <latexit sha1_base64="tzgXEMTARup4IKe9u1o/cyvdQtQ=">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</latexit> <latexit sha1_base64="j1pQvRUbV7OCaYQRxUkAWyWMy1M=">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</latexit> <latexit sha1_base64="j1pQvRUbV7OCaYQRxUkAWyWMy1M=">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</latexit> <latexit sha1_base64="rHF0abh+71EBtN7lt/MqzAqRho=">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</latexit> Vague Statements u ( x, t ; µ ) In order to determine : what do we have at end ? u ( x i , t k , µ ) a) possibly measures, either pointwize

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µ ) In order to determine : what do we have at end ? u ( x i , t k , µ ) a) possibly measures, either pointwize R ϕ i,k ( x, t ) u ( x, t, µ ) dxdt or moments

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