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

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

Reduced Basis Methods is one of the way for Model Reduction

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

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

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

an example

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 .

How to get the Kolmogorov best space X n ??

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

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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

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

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

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