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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Dynamic mode decomposition ： ウィキペディア英語版 Dynamic mode decomposition Physical systems, such as fluid flow or mechanical vibrations, behave in characteristic patterns, known as modes. In a recirculating flow, for example, one may think of a hierarchy of vortices, a big main vortex driving smaller secondary ones and so on. Most of the motion of such a system can be faithfully described using only a few of those patterns. The dynamic mode decomposition (DMD) provides a means of extracting these modes from numerical and experimental pairs of time-shifted snapshots. Each of the modes identified by DMD is associated with a fixed oscillation frequency and growth/decay rate, determined by DMD without requiring knowledge of the governing equations. This is to be contrasted with methods, such as the proper orthogonal decomposition, which produce a set of modes without the associated temporal information. == Description == A time-evolving physical situation may be approximated by the action of a linear operator $e^$ to the instantaneous state vector. : $\backslash approx\; e^\; q(t)$ The dynamic mode decomposition strives to approximate the evolution operator $\backslash tilde\; A\; :=\; e^$ from a known sequence of observations, $V\_=\backslash$. Thus, we ask the following matrix equation to hold: : $$ V_=\tilde A V_ Generally, the vectors $\backslash$, and subsequently $\backslash tilde\; A$, are very-high-dimensional, and so a strict eigendecomposition of $\backslash tilde\; A$ is computationally difficult. However, in DMD it is assumed that the set of $\backslash$ does not span the entire vector space (a good assumption, especially if there is spatial structure in the signal). Thus, after a given time $n$, where $n$ is much less than the dimensionality of the system, one can write $q\_$ as a linear combination of the previous vectors, i.e., $q\_\; =\; c\_0q\_0\; +\; \backslash cdots\; +\; c\_nq\_n\; =\backslash c$. In matrix form, we then have: : $$ V_= V_ S where ''S'' is the companion matrix :$S=\backslash begin$ 0 & 0 & \dots & 0 & c_0 \\ 1 & 0 & \dots & 0 & c_1 \\ 0 & 1 & \dots & 0 & c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & c_n \end. The eigenvalues of ''S'' then approximate some of the eigenvalues of $\backslash tilde\; A$. However, since the ''S'' is small (with dimensions (''n'' + 1) × (''n'' + 1) as compared to $\backslash tilde\; A$), the eigenvalues and eigenvectors of ''S'' can be computed with ease.〔Schmid, Peter J. "Dynamic mode decomposition of numerical and experimental data." Journal of Fluid Mechanics 656.1 (2010): 5–28.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dynamic mode decomposition」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.