Dynamic Mode Decomposition - Description

Description

A time-evolving physical situation may be approximated by the action of a linear operator to the instantaneous state vector.

The dynamic mode decomposition strives to approximate the evolution operator from a known sequence of observations, . Thus, we ask the following matrix equation to hold:

V_{1 \dots n+1}=\tilde A V_{0\dots n}

The right hand side states that is a linear combination of the columns of, which can be expressed as

V_{1 \dots n+1}= V_{0\dots n} S

where S is the companion matrix

S=\begin{pmatrix} 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{pmatrix}.

The matrix S is small as compared to the sample data V. Therefore eigenvalues and eigenvectors can be computed with ease.

Read more about this topic:  Dynamic Mode Decomposition

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