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

Famous quotes containing the word description:

    “The next Augustan age will dawn on the other side of the Atlantic. There will, perhaps, be a Thucydides at Boston, a Xenophon at New York, and, in time, a Virgil at Mexico, and a Newton at Peru. At last, some curious traveller from Lima will visit England and give a description of the ruins of St. Paul’s, like the editions of Balbec and Palmyra.”
    —Horace Walpole (1717–1797)

    “As they are not seen on their way down the streams, it is thought by fishermen that they never return, but waste away and die, clinging to rocks and stumps of trees for an indefinite period; a tragic feature in the scenery of the river bottoms worthy to be remembered with Shakespeare’s description of the sea-floor.”
    —Henry David Thoreau (1817–1862)

    “A sound mind in a sound body, is a short, but full description of a happy state in this World: he that has these two, has little more to wish for; and he that wants either of them, will be little the better for anything else.”
    —John Locke (1632–1704)