MATLAB Function to perform the dynamic mode decomposition (DMD) on spatio-temporal data spaced evenly in time.

In simple terms, it decomposes the data into oscillating spatio-temporal patterns, with a fixed frequency and growth/decay rate.

This script is based on the techniques and codes presented in the book 'Data-Driven Science and Engineering' by Steven L. Brunton and J. Nathan Kutz, as well as codes available on their DMD book website.

See Steve's video below for an excellent description of the method.

The script for finding the optimal threshold for the modes is developed by D. L. Donoho and M. Gavish in "The Optimal Hard Threshold for Singular Values is 4/sqrt(3)"

This example shows how to compute DMD on 2D velocity field data. Assume you have velocities uxand uy stored on grids of size (NX,NY) at times 1 through m+1, equally spaced in time. The data matrix would be of size (n,m+1) with n=2*n0, and n0=NX*NY, with data(1:n0,k) being ux at time t_k, flattened as a vector, and similarly data(n0+1:2*n0,k) being uy. Then, the DMD can be computed using:

X = data(:,1:end-1) % Size (n,m)   
Xprime = data(:,2:end) % Size (n,m)

% If all the optimal thresholded modes are desired

[U,Sigma,V,Phi,Lambda,b,r_optimal] = DMD(X,Xprime)

% If only r modes are desired

r = 21; % Set desired value

[U,Sigma,V,Phi,Lambda,b,r_optimal] = DMD(X,Xprime,r)