(c) Repeat the above experiment comparing two A matrices with different singular value distributions: the first will have values spaced linearly from 100 to 1 and the second will have values spaced logarithmically from 100 to 1 . Does anything change? Please explain why yes or no. (d) Repeat the above experiment, but now with an A matrix that has size 10010. Explain any changes. Exercise 1.7 Load the data set for fluid flow past a cylinder (you can either download this from our book http:/IDMDbook.com or generate it using the IBPM code on GitHub). Each column is a flow field that has been reshaped into a vector. (a) Compute the SVD of this data set and plot the singular value spectrum and the leading singular vectors. The U matrix contains eigenflow fields and the V represents the amplitudes of these eigenflows as the flow evolves in time. (b) Write a code to plot the reconstructed movie for various truncation values r. Compute the r value needed to capture 90%,99%, and 99.9% of the flow energy based on the singular value spectrum (recall that energy is given by the Frobenius norm squared). Plot the movies for each of these truncation values and compare the fidelity. Also compute the squared Frobenius norm of the error between the true matrix X and the reconstructed matrix X~, where X is the flow field movie. (c) Fix a value r=10 and compute the truncated SVD. Each column wkR10 of the matrix W=~V~ represents the mixture of the first 10 eigenflows in the k th column of X. Verify this by comparing the k th snapshot of X with U~wk. (d) Now, build a linear regression model for how the amplitudes wk evolve in time. This will be a dynamical system: wk+1=Awk. Create a matrix W with the first 1 through m1 columns of V and another matrix W with the 2 through m columns of V. We will now try to solve for a best-fit A matrix so that WAW Compute the SVD of W and use this to compute the pseudo-inverse of W to solve for A. Compute the eigenvalues of A and plot them in the complex plane. (e) Use this A matrix to advance the state wk=Ak1w1 starting from w1. Plot the reconstructed flow field using these predicted amplitude vectors and compare with the true values. This exercise derived the dynamic mode decomposition from Section 7.2. (c) Repeat the above experiment comparing two A matrices with different singular value distributions: the first will have values spaced linearly from 100 to 1 and the second will have values spaced logarithmically from 100 to 1 . Does anything change? Please explain why yes or no. (d) Repeat the above experiment, but now with an A matrix that has size 10010. Explain any changes. Exercise 1.7 Load the data set for fluid flow past a cylinder (you can either download this from our book http:/IDMDbook.com or generate it using the IBPM code on GitHub). Each column is a flow field that has been reshaped into a vector. (a) Compute the SVD of this data set and plot the singular value spectrum and the leading singular vectors. The U matrix contains eigenflow fields and the V represents the amplitudes of these eigenflows as the flow evolves in time. (b) Write a code to plot the reconstructed movie for various truncation values r. Compute the r value needed to capture 90%,99%, and 99.9% of the flow energy based on the singular value spectrum (recall that energy is given by the Frobenius norm squared). Plot the movies for each of these truncation values and compare the fidelity. Also compute the squared Frobenius norm of the error between the true matrix X and the reconstructed matrix X~, where X is the flow field movie. (c) Fix a value r=10 and compute the truncated SVD. Each column wkR10 of the matrix W=~V~ represents the mixture of the first 10 eigenflows in the k th column of X. Verify this by comparing the k th snapshot of X with U~wk. (d) Now, build a linear regression model for how the amplitudes wk evolve in time. This will be a dynamical system: wk+1=Awk. Create a matrix W with the first 1 through m1 columns of V and another matrix W with the 2 through m columns of V. We will now try to solve for a best-fit A matrix so that WAW Compute the SVD of W and use this to compute the pseudo-inverse of W to solve for A. Compute the eigenvalues of A and plot them in the complex plane. (e) Use this A matrix to advance the state wk=Ak1w1 starting from w1. Plot the reconstructed flow field using these predicted amplitude vectors and compare with the true values. This exercise derived the dynamic mode decomposition from Section 7.2
