4. Reduced order modeling with respect to the Euclidean norm, is a big deal, and ubiquitious as an application of SVD. (Just say YES!

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1 EE520, Bollt, HW 1, SVD, PCA, Re Ch 15, and into Ch16, due F Sept Image of a circle. (Matlab). Make a drawing of a circle of radius r. Plot the image of a unit circle in R^2 when each point is multiplied by A = (3-2; -1 5). Also overlay the scaled left singular vectors σ1u1 and σ2u2 on your plot and verify that they line up with the axes of the ellipse. Does the placement of the (center) of the circle matter? Discuss. 2. Very simple linear regression. Consider the three points (0,1), (1,-1), (2,-2). What is the equation of the line that best fits those three points, in the sense of least squares? State the solution of this problem using methods of SVD. We have seen that the least-squares problem can be stated in terms of the pseudoinverse which is then in terms of the SVD. 3. Repeat on your own computer, and then report with graphs, pictures, words and equations, (so in other words, discuss) the Example 2, regarding the spatiotemporal function f(x,t) in Eq The point of the discussion is how the POD modes are more efficient than other modal decomposition, such as a Fourier decomposition. Can you expand the presentation of the example to emphasize this point? 4. Reduced order modeling with respect to the Euclidean norm, is a big deal, and ubiquitious as an application of SVD. (Just say YES!) 5. Dimension reduction. Load the file sdata.csv which contains a matrix of data. Each row of the matrix is a point (x_i, y_i, z_i) in R^3. We will approximate this data set as an affine one- dimensional space (a line that doesn t pass through the origin). a) Find the line that best approximates the data in the sense of minimizing the sum of the squares of the projections of all points onto the line. Plot the line and the data on the same axes and verify that the line approximates the points. Hint: before finding the line, shift every point so that the data has zero mean. You can make 3D scatter plots in Matlab by using plot3 b) Instead of using three numbers (x_i, y_i, z_i) to describe each data point, we can now use a singlenumber w_i, which is the position along the line of the projected data point. Give a formula that converts (x,y,z) to w and the reverse formula, which converts w to a point (x,y,z). c) Convert the data set to w_i coordinates, and plot a histogram of the {w_i} to see how the points are distributed. Use 20 equally spaced bins for the histogram. Big Hint:

2 6. Compute the SVD in closed form, by hand, by the method discussed in 15.1, of the matrix, A=[1-1; 0 1; 1 0]; 7. 8.

3 9. Compress your professor. Reduce order professor. There is an image of me giving a lecture locally at google images. Download it and save it in an appropriate place. In Matlab you can open this image by typing H = imread( whateveryoucalledit.jpg ); imshow(h) allows you to look at it from Matlab Then try size(h) to see how big the array is? There are three color channels. This work will be easier if we work with a grayscale version of the image. I = rgb2gray(h) I = double(i); allows you to make an array you can manipulate with mathematical operations like SVD, and so forth. Use the singular value decomposition to compress the image in Matlab. In particular replace I with nearest matrices with respect to the matrix 2-norm whose ranks are 10, 50, 100, 300. Show the result as pictures. Also, show a power spectrum plot. HW #2, due F Sept Eigenfaces. Perform an eigenface analysis of the data set provided. Here is an example of the kind of questions such an analysis would include, but this question is mostly open ended for you to discuss and code yourself Do a POD analysis for a clamped beam PDE. a. Get this code running, and then decompose the solutions so as to illustrate the dominant empirical modes. b. Plot those. c. Also show the time averaged power spectrum. d. Also, show the time varying Fourier coefficients for the first 5 modes. e. And show the residuals using each of the first 5 modes. beam-forced-damped-vibration

4 3. Download the data for images of the noisy professor. Attempt denoising by using the Robust PCA method BBerik-bolltnoisy.mat 4. Adjust the ICA codes from your book which were designed for images to do a linear BSS for a pair of sound files. I have provided some typical soonds. Moo. Bock. Etc. First combine the sounds. Then separate the sounds. Your solution as reported to me should include, discussion, graphs of before, (eg amplitude) and after combining, and then separation. AND for this one, I want you also to me the.wav files so I can hear with my own ears how well it went. Please comment what you think. There is a write.wav files command in matlab. HW #3, Due W Oct 4. In the spirit of Classification, Ch 17, Dogs vs Cats. 1) I will supply data of dogs and cats pictures. Arrange the data appropriately, and then try out the codes in the book to classify these report all the pictures as in the book and discuss accordingly. 2) Go to the website on sounds: Use the sound import routines in matlab to import at least a dozen sounds in each of two classes and then try out the LDA classification method, (recycling some of the codes above), after projecting onto an appropriate basis set (e.g.is the dwt2 useful in this setting? Discuss, and choose appriately). Write a small report style discussion of this using appropriate mathematical formulas, language and supporting figures. So choose two different labeled groups of sounds, to your taste. I choe the didgeridoo music versus folk music, but only because I like to say didgeridoo since it is a funny word. You choose as you wish. 3) For the sound data, or the image data (you choose) use another classification method of your choosing, and report success. E.g., knn method, k-means method, Gaussian mixture models, statistical, and on and on. You choose. 4) Discuss (no need to do) how the following adjustments impact the concepts in this homework. A) What if you needed to make a partition of k>2 elements? B) What if you want the method itself to suggest a good

5 number of partition elements? Discuss generally and then in the context of at least one of the methods. HW #4, due Fri Oct 20. Reconsider that data set you have already viewed in HW#2 of a noisy professor in the data, BBerik-bolltnoisy.mat 1. In terms of a discrete Fourier transform using the commands built into Matlab, akin to Fig in your book, showing the result pictures of reconstruction using a similar level of modes, and you may inspect the codes on pages to help. ALSO show the L2-error between the fitted reconstruction error versus the number N of modes used. That is, if I N (x i ) = a n φ n (x i ), a fine sum and truncation of the i= Fourier series at N terms, (expand upon this using notes from class, and material from pages so write this out further explicitly showing what is the specific form of φ n (x i ), and then how the matlab commands are used to find the FFT to get there. Then, with this, let e n = I(x i ) I N (x i ) 2 where (x i ) are the pixel positions, for a 1 i q,1 j p with q by p pixelated image, and. 2 is the usual L2 norm over the domain of the image. So finally there will be several images you will show me of decreasing quality of reproduction due to fewer and fewer Fourier modes used. And you will show me a plot of error vs n. 2. Now for compressed sensing. First we will sparsely sample the image in two different ways. a. A random sampling (akin to Fig ), and show me the binary sampling mask, and the resulting subsampled image. Use several levels of sparsity. (To produce your own binary mask, try A=1.5*rand(100,100); B=floor(A); spy(b); what level of sparsity does 1.5 produce? Then >I.*A will be the subsampled image, by array arithmetic that you can view by imagesc give it a nice gray color scale too). b. Second kind of sparse binary mask try something like the Candes- Romberg mask mentioned on page 470, and shown in 18.15b where lots of carefully selected diagonal lines are sampled. You can build your own easily, or they also exist on the internet for you to find. In any case, use several levels of sparsity (several masks), and show them in spy plot, and also show the resulting sparsified image. 3. Now for compressed sensing. First you will need to be able to realize that the image is really some form of Ax=b in disguise. The book discusses the nice localization properties of wavelets on page 471 as a hunt. But this can also be done

6 via Fourier. In any case, this part is paper and pencil argue why and how solving Ax=b is related to the image, and recovering the image. 4. Do it now that you have realized that solving Ax=b is relevant and you have subsampled the image in part 2, can you recover the image by solving min x 1? Ax=b Hint you will want to use the suggested package, cvx described on page 453. Further hint I find it does not work on the latest, matlab 2017b on my machine, but it does work on an older version such as the 2015 version of matlab and you should be able to install old versions. So page 453 for hints on syntax. 5. a. Required: show me (for the levels of sparsity you chose) a picture akin to Fig showing the sparsity structure using L1-optimization versus the sparsing but incorrect solution using the \ operator ( that is in fact a QR-recomposition solution). b. Optional even better than Fig. 18.3, would be if you shower a fuller compare and contrast, to the L2 optimal solutions, analogous to Fig Summary and all this compressed sensing is for the noising professor image. HW#5, due W Nov 8. Consider an Euler beam, with pinned boundary conditions at both ends. Also consider that it has a chaotic element bouncing in the middle. The equation is generally of the form, EI(d^4w/dx^4)=-mu(d^2w/dt^2)+q(x,t), with boundary conditions, w(0,t)=w(l,t)=0, and initial conditions, w(x,0)=f(x). And q(x,t) comes from a chaotic element, hanging from the beam, derived from the solution of a Rossler equation. A code that solves such a problem is provided as beamchaotic.m and a sample solution (downsampled) is provided in beamchaotic.mat. 1) Read the code and more clearly write the specific equation being solved; write out the PDE specifically. 2) The goal of this HW will be to perform a DMD analysis of this solution w(x,t). That is, considering Eq and in your book, produce a figure analogous to Fig 20.1 in your book, a DMD presentation of the dyamics projected onto just a few modes. How many did you use and why? I expect to see the same 6 panels, but now for describing this beam problem. 3) So for this to be complete, fully describe what you interpret, your choice of dimensionality and why, and also rewrite some of the mathematical background material from Sec 20.1 on this topic. HW#6, due W Nov 15

7 On Data assimilation. Consider the Rossler equations, x =-y-z, y =x+a y, z = b+ z(x-c), and choose the parameter values that give a very nice chaotic attractor. Let a=0.2, b=0.2, and c= ) Use ODE45, and an initial condition to integrate these equations, and show time series for each of x(t), y(t), and z(t); choose x(0)=1; y(0)=1; z(0)=1, and a time range t in [0,100]; Also show a 3d plot of the attractor, in a parametric plot of (x(t),y(t),z(t)) using the command plot3. 2) Now illustrate how this equation shows sensitive dependence to initial conditions; Choose x(0)=1.001; y(0)=1; z(0)=1, and redraw all of the plots from 1) but now show both solutions (in a different color) thus highlighting the difference; please describe sensitive dependence to initial conditions (as you read it from other sources) and interpret what you see in 2) vs 1) in this context. 3) Plot error e(t)=norm([x_1(t)-x_2(t), y_1(t)-y_2(t), z_1(t)-z_2(t)]) where 1 and 2 mean the answers from each of the above. So plot e(t) (growing Euclidean norm of the error) but as t vs e(t), and then t vs log(e(t)). What do you see? 4) Now Repeat all the computations of data assimulation by the ensemble Kalman Filter method (EnKF) demonstrated in your book, that was designed in 21.3, but for the Lorenz equations. You may adjust those equations, and those codes from the book section, to be correspondingly for the Rossler attractor. Include all the relevant equations (noted in the book in that section), and codes and graphs, and your own words describing what you see. HW#7/8. due Th Dec 14 (last homework/project). I Consider the simple equation free scenario from, which is in turn from, 10 Let a) State how this equation can be interpreted in the formalism as a fast-slow system, and declare which are the fast variables and which are the slow variables and what is the slow manifold. b) Before doing anything equation free use the full equations and integrate them with the intitial condition, and be sure to show not only a time series of each variable, but a time series of error from the slow manifold.

8 c) In b) use two different kinds of solvers, the standard adaptive RK solver built into ODE45 with an appropriate step size, and then contrast to performance when you use a stiff solver such as ODE115s.. By performance we mean how well it integrates the fast and slow variables despite the large derivative changes during approach to slow manifold. The way to illustrate that is to show error from slow manifold with respect to a given step size. Read in as well. d) Now on to equation free. First the setup: State your coarse variable(s), your fine variable(s), your projection operator and your lift operator. e) Build your own eqfree solver analogous to what is shown in the references, or your book and illustrate performance of your solver versus the result you got above using a standard solver but knowing the full system. f) Interpret why this is interesting relevant to being able to integrate a system despite not necessarily having all the variables. II See associated.zip file of codes from the book, Sec A.2, [age 314, Fig A.1 And A.3 Fig A.6, a Ulam-Galerkin estimation based on the Frobenius-Perron operator for the standard map and the Lorenz equations respectively. 1. Run the codes as they are and include the results the produce and interpret in your own words, but it is ok to interpret based on what is written in the book. 2. Look at A Glance at the Standard Map.pdf attached and use the codes from A.2 to study the parametric changes to the almost invariant sets as you adjust the critical parameter k from steps in 0.9 increasing through 1.5. Describe what you see. 3. Use the codes presented from A.3 and modify to describe an outer cover of the chaotic attractor associated with the Rossler equations; with the usual parameters associated with popular presentations of a chaotic attractor show me those equations and parameters.

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