Problem Set 5: Iterative methods with generalized least squares

Transcription

1 Problem Set 5: Iterative methods with generalized least squares GEOS 627: Inverse Problems and Parameter Estimation, Carl Tape Assigned: February 20, 2017 Due: March 6, 2017 Last compiled: July 7, 2017 Overview and instructions 1. Reading: Section 6.22 of Tarantola (2005) 2. Problem 1. In HW4 you explored Gaussian random fields with prescribed covariance (exponential or Gaussian) for a 1D spatial field. This problem deals with 2D Gaussian random fields. Your goal is to understand how these fields are generated, and to get some intuitive sense for how certain parameters change the appearance of the 2D fields. The Matlab scripts for computing Gaussian random fields in the frequency domain were provided by Miranda Holmes, who wrote them as part of the study Bühler and Holmes- Cerfon (2009). She also wrote a useful set of notes, Generating stationary Gaussian random fields, which is available as ft_summary.pdf in the same directory containing the Matlab scripts (~/inv2017/util_grf/). Why do we care about the statistical characteristics of spatial random fields? Because real systems have complex variations that we would like to quantify in simpler terms. Gneiting et al. (2012) examined several different covariance functions, including the Matern family, which includes the Gaussian and exponential functions, and they illustrate their technique with line transects of arctic sea ice. 3. Problem 2. Iterative methods are needed for nonlinear problems, which are characterized by a nonlinear forward model g(m). In many problems it is not computationally feasible to evaluate the misfit function dozens of times, let alone millions (or billions) of times needed to cover the M-dimensional model space. The basic strategy is to evaluate the misfit function at one point, then use the gradient (and Hessian) at that point to guide the choice of the next point. This problem will build upon the lab exercie lab_iter.pdf. 4. Problem 3. Here we revisit Tarantola (2005), Problem 7-1. See HW3 for background. 1

2 Problem 1 (3.0). 2D Gaussian random fields In this problem, a sample is a 2D spatial field that is n x n y. 1. (0.2) Make a copy of the template code: cp covrand2d_template.m covrand2d.m Run covrand2d.m, which will generate 1000 samples of a 2D Gaussian random field with prescribed covariance; eight samples are plotted. Compute and plot the sample mean µ 1000 with a colorbar. Throughout this problem, when plotting samples, use the same plotting commands 1 : axis equal, axis(ax1aex), caxis(3*sigma*[-1 1]) 2. (0.3) Open xy2distance.m 2, set idisplay=1, then run the example at the bottom. (a) (0.0) Explain how the points are ordered within the distance index matrix id. (b) (0.1) What kind of matrix structure does id have? (Be as specific as possible.) (c) (0.1) How can you compute the actual distances between points, D (or D)? (Recognize the difference between the distance matrix D and the index distance matrix.) (d) (0.1) What is the maximum distance between two points in the example grid? Note that you need to consider x, which is not represented in the plotted example grid. After you are done, be sure to reset idisplay=0. 3. (0.0) What is the maximum distance between two points in the default grid in covrand.m? 4. (0.5) Compute the sample covariance matrix C 1000, which we label as Csamp. (a) (0.2) Include a plot of C 1000 figure; imagesc(csamp); axis equal, axis tight (b) (0.1) Plot the points of C 1000 vs D. We denote this as C 1000 (d). (c) (0.1) Superimpose the covariance function, C(d). Hint: plot(id*dx,c, r. ) is one way to plot C(d). (d) (0.1) Show that the length scale is consistent with the input value L (Matlab variable L) (hint: see HW4 solutions). 5. (0.0) Now set ifourier=1 and idouble=1 and convince yourself that the FFT method gives the same result. Check that your scatterplot of estimated C 1000 (d) is about the same. 6. (0.3) The Cholesky decomposition is extremely unstable. In order to consider denser grids, or covariance functions with large L length scales, we need to use Fourier methods (ifourier=1). (a) Explain what happens in the code if idouble=0 instead of idouble=1. 1 If you are not using Matlab, then they key points when plotting are to not distort the shape of the samples and to use a color scale that ranges between 3σ and +3σ. 2 This is in the subfolder util grf, which will be in your Matlab path (type open xy2distance) if you ran covrand2d.m in the previous problem. 2

3 (b) Include a plot of C 1000 for idouble=0. (c) Explain the impact of changing idouble on the samples. (After you are done, be sure to reset idouble=1.) From here on out, we will use idouble=1. 7. (0.2) Compute the mean and standard deviation of each sample from the set of 1000 samples. Plot the means and standard deviations as two histograms. 8. (1.0) Generate a Gaussian random field with (a) Gaussian covariance, (b) exponential covariance, and (c) circular covariance. In each case, use the same Gaussian random vector (note: this requires coding). Keep all other parameters fixed (as defaults), and use nx = 2^7, ichol=0, ifourier=1, idouble=1. (a) Show the GRFs in a 3 1 subplot figure. In all plots use the same the color range caxis(3*sigma*[-1 1]). (b) Show your modified lines of code. Hint: Use the template code provided at the end of covrand2d.m and see grf2.m. 9. (0.5) Choose a covariance function (pick any icov), then plot six Gaussian random fields, each with a different length scale L. Use the same Gaussian random vector to generate each GRF. Include a 3 2 subplot figure showing your GRFs. Problem 2 (4.0). Implementation of iterative methods See the lab exercise on the quasi-newton method (lab_iter.pdf). There you were asked to adapt genlsq_template.m to write a functioning version of the quasi-newton algorithm for the 4-parameter epicenter problem. Moving forward, it is important that your code is correct. Start with genlsq_sol_template.m cp genlsq_sol_template.m genlsq_iter.m Alternatively you can use your genlsq.m script, but, whichever you use, check your results for the quasi-newton method with those listed in Table 1. In this problem you will replace the quasi-newton method with three other methods. You will not need to touch the code associated with the forward model (forward_epicenter.m). 1. (1.5) Implement the steepest descent method (Eq ). Use Eq for µ n. Use 8 iterations (niter=8). Include the following: (a) your code (b) a plot of the misfit reduction with iteration (note: this plot is produced by default) (c) a plot showing epicenter samples of the prior and posterior models (with m target and m initial ) (note: this plot is produced by default) (d) the posterior model (in Table 1; list numbers to precision) 2. (1.5) Repeat Problem 2-1 for the conjugate gradient method (Eq ). Use Eq for µ n. 3

4 Use Eq for α n. Use F 0 = I such that λ n = F 0 γ n = γ n. Note that the search direction is initialized as φ 0 = λ 0 (= γ 0 ). 3. (0.5) Repeat Problem 2-1 for the variable metric method (Section ). Use Eq for µ n. Use F 0 = I. Use Eq for F n+1, but note that there is a typo: there should be no transpose on the last δγ term in the denominator. Hint: Write the equations in non-hat notation, such as in Eq For example, note that ˆFˆγ = Fγ. 4. (0.5) (a) (0.4) Compare and contrast these three methods (see Tarantola, 2005). NOTE: This problem can be answered even if your implementations in 2-1, 2-2, 2-3 were unsuccessful. (b) (0.1) Compare the performance of each method for our problem. Problem 3 (3.0). Revisiting Tarantola (2005), Problem 7-1 Your goal is to implement Tarantola (2005), Problem 7-1. Prepare for this example by copying two files: cp genlsq_sol_template.m genlsq_crescent.m cp forward_epicenter.m forward_epicenter_crescent.m Run genlsq_crescent.m and check your quasi-newton results with those listed in Table 1. The file forward_epicenter_crescent.m will serve as a template for this problem. genlsq_crescent.m contains the iterative inverse problem, and it also calls the forward model forward_epicenter.m. In genlsq_crescent.m, replace the call to forward_epicenter.m with forward_epicenter_crescent.m. 1. (2.5) Adapt forward_epicenter_crescent.m for Problem 7-1 of Tarantola (2005) (HW3). In addition to using the values used in Problem 7-1, make the following additional choices: Define m prior to be the center of the plotting grid used for HW3. The code you want to use is this: % range of model space (Tarantola Figure 7.1) xmin = 0; xmax = 22; ymin = -2; ymax = 30; xcen = (xmax+xmin)/2; ycen = (ymin+ymax)/2; % prior model mprior = [ xcen ycen ] ; 4

5 Use the same σ values for the prior epicenter as inforward_epicenter.m (σ xs = σ ys = 10 km). This prior model is chosen as an analog for the perspective that the epicenter could be anywhere within a large region (such as the plotting grid). m initial = (15,20). m target = (15,5). (This is probably what Tarantola used, though we can t be certain.) For the case of fixed data errors, set tobs = [ ] ; eobs = tobs - dtarget; This allows us to remove the errors added to the arrival time data listed in Tarantola. (It gets added back in as dobs = dtarget + eobs, which is tobs.) axepi = [xmin xmax ymin ymax]; Solve the problem using the quasi-newton method with eight iterations. (a) Include figures showing (a) the misfit reduction; (b) the prior and posterior samples, along with the initial model. (Note that these figures are automatically generated.) (b) List the solution after eight iterations: m post, C post, and the correlation matrix ρ post. Complete Table 2. (c) How many iterations are needed for convergence? 2. (0.5) See Aster et al. (2013, p ) for how to compute confidence regions. The key concept is that the inequality (m m post ) T C 1 post (m m post) 2 describes the interior region of an M-dimensional ellipsoid (in our case, M = 2). For example, 2 can be chosen to represent the boundary of the 95% confidence region. (a) (0.1) Use eig to compute the eigen-decomposition of C post. Compute the quantity λ max /λ min. (b) (0.1) Use delta2 = chi2inv(0.95,2) to compute 2. Hint: See ex_2_1_1_uaf.m for an example of using chi2inv. (c) (0.1) Compute the lengths of the semi-major axis and semi-minor axis of the ellipse, where the length of the kth axis is λ k. (d) (0.2) Plot the confidence region using plot_ellipse. (You should see agreement between the locations of your samples of C post and the ellipse.) Also include the ellipse axes in your plot (plus m post, m initial, m target, samples of prior and posterior, etc). NOTE: Even if you did not successfully implement forward_epicenter_crescent.m, you can still do the confidence region for the epicenter associated with the forward problem of forward_epicenter.m Problem Approximately how much time outside of class and lab time did you spend on this problem set? Feel free to suggest improvements here. 5

6 References Aster, R. C., B. Borchers, and C. H. Thurber (2013), Parameter Estimation and Inverse Problems, 2 ed., Elsevier, Waltham, Mass., USA. Bühler, O., and M. Holmes-Cerfon (2009), Particle dispersion by random waves in rotating shallow water, J. Fluid Mech., 638, Gneiting, T., H. Sevcikova, and D. P. Percival (2012), Estimators of fractal dimension: Assessing the roughness of time series and spatial data, Statistical Science, 27(2), Tarantola, A. (2005), Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, Penn., USA. Table 1: Summary of results for the four iterative methods. Posterior models are listed for the eighth iteration: m post = m 8. QN = quasi-newton; SD = steepest descent; CG = conjugate gradient; VM = variable metric. List numbers to precision. prior initial target QN SD CG VM x s, km y s, km t s, s v Table 2: Epicenter problem after eight iterations with the quasi-newton method. List numbers to precision. x s, km y s, km prior initial target QN 6