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MA 590  Homework 3 

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MA 590
Homework 3
Problems
1 (30 points) The goal of this problem is to explore the use of the truncated singular value
decomposition (TSVD) in solving linear discrete inverse problems. Consider the Phillips’
test problem, which is a discretization of a Fredholm integral equation of the first kind; for
the problem description, see the ‘Regularization Tools’ manual (by P. C. Hansen, link posted
on Canvas). The true solution, operator matrix, and noiseless data can be generated using
the .m-file phillips.m (also posted on Canvas).
(a) Use phillips.m with n = 64 to generate the true solution xtrue, operator matrix G, and
noiseless data ytrue. Report the condition number of G. Generate noisy data y by adding
a normally distributed perturbation e with zero mean and standard deviation 10−4
, so that
||e||2 ≈ 8 × 10−4
. Make figures plotting the true solution xtrue and the data (show ytrue and
y in the same plot).
(b) Compute the singular value decomposition of G using MATLAB’s svd function. Plot the
singular values on a semilogy plot, and discuss the range of the spectrum (i.e. report the
largest and smallest singular value). How many nonzero singular values does G have; i.e.
what is p in the compact SVD for this problem?
(c) Make a semilogy plot of the coefficients |u
T
i y|, singular values σi
, and ratios |u
T
i y|/σi vs.
the index i for i = 1, . . . , n. Does the Discrete Picard Condition hold for y here? Discuss
your findings.
(d) Compute the generalized inverse solution using the compact SVD matrices to compute the
pseudoinverse, and plot your estimated solution compared to the true solution. Does the
generalized inverse solution give a reasonable result for this problem? Discuss.
(e) Consider the TSVD with truncation parameter k. Generate an “L-curve” using a loglog
plot comparing the 2-norm of the TSVD solution to the 2-norm of the residual for different
values of k between k = 2 and k = 62. From this plot, what is the “optimal” value of k? Use
the “optimal” k to compute the TSVD solution, and plot your estimated solution compared
to the true solution. Does the TSVD give a reasonable result? Discuss.
(f) Repeat this problem increasing the level of noise added in part (a) (i.e. increase the standard
deviation of the perturbation) and discuss how your results change. In particular, how does
the “optimal” k value differ when the noise in the data increases?
Note: For any of the above problems for which you use MATLAB to help you solve, you must
submit your code/.m-files as part of your work. Your code must run in order to receive full credit.
If you include any plots, make sure that each has a title, axis labels, and readable font size, and
include the final version of your plots as well as the code used to generate them.
2

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