MA 590  Homework 4

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MA 590
Homework 4
Problems
1 (10 points) The goal of this problem is to explore the use of Tikhonov regularization in
solving linear discrete inverse problems. Consider again the Phillips’ test problem, as used in
HW3, which can be generated with phillips.m (posted on Canvas; see the ‘Regularization
Tools’ manual by P. C. Hansen for the problem description).
(a) Use the phillips function with n = 64 to generate the true solution xtrue, operator matrix
G, and noiseless data ytrue. Generate noisy data y by adding a normally distributed perturbation to ytrue with zero mean and standard deviation 10−4
. For a number of different
regularization parameters α in the range of 10−4
to 1, compute the filter factors
fi =
σ
2
i
σ
2
i + α2
along with the corresponding zeroth-order Tikhonov solution
xα =
Xn
i=1
fi
u
T
i y
σi
vi
using of the SVD of G. Use a logarithmic distribution of α values, as generated by MATLAB’s
logspace function. For each α, plot both the filter factors and the solution, and comment
(b) BONUS: Explore higher-order Tikhonov solutions using the finite difference smoothing
operators L1 and L2 to regularize with respect to the first and second derivatives of x, respectively. You may make use of MATLAB’s gsvd function and the ‘Regularization Tools’
get l function (posted on Canvas) in computing your solutions. Comment on the “optimal”
choice of α and the corresponding solution xα in each case.
2 (10 points) The aim of this problem is to demonstrate how the Tikhonov solution xα,
its norm ||xα||2, and the residual norm ||Gxα − y||2 change as α goes from large values
(oversmoothing) to small values (undersmoothing). Use the ‘Regularization Tools’ shaw
test problem (posted on Canvas) with n = 32, adding Gaussian noise with zero mean and
standard deviation 10−3
to generate the data y.
(a) Use MATLAB’s logspace to generate 20 logarithmically distributed values of α from 10−5
to 10. For each α, compute the corresponding zeroth-order Tikhonov solution xα. Store the
solutions xα as the columns of a matrix X, and inspect X (e.g., by means of mesh or surf)
to study the progression of xα as α varies from oversmoothing to undersmoothing. Discuss
(b) For the α values in part (a) (or even more values in the same interval), compute the error
norm ||xtrue − xα||2 and plot it versus α. Determine the optimum value of α (i.e., the one
that leads to the small error norm) and locate its position on the corresponding L-curve –
is it near the “corner”? Discuss.
3 (10 points) This problem illustrates the difficulty of the TSVD and Tikhonov methods
in computing regularized solutions when the true solution is discontinuous. Consider the
‘Regularization Tools’ wing test problem (posted on Canvas), where the solution has two
discontinuities. Generate the model problem using wing with n = 100, plot the exact solution
xtrue, and notice its form. Using the noiseless data ytrue, compute the TSVD and zerothorder Tikhonov solutions for various regularization parameters. Monitor the solutions and
try to find the “best” values of the regularization parameters k and α. Discuss the difficulties
in reconstructing the discontinuities of the true solution in each case.
Note: For any of the above problems for which you use MATLAB to help you solve, you must