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2.3 Spectral Filtering 9
2.3 Spectral Filtering
Due to the difficulties associated with the discrete inverse problems the naive
solution x = A −1 b is useless since it is becomes dominated by the rounding
errors. We will in this section introduce two spectral filtering methods, which
can be expressed as a filtered SVD expansion on the form:
xfilter =
min{m,n}
i=1
ϕi
〈ui, b〉
vi,
where ϕi are the filter factors for the corresponding method. We will first
introduce the truncated SVD method (TSVD).
We realised that the large errors in the naive solution came from the noisy SVD
coefficients corresponding to the smallest singular values but we also noticed that
the SVD coefficients for large singular values were useful, since these coefficients
fulfilled 〈ui,b〉
σi ≃ 〈ui,¯b〉 , where b is the noisy right-hand side and σi
¯b is the righthand
side without noise. This leads to the truncated SVD (TSVD) method
where we choose only to include the first k components of the naive solution
to x. With this method we therefore cut off those SVD coefficients that are
dominated by inverted noise. We define the TSVD solution as
xk =
σi
k 〈ui, b〉
vi,
i=1
where we call k the truncation parameter and k must be chosen such that all
the noise-dominated SVD coefficients are discarded. This leads to the following
filter factors for the TSVD method:
ϕi =
σi
1 i ≤ k
0 i > k.
The second method we will introduce the Tikhonov regularization. For this
method the filter factors is defined as
ϕi =
σ 2 i
σ 2 i
+ ω2 , i = 1, · · · , n,
where ω is the regularization parameter, which in a sense corresponds to the
truncation parameter k. The Tikhonovs regularization corresponds to the following
minimization problem
min
x {Ax − b 2 2 + ω2 x 2 2 }.