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Unformatted text preview: the optimal L2 approximation and (discretized) L1 optimal approximation for K = 10. You
can ﬁnd the L2 optimal approximation analytically, or by solving a leastsquares problem associated
with the discretized version of the problem. Since y is even, you can take the sine coeﬃcients in
your approximations to be zero. Show y and the two approximations on a single plot.
In addition, plot a histogram of the residuals (i.e., the numbers f (ti ) − y (ti )) for the two approximations. Use the same horizontal axis range, so the two residual distributions can easily be compared.
(Matlab command hist might be helpful here.) Make some brief comments about what you see.
5.4 Penalty function approximation. We consider the approximation problem
minimize φ(Ax − b)
where A ∈ Rm×n and b ∈ Rm , the variable is x ∈ Rn , and φ : Rm → R is a convex penalty
function that measures the quality of the approximation Ax ≈ b. We will consider the following
choices of penalty function:
40 (a) Euclidean norm. m φ(y ) = y 2 2
yk )1/2 . =(
k=1 (b) ℓ1 norm. m φ(y ) = y 1 =
k=1  yk  . (c) Sum of the largest m/2 absolute values.
⌊m/2⌋ y [k]...
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 Fall '13
 F.Borrelli
 The Aeneid

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