Question 2 and Question 3
This question, as well as the next one, are concerned with recovering a function u(t) on the interval [0, 1] given noisy data bi at points ti = ih, i = 0, 1, . . . , N, with N = 1/h. Because the data values are noisy, we cannot simply set u(ti) = ui = bi : knowing that u(t) should be piecewise smooth, we add a regularization term to penalize excessive roughness in u. For the unknown vector u = (u0, u1, . . . , uN ) T we therefore solve
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