A blurred, noisy measurement is deblurred by Tikhonov regularisation, as in Section…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics
Error, Chance and the Machine, Ch Linear systems, least squares and the singular value decomposition
Problem
A blurred, noisy measurement b=Ax+e is deblurred by Tikhonov regularisation, as in Section 2.7. The data are synthetic, so the true signal x is known. Only three singular components matter: σi=1.00,3.00×10−2,1.00×10−3, with true coefficients viTx=1.00,0.500,0.200 (arbitrary units). The noise is zero-mean and independent, and each uiTe has variance δ2 with δ=1.00×10−3.
((a)) The error of xλ is a bias part plus a noise part, xλ−x=−i∑(1−fi)(viTx)vi+i∑fiσiuiTevi. Show that its expected squared length is B(λ)+N(λ) with B=i∑(1−fi)2(viTx)2,N=i∑fi2σi2δ2, and say why no cross term appears.
((b)) Evaluate B, N and B+N at λ=1.00×10−3, 1.00×10−2 and 1.00×10−1. Give also the relative error B+N/∥x∥2 as a percentage, where ∥x∥22=∑i(viTx)2.
((c)) Which of the three values of λ is best? At each, say which part dominates, and which singular component is responsible.
((d)) Evaluate B+N at λ=5.00×10−3. What does this show about the position of the best λ and the sharpness of the minimum? Say which quantity in part 1 a real measurement would not supply, and so why this calculation cannot be used to pick λ there.