Tikhonov regularization Solve the Tikhonov minimization problem x { - - PowerPoint PPT Presentation

Tikhonov regularization

Solve the Tikhonov minimization problem min

x {Ax − b2 + µLx2}

= ⇒ xµ, where

• A ∈ Rm×n;
• L ∈ Rp×n, p ≤ n, is the regularization operator.

Common choices: L = I or a finite difference

• perator;
• µ > 0 is the regularization parameter. It is important

to determine a suitable value; see Engl, Hanke, Neubauer; Hansen; Kilmer; O’Leary; ...

• N(L)∩N(A) = {0}

= ⇒ xµ unique for any µ > 0.

Example 2: Fredholm integral equation of the 1st kind 1 k(s, t)x(t)dt = 1 6(s2 − s), 0 ≤ s ≤ 1, k(s, t) =    s(t − 1), s < t, t(s − 1), s ≥ t. Solution x(t) = t. Code deriv2 from Regularization Tools by Hansen discretizes by Galerkin method with 200 piecewise constant test and trial functions. Relative error (noise) in rhs 0.1%.

Solution for problem with L = I, µ determined by L-curve

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.01 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Solution obtained with square invertible regularization

• perator L = tridiag[−1, 2, −1], µ determined by L-curve

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08