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Geology 5670/6670 Inverse Theory 17 Feb 2015 Last time: Nonlinear Inversion • Complicating factors for solution of nonlinear problems, i.e. of the form F(m) = d, using a standard gradient method (i.e., Gauss-Newton) include multiple minima and nonconvergence • Divergence results when the sensitivity matrix G is ill-conditioned. Stabilization of the problem can use the generalized inverse of order p: 1 T m k1 m k V p p U p d k Or a truncated Gauss step: T 1 T m k 1 m k k G k G k G k d k Read for Wed 18 Feb: Menke Ch 9 (163-188) © A.R. Lowry 2015 Last time (cont’d): Nonlinear Problems • Other approaches to stabilization of a nonlinear problem use damping, and these include Levenburg damping: T 1 T m k 1 m k G k G k k I G k d k And Levenburg-Marquardt damping: 1 T T T mk 1 mk Gk Gk kdiagGk Gk Gk d k Note the latter is “intermediate”, in a sense, between Levenburg damping and a truncated Gauss step. Assignment I is now posted on the course website… Due Friday, Feb 27 at the beginning of class Criteria for stopping iteration: (Are we there yet?) 1. Reduced 2: Stop when 1 2 N M N ei2 i1 2 i 1 Note that if measurement uncertainties are overestimated, 2 for the global minimum is < 1 and final model depends on your initial guess: 2 = 1 contour m0 m0 Criteria for stopping iteration: (Are we there yet?) 2. Residual norm less than some tolerance: d Gm d Gm 2 d Gm T 3. Parameter step less than some tolerance: m k 1 m k mk 4. Gradient less than some tolerance: T E 2G d 5. Number of iterations: k > kmax Solution Appraisal: For the nonlinear problem, 1 T C m 2 G G or T 1 1 C m G C G is only approximately true, and then only when m is “near” mtrue (and higher order derivatives are “small”, & there are no other minima). That this approach works as an approximation is because errors arising from truncation of the Taylor series approximation (used to derive the sensitivity matrix G in the iterative approach) become small near mtrue. However, generally more robust to use an analytic confidence region… The Likelihood Ratio Method can be used to generate confidence regions from contours of E: d F m N E 2 i i i1 Assume measurement errors are jointly normal, zero-mean, constant variance and uncorrelated E is 2 with N degrees of freedom (if known), or 2 with N-M degrees of freedom if must be inferred from 2 of the minimum misfit.