Course on Inverse Problems Albert Tarantola Lesson X: Optimization - - PowerPoint PPT Presentation

Princeton University

Department of Geosciences

Course on Inverse Problems

Albert Tarantola

Lesson X: Optimization

Optimization

• If the volumetric probability fpost(M) is expected to have a

small number of maxima (say one, or two, or three),

• we may try to locate them by using standard optimization

methods (simplex methods, gradient-based methods),

• and we may try to study fpost(M) in the neighborhood of

each optimum. Practical tip: simplex methods and gradient-based methods work much better with the function ψ(M) = log( fpost(M)/ f0) than with the function fpost(M) .

Least-squares theory

• The model parameter manifold may be a linear space, with

vectors denoted m, m′, . . . , and the a priori information may have the Gaussian form fprior(m) = k exp

• 1

2 (m − mprior)t Cm

• 1 (m − mprior)
• .
• The observable parameter manifold may be a linear space,

with vectors denoted o, o′, . . . , and the information brought by measurements may have the Gaussian form gobs(o) = k exp

• 1

2 (o − oobs)t Co

• 1 (o − oobs)
• .
• The forward modeling relation becomes, with these nota-

tions,

• = o(m)

.

Then, the posterior volumetric probability for the model pa- rameters, whose general expression is fpost(m) = 1 ν fprior(m) gobs( o(m) ) . here becomes fpost(m) = k exp( −S(m) ) , where the misfit function S(m) is the sum of squares 2 S(m) = (m − mprior)t Cm

• 1 (m − mprior)

+ (o(m) − oobs)t Co

• 1 (o(m) − oobs)

.

The maximum likelihood model is the model m maximizing fpost(m) . It is also the model minimizing S(m) . It can be

• btained using a quasi-Newton algorithm,

mn+1 = mn − H-1

n γn

, where the Hessian of S is Hn = Ot

n C-1

• On + C-1

m

, and the gradient of S is γn = Ot

n C-1

• ( o(mn) − oobs ) + C-1

m ( mn − mprior )

.

Here, the tangent linear operator On is defined via

• ( mn + δm ) = o(mn) + On δm + . . .

When the notations m = {mα} = {m1, m2, . . . , mp}

• = {oi} = {o1, o2, . . . , oq}
• i = oi(m1, m2, . . . , mp)

apply, then On is the matrix of partial derivatives Oiα = ∂oi ∂mα (evaluated at point mn ).

As we have seen, the model m∞ at which the algorithm con- verges maximizes the posterior volumetric probability fpost(m) . To estimate the posterior uncertainties: the covariance oper- ator of the Gaussian volumetric probability that is tangent to fpost(m) at m∞ is

• Cm = H-1

∞

, while the covariance operator of the Gaussian volumetric prob- ability that is tangent to gpost(o) at o∞ = o(m∞) is

• Co = O∞

Cm Ot

∞

. Example: plants leafs and the radiative transfer model.

⇒ mathematica notebook