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CHAPTER 3. THE ADJOINT PROBLEM
3.2
19
How to find an optimum model
Figure 3.2 shows a schematic sketch of the residual energy at one point in space as a function of two model parameters
λ and µ. The colors represent different values of the residual energy. Red areas represent models with high residual
energy which do not fit the data, while the blue parts are good fitting models with low residual energies. The aim is to
find the minimum of the residual energy marked by the red cross. Starting at a point m1 = (λ1 (x), µ1 (x), ρ1 (x), ) in
the parameter space we want to find the minimum by updating the material parameters in an iterative way
m2 = m1 + µ1 δm1 ,
(3.4)
along the search direction δm1 with the step length µ1 . To find the optimum search direction δm1 we expand the
residual energy E(m1 + δm1 ) near the starting point in a Taylor series:
2 ∂E
1
∂ E
δmT
(3.5)
E(m1 + δm1 ) ≈ E(m1 ) + δm1
+ δm1
∂m 1 2
∂m2 1 1
and set the derivative of Eq. 3.5 with respect to δm1 zero
2 ∂E(m1 + δm1 )
∂ E
∂E
+ δm1
=0
=
∂δm1
∂m 1
∂m2 1
(3.6)
Which finally leads to
δm1 = −
∂2E
∂m2
−1 1
∂E
∂m
= −H1 −1
1
∂E
∂m
(3.7)
1
where (∂E/∂m)1 denotes the steepest-descent direction of the objective function and H1 −1 the inverse Hessian
matrix. The inverse Hessian matrix for the elastic problem is often singular and can only be calculated with high
computational costs. Therefore the inverse Hessian matrix is approximated by a preconditioning operator P. There is
u
−
u
obs
=
δu
¬ time
mod
channel ®
channel ®
Figure 3.1: Definition of data residuals δu.
channel ®