1. No category

ALL-AT-ONCE INVERSION RESULTS FOR THREE-DIMENSIONAL
MAGNETOTELLURICS
Wenke Wilhelms, Ralph-Uwe Börner, and Klaus Spitzer
KKT system
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ABSTRACT
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The all-at-once inversion approach requires no explicit
forward calculation, because the forward modelling equations
are incorporated in the objective function as constraints. This
leads to a huge, so-called Karush-Kuhn-Tucker (KKT) system,
which is solved in each step of the iteration procedure to
update model parameters m, Lagrangian multipliers λ, and
data u - all at once. Forming the Hessian, i.e., the matrix
containing second derivatives required in a Newton step, is
key in the all-at-once approach. The resulting KKT system can
be solved using Krylov subspace projection techniques. In our
case preconditioning is indispensable and needs to be
implemented in order to achieve a better conditioned system
of equations.

Lu
δu
Q Q
K
A
 K> βW>W + R − D (G − B)> δm = − Lm
δλ
Lλ
A
G−B
0
Regularisation
W is a regularisation matrix:
k1W1
W=
k2W2
W1 is the identity matrix with a weighting factor k1. W2
contains the first derivative with a weighting factor k2.
The regularisation parameter β controls the influence of
the regularisation.
Objective function L
L(u, m, λ) = 12 ||Q(u + up)||22 + β2 ||W(m − mref)||22 + λ>[A(m)u − b(m)]
Derivatives of the objective function:
Lu = Q>Q(u + up) + A>λ
Lm = βW>W(m − mref) + (G − B)>λ
Lλ = Au − b
sQMR and Preconditioning
- sQMR stand for simplified (symmetric) quasi-minimal residual
method.
- sQMR is a Krylov subspace method, an iterative solver for
large and sparse systems of equations, e.g. KKT system.
- Without preconditioning sQMR does not converge in a
reasonable number of iterations (see Fig. 1). Reasons are
shown in Fig. 2 and 3.
- Choosing the preconditioning matrix is a trade off between the
computational effort for calculating the preconditioned system
and the computational time for solving it.
Characteristics of the KKT matrix
8
6
8
10
x 10
6
6
10
4
10
4
2
2
10
0
0
10
−2
−2
10
−4
−4
10
−6
0
10
−8
0
without preconditioning
with preconditioning
−6
1000
2000
3000
4000 5000 6000
number of eigenvalue
7000
8000
9000
−1
10
10
10000
Fig. 2: Eigenvalues of the KKT matrix Hkkt.
0
1000
2000
3000
4000
5000 6000
number of eigenvalue
7000
8000
9000
10000
Fig. 3: Log scale plot of eigenvalues of the KKT
matrix.
−2
10
residual
- Hkkt ∈ Rn×n is symmetric and indefinite.
- KKT matrix has positive and negative eigenvalues, as shown in Fig. 2.
- Eigenvalues that seem to be zero in Fig. 2 are not exactly zero but
very small (Fig. 3).
- Therefore preconditioning is indispensable.
−3
10
−4
10
−5
10
0
2
4
6
8
10
12
14
number of inner (sQMR) iterations
16
18
20
Fig. 1: Convergence curves for sQMR with and without
preconditioning.
Inversion Results
4
4
1
1
0.5
5
z in km
We developed a three-dimensional
magnetotelluric inversion using the all-at-once
approach. We worked on sQMR, a Krylov
subspace method, to solve the huge and sparse
system of equations. Preconditioning is inevitable
to enhance the eigenvalue distribution of the KKT
matrix. To improve the inversion results we will
also include more data for a bigger variety of
frequencies.
0
−0.5
10
−1
log10(σ in S/m)
z in km
0.5
CONCLUSION & OUTLOOK
5
0
−0.5
10
−1
−1.5
−1.5
15
15
−2
−2
−5
0
y in km
5
4
x 10
−5
0
y in km
5
4
x 10
Fig. 4: Left: true model. Right: inversion results.
[1] R. W. Freund and N. M. Nachtigal. “A new Krylov-subspace method for symmetric indefinite linear systems”. In: Proc. 14th
imacs world congress on computational and applied mathematics (1994), pp. 1253–1256.
[2] E. Haber and U. M. Ascher. “Preconditioned all-at-once methods for large, sparse parameter estimation problems”. In:
Inverse Problems 17 (2001), pp. 1847–1864.
[3] E. Haber, U. M. Ascher, and D. W. Oldenburg. “On optimization techniques for solving nonlinear inverse problems”. In:
Inverse Problems 16 (2000), pp. 1263–1280.
[4] Y. Saad. Iterative methods for sparse linear systems. SIAM, 2000.
TU Bergakademie Freiberg
Institute of Geophysics and Geoinformatics
Gustav-Zeuner-Str. 12
09599 Freiberg
Germany
x 10
log10(σ in S/m)
x 10
A
W
Q
b
d
u
m
λ
...
...
...
...
...
...
...
...
forward operator
regularisation matrix
interpolation operator
right hand side of the forward problem
measured data
electric field
model parameters
Lagrange parameters
contact:
Wenke Wilhelms
mail: [email protected]
β
... regularisation parameter
mref ... reference model
up ... primary electric field
Derivatives of the forward problem:
K = ∂ m A> · λ
R = ∂mG> · λ
G = ∂mA · u
D = ∂mB> · λ
B = ∂mb