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If the data are represented by vector
, model parameters by
vector
, and their functional relationship is defined by the
forward modeling operator
, the least-squares optimization
approach amounts to minimizing the least-squares norm of the residual
difference
. In Tikhonov's regularization approach, one
additionally attempts to minimize the norm of
, where
is the regularization operator. Thus, we are looking for the model
that minimizes the least-squares norm of the compound vector
,
where
is a scalar scaling parameter. The formal solution has the
well-known form
![\begin{displaymath}
\widehat{\mathbf{m}} =
\left(\mathbf{L}^T\,\mathbf{L} +
...
...bf{D}^T\,\mathbf{D}\right)^{-1}\,\mathbf{L}^T\,\mathbf{d}\;,
\end{displaymath}](img12.png) |
(1) |
where
denotes the least-squares estimate of
,
and
denotes the adjoint operator. One can carry out the
optimization iteratively with the help of the conjugate-gradient method
(Hestenes and Steifel, 1952) or its analogs. Iterative methods have computational
advantages in large-scale problems when forward and adjoint operators are
represented by sparse matrices and can be computed efficiently
(Saad, 2003; van der Vorst, 2003).
In an alternative approach, one obtains the regularized estimate by
minimizing the least-squares norm of the compound vector
under the constraint
![\begin{displaymath}
\epsilon \mathbf{r = d - L m = d - L P p}\;.
\end{displaymath}](img16.png) |
(2) |
Here
is the model reparameterization operator that
translates vector
into the model vector
,
is the scaled residual vector, and
has the
same meaning as before. The formal solution of the preconditioned
problem is given by
![\begin{displaymath}
\widehat{\mathbf{m}} =
\mathbf{P}\,\widehat{\mathbf{p}} =...
...f{L}^T +
\epsilon^2\,\mathbf{I}\right)^{-1}\, \mathbf{d}\;,
\end{displaymath}](img20.png) |
(3) |
where
is the identity operator in the data space.
Estimate 3 is mathematically equivalent to
estimate 1 if
is invertible
and
![\begin{displaymath}
\left(\mathbf{D}^T\,\mathbf{D}\right)^{-1} =
\mathbf{P}\,\mathbf{P}^T = \mathbf{C}\;.
\end{displaymath}](img23.png) |
(4) |
Statistical theory of least-squares estimation connects
with the model covariance operator (Tarantola, 2004). In a more
general case of reparameterization, the size of
may be
different from the size of
, and
may not have
the full rank. In iterative methods, the preconditioned formulation
often leads to faster convergence. Fomel and Claerbout (2003) suggest
constructing preconditioning operators in multi-dimensional problems
by recursive helical filtering.
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2013-03-02