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This book does not solve giant problems,
but it does solve personal-computer-sized problems
in the manner of giant problems.
There is big money in solving giant problems.
Big money brings specialist solutions beyond the scope of this book.
But let us take a look.
Closest to me is the seismic survey industry.
Model space is 3-D, a cube, roughly a thousand 2-D screen fulls,
a screen full being roughly 1,000
1,000,
a gigabyte in total.
Data space is 5-dimensional.
A seismogram is a thousand time points.
Our energy source lies in two dimensions on the Earth surface plane,
as do our receivers.
1+2+2=5.
All this compounds roughly to 1,000 to the
power,
a thousand terabytes, a petabyte.
Fully convergent solutions needing
iterations
of operators is ridiculous, while more than a handful are nearly so.
We think mainly of using only the adjoint.
Theory and experimentation offer some guidance.
Remember that adjoints are great when unitary (already an inverse).
Adjoints are improved if they can be made more unitary.
The basic strategy for improving an adjoint is
finding one good diagonal-weighting function before the adjoint and another after it.
Recalling ``IID,'' adjoints are also made more unitary
by filter matrices that have the effect of whitening output.
Simple filter matrices are the gradient and the Laplacian.
More generally,
a compact way to whiten spectra
is multidimensional autoregression,
expounded in Chapter .

**Subsections**

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**Up:** Preconditioning
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2015-05-07