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![]() | Geophysical applications of a novel and robust L1 solver | ![]() |
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The hybrid norm is defined as
The Conjugate Direction method is commonly used for solving
immense linear regressions in exploration geophysics. The idea of the CD
method is to search the plane determined by the gradient and the
previous step for the best step direction and length, instead of
moving along the gradient direction. The best direction in that
plane is the linear combination of the gradient and previous step
vector that decreases the measure of the residual the most. Traditionally,
the measure is chosen to be L2, for its simplicity; however, we
generalize the CD method for any arbitrary convex measure
. Readers
can determine which measure to use to satisfy their own
objectives.
Now let us examine the generalization of the CD method in detail. At each
iteration, we have the residual vector
, the gradient
and the
previous step
. Therefore, the updated residual can be written as:
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(3) |
Now, taking the derivatives of the parabolic function in (5)
with respect to
and
and setting them to zero, we end
up with a linear system of
and
:
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(6) |
Then we can obtain
by simply solving a set of 2
2 linear equations.
Notice that the
we get here is minimizing the
approximated function (5), not the original objective
function. Therefore, it is necessary to solve for
multiple times within each CD iteration. By doing this relatively cheap
plane-search loop, we expect to save the number of iterations for
the outer loop (Conjugate Direction), which is usually much more
computational intensive (requiring application of both the forward and
adjoint operator).
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![]() | Geophysical applications of a novel and robust L1 solver | ![]() |
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