Alternatives to conjugate direction optimization for sparse solutions to geophysical problems |
This maps to our conventional model-fitting treatment using optimization theory according to the following:
parameter space | model space | (6) | |
objective function | regularization | (7) | |
constraint functions | forward operator | (8) | |
constraint vector | data space | (9) |
(10) | |||
(11) |
I have introduced the scalar, , which is the value of the objective. This will be either minimized or maximized depending on the particular geophysical problem. Effectively, this means finding a value for that optimally aligns with the model-fitting and data-fitting goals.
Correspondingly, for a simple example:
(12) |
(13) |
To solve according to the norm, we need to take account of the absolute value in the definition of the error criteria, noting that the element of the residual is defined as an absolute value of the error term:
(14) |
To account for this, we extend the linear programming matrix representation to augmented form. This requires an extension of the vector. The approach is not entirely dissimilar to the placement of a regularization in the model vector in a conventional setup, in that it is reformulating the equations to provide us with a fit in compliance with our a priori geophyiscal knowledge.
The augmented representation is written in matrix form as
The first row of the augmented form results in the optimization criteria:
A large value of Z maximizes the original objective function, subject to the constraints in the matrix. The goal is to put as much ``energy'' in the Z (objective function) with as little energy in all other rows of the augmented matrix; this is accomplished with the Simplex Algorithm, simultaneously satisfying the minimization of the pure norm criteria.
Alternatives to conjugate direction optimization for sparse solutions to geophysical problems |