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if adjoint
then erase x
if operator itself
then erase y
do iy = 1, ny {
do ix = 1, nx {
if adjoint
x(ix) = x(ix) + b(iy,ix)y(iy)
if operator itself
y(iy) = y(iy) + b(iy,ix)x(ix)
}}
Notice that the ``bottom line'' in the program is that and
are simply interchanged.
The preceding example is a prototype of many to follow;
therefore observe carefully the similarities and differences between
the operator and its adjoint.
Next, we restate the matrix-multiply pseudo code in real code.
The module matmult for matrix multiply along with its adjoint exhibits the style we use repeatedly. At last count there were 53 such routines (operator with adjoint) in this book alone.
void matmult_lop (bool adj, bool add, int nx, int ny, float* x, float*y) /*< linear operator >*/ { int ix, iy; sf_adjnull (adj, add, nx, ny, x, y); for (ix = 0; ix < nx; ix++) { for (iy = 0; iy < ny; iy++) { if (adj) x[ix] += B[iy][ix] * y[iy]; else y[iy] += B[iy][ix] * x[ix]; } } } |
We now have a module with two entries/
One is named _init
for the physical scientist who defines the physical problem by
defining the operator.
The other is named _lop
for the least-squares problem solvers,
computer scientists not interested
in how we specify . They will be iteratively computing
and
to optimize the model fitting.
To use matmult, two calls must be made, the first one
matmult_init( bb);is done by physical scientists to set up the operation. Most later calls are done by numerical analysts in solving code like in Chapter
matmult_lop( adj, add, nx, ny, x, y);where adj is the logical variable saying whether we desire the adjoint or the operator itself, and where add is a logical variable saying whether we want to accumulate like
We split operators into two independent processes; the first is used for geophysical set up, while the second is invoked by mathematical library code (introduced in the next chapter) to find the model that best fits the data. Here is why we do so. It is important that the math code contain nothing about the geophysical particulars. This independence enables us to use the same math code on many different geophysical applications. This concept of ``information hiding'' arrived late in human understanding of what is desirable in a computer language. Subroutines and functions are the way new programs use old ones. Object modules are the way old programs (math solvers) are able to use new ones (geophysical operators).
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