Basic operators and adjoints |

A huge volume of literature establishes theory for two estimates of the model, and , where

Some reasons for the literature being huge are the many questions about the existence, quality, and cost of the inverse operators. Let us quickly see why these two solutions are reasonable. Inserting equation (42) into equation (43), and inserting equation (44) into equation (42), we get the reasonable statements:

Equation (45) says the estimate gives the correct model if you start from the modeled data. Equation (46) says the model estimate gives the modeled data if we derive from the modeled data. Both these statements are delightful. Now, let us return to the problem of the inverse matrices.

Normally, a rectangular matrix does not have an inverse.
Surprising things often happen, but commonly,
when is a tall matrix
(more data values than model values),
then the matrix for finding
is invertible while that for finding
is not;
and when the matrix is wide instead of tall
(the number of data values is less than the number of model values),
it is the other way around.
In many applications neither
nor
is invertible. This difficulty is
solved by ``**damping**'' as we see in later chapters.
If it happens that
or
equals (unitary operator),
then the adjoint operator is the inverse
by either equation (43) or (44).

Current computational power limits matrix inversion jobs to about variables. This book specializes in big problems, those with more than about variables. The iterative methods we learn here for giant problems are also excellent for smaller problems; therefore we rarely here speak of inverse matrices or worry much if neither nor is an identity.

- Consider the matrix

(47) - Modify the calculation in Figure 5 so that there is a triangle waveform on the bottom row.
- Notice that the triangle waveform is not time aligned
with the input
`in2`. Force time alignment with the operator or . - Modify
`leakint`by changing the diagonal to contain 1/2 instead of 1. Notice how time alignment changes in Figure 5. - Suppose a linear operator has
its input in the discrete domain and
its output in the continuum.
How does the operator resemble a matrix?
Describe the operator that has
its input in the discrete domain and
its output in the continuum.
To which do you apply the words
``scales and adds some functions,''
and to which do you apply the words
``does a bunch of integrals''?
What are the integrands?

Basic operators and adjoints |

2014-09-27