  The classical approach to solvability of a mathematical problem is to define
a method which includes certain rules of operation or algorithms. Then using
the defined method, one can show that some problems are solvable or not
solvable or undecidable depending on the particular method. With numerical
solutions implemented in a computer, it might be more practical to define
solvability of a mathematical problem as a complex decomposition problem. The
decomposition breaks the data into a simple part and a complex part. The simple
part is the solvable part by the method prescribed in the problem definition.
The complex part is the leftover of the simple part. Complex part can be viewed
as the "residual" of data or operator. It should be interpreted and not to be
discarded as useless. We will give different examples to illustrate the more
practical definition of solvability. The complex part is not noise and should
not be viewed as useless part of the data. It has its own merit in terms of
topological or geological interpretation. We have de-emphasized absolute
solvability and have emphasized the practical solvability where the simple and
complex parts both play important roles.
