e
continue research of the previous section. The procedure has several problems:
1. For weak penalty term (high
)
the procedure takes several time steps to converge to
.
For strong penalty term (low
)
it overshoots at the spacial boundaries of
,
taking the solution far beyond desired minimal installation at
.
2. It picks up and accumulates discrepancy from correct solution while
converging into
.
3. Boundary probing functions introduce discrepancy beyond
area.
Therefore, we propose to stop advancement in time until we iteratively and
minimalistically converge into
using collections of internal probing functions of increasing precision.
Without time increment the evolution
looks like
this:
where the vector
is input parameter,
is a control parameter at our disposal,
is a well conditioned matrix and
has the
form
for some wavelet basis
and selection of probing functions
.
We exploit the freedom of selection of probing functions by inserting control
parameters
:
Furthermore, we place probing functions conservatively within the area
to make sure that no support of any
intersects with
.
The precision loss that comes from such requirement may be mitigated by
increasing scale of the probing functions. The penalty function
now takes the
form
We introduce the
notations
then
Let
The control parameters
are optimized to
achieve
with
constraints
We perform elementary
transformations:
and arrive to the
problem
This is a quadratic optimization problem with linear constraints. To make this
evident, we transform the problem to canonical
form:
The problem takes the
form
where
The gradients for the problem
are
