4.6 Advanced Features for Set-Theoretic Microstructure Quantification
115
4.6.1 A Heuristic Iterative Scheme to Determine a Zero-Cutoff
Threshold
Our problems are largely due to errant
small values of J and E, so what follows is an approach to resolve
this problem, all
using the LMS estimator on the 10GHz data.
1) Run the calculation throwing out all (E,J) pairs for which
E is less than 20\% of the maximum size for each cell and
each experiment. We’ll call the outputs ’sigmaxx_0.txt’
and ’sigmayy_0.txt’.
2) Rerun the calculation throwing out (E,J) pairs with E less
than the 20\% threshold, and also pinning to zero all currents
associated with conductivity values in ’sigmaxx_0.txt’ and
’sigmayy_0.txt’ that are less than or equal to zero. We’ll
call the outputs ’sigmaxx_1.txt’ and ’sigmayy_1.txt’
3) Repeat 2) substituting ’sigmaxx_1.txt’ for ’sigmaxx_0.txt’
and ’sigmayy_1.txt’ for ’sigmayy_0.txt’ to obtain
’sigmaxx_2.txt’ and ’sigmayy_2.txt’
4) Continue in this manner until there are no negative
conductivity values in the output.
This converged after 4 iterations, the outputs for which are
attached. You can see that the first step (zeroth iteration)
gets us into the ball park (it’s much better than with no
threshold on the size of E). The results of the first iteration
are pretty good. The fourth iteration is very good.
This, of course, assumes that we know that there are no negative
anomalous conductivities, which will not always be the case,
so this is not a general algorithm. It also assumes that we
can throw out the small (E,J) pairs at a 20\% threshold, which
may also not be the case in general, so again it is not general
algorithm. But it may be that for any given problem a threshold
for (E,J) size and conductivity value can be found that will work.
If our threshold on conductivity is nonzero, we cannot pin the
currents to zero, but will have to rely on a constraint equation,
which is in development.
115
4.6.1 A Heuristic Iterative Scheme to Determine a Zero-Cutoff
Threshold
Our problems are largely due to errant
small values of J and E, so what follows is an approach to resolve
this problem, all
using the LMS estimator on the 10GHz data.
1) Run the calculation throwing out all (E,J) pairs for which
E is less than 20\% of the maximum size for each cell and
each experiment. We’ll call the outputs ’sigmaxx_0.txt’
and ’sigmayy_0.txt’.
2) Rerun the calculation throwing out (E,J) pairs with E less
than the 20\% threshold, and also pinning to zero all currents
associated with conductivity values in ’sigmaxx_0.txt’ and
’sigmayy_0.txt’ that are less than or equal to zero. We’ll
call the outputs ’sigmaxx_1.txt’ and ’sigmayy_1.txt’
3) Repeat 2) substituting ’sigmaxx_1.txt’ for ’sigmaxx_0.txt’
and ’sigmayy_1.txt’ for ’sigmayy_0.txt’ to obtain
’sigmaxx_2.txt’ and ’sigmayy_2.txt’
4) Continue in this manner until there are no negative
conductivity values in the output.
This converged after 4 iterations, the outputs for which are
attached. You can see that the first step (zeroth iteration)
gets us into the ball park (it’s much better than with no
threshold on the size of E). The results of the first iteration
are pretty good. The fourth iteration is very good.
This, of course, assumes that we know that there are no negative
anomalous conductivities, which will not always be the case,
so this is not a general algorithm. It also assumes that we
can throw out the small (E,J) pairs at a 20\% threshold, which
may also not be the case in general, so again it is not general
algorithm. But it may be that for any given problem a threshold
for (E,J) size and conductivity value can be found that will work.
If our threshold on conductivity is nonzero, we cannot pin the
currents to zero, but will have to rely on a constraint equation,
which is in development.