Blurb::
Variance of mean estimator within multilevel polynomial chaos
Description::
Multilevel Monte Carlo performs optimal resource allocation based on a
known estimator variance for the mean statistic:

.. math::  Var[\hat{Q}] = \frac{\sigma^2_Q}{N} 

Replacing the simple ensemble average estimator in Monte Carlo with a
polynomial chaos estimator results in a different and unknown
relationship between the estimator variance and the number of samples.
In one approach to multilevel PCE, we can employ a parameterized
estimator variance:

.. math::  Var[\hat{Q}] = \frac{\sigma^2_Q}{\gamma N^\kappa} 

for free parameters :math:`\gamma`  and :math:`\kappa` , with default values
that may be overridden as part of this specification block.

This approach is supported for regression-based PCE approaches
(over-determined least squares, under-determined compressed
sensing, or othogonal least interpolation).

In practice, it can be challenging to estimate a smooth convergence
rate for estimator variance in the presence of abrupt transitions in
the quality of sparse recoveries.  As a result, sample allocation by
greedy refinement is generally preferred.

Topics::

Examples::
This example starts with sparse recovery for a second-order candidate
expansion at each level.  As the number of samples is adapted for
each level, as dictated by the convergence of the estimator variance,
the candidate expansion order is incremented as needed
in order to synchronize with the specified collocation ratio.


.. code-block::

    method,
     model_pointer = 'HIERARCH'
     multilevel_polynomial_chaos
       orthogonal_matching_pursuit
       expansion_order_sequence = 2
       pilot_samples = 10
       collocation_ratio = .9
       allocation_control
         estimator_variance estimator_rate = 2.5
       seed = 1237
       convergence_tolerance = .01


Theory::

Faq::

See_Also::
