.. _method-multilevel_polynomial_chaos-allocation_control-estimator_variance-estimator_rate:

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estimator_rate
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Rate of convergence of mean estimator within multilevel polynomial chaos


.. toctree::
   :hidden:
   :maxdepth: 1



**Specification**

- *Alias:* None

- *Arguments:* REAL

- *Default:* 2


**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` .

The default values are :math:`\gamma = 1`  and :math:`\kappa = 2`  (adopts a
more aggressive sample profile by assuming a faster convergence rate
than Monte Carlo).  This advanced specification option allows to user
to specify :math:`\kappa` , overriding the default.



