1Blurb:: Variance of mean estimator within multilevel polynomial chaos
2
3Description::
4Multilevel Monte Carlo performs optimal resource allocation based on a
5known estimator variance for the mean statistic:
6
7\f[ Var[\hat{Q}] = \frac{\sigma^2_Q}{N} \f]
8
9Replacing the simple ensemble average estimator in Monte Carlo with a
10polynomial chaos estimator results in a different and unknown
11relationship between the estimator variance and the number of samples.
12In one approach to multilevel PCE, we can employ a parameterized
13estimator variance:
14
15\f[ Var[\hat{Q}] = \frac{\sigma^2_Q}{\gamma N^\kappa} \f]
16
17for free parameters \f$\gamma\f$ and \f$\kappa\f$, with default values
18that may be overridden as part of this specification block.
19
20This approach is supported for regression-based PCE approaches
21(over-determined least squares, under-determined compressed
22sensing, or othogonal least interpolation).
23
24In practice, it can be challenging to estimate a smooth convergence
25rate for estimator variance in the presence of abrupt transitions in
26the quality of sparse recoveries.  As a result, sample allocation by
27greedy refinement is generally preferred.
28
29Topics::
30
31Examples::
32
33This example starts with sparse recovery for a second-order candidate
34expansion at each level.  As the number of samples is adapted for
35each level, as dictated by the convergence of the estimator variance,
36the candidate expansion order is incremented as needed
37in order to synchronize with the specified collocation ratio.
38
39\verbatim
40method,
41	model_pointer = 'HIERARCH'
42	multilevel_polynomial_chaos
43	  orthogonal_matching_pursuit
44	  expansion_order_sequence = 2
45	  pilot_samples = 10
46	  collocation_ratio = .9
47	  allocation_control
48	    estimator_variance estimator_rate = 2.5
49	  seed = 1237
50	  convergence_tolerance = .01
51\endverbatim
52
53Theory::
54
55Faq::
56See_Also::
57