Blurb::
Gaussian Process Adaptive Importance Sampling
Description::
``gpais`` is recommended for problems that have a relatively small
number of input variables (e.g. less than 10-20). This method, Gaussian
Process Adaptive Importance Sampling,
is outlined in the paper :cite:p:`Dalbey2014`.

This method starts with an initial set of LHS samples and adds samples
one at a time, with the goal of adaptively improving the estimate of
the ideal importance density during the process. The approach uses a
mixture of component densities. An iterative process is used
to construct the sequence of improving component densities. At each
iteration, a Gaussian process (GP) surrogate is used to help identify areas
in the space where failure is likely to occur. The GPs are not used to
directly calculate the failure probability; they are only used to approximate
the importance density. Thus, the Gaussian process adaptive importance
sampling algorithm overcomes limitations involving using a potentially
inaccurate surrogate model directly in importance sampling calculations.
Topics::
uncertainty_quantification
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Theory::

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