  We develop adaptive replicated designs for Gaussian process metamodels of
stochastic experiments. Adaptive batching is a natural extension of sequential
design heuristics with the benefit of replication growing as response features
are learned, inputs concentrate, and the metamodeling overhead rises. Motivated
by the problem of learning the level set of the mean simulator response we
develop four novel schemes: Multi-Level Batching (MLB), Ratchet Batching (RB),
Adaptive Batched Stepwise Uncertainty Reduction (ABSUR), Adaptive Design with
Stepwise Allocation (ADSA) and Deterministic Design with Stepwise Allocation
(DDSA). Our algorithms simultaneously (MLB, RB and ABSUR) or sequentially (ADSA
and DDSA) determine the sequential design inputs and the respective number of
replicates. Illustrations using synthetic examples and an application in
quantitative finance (Bermudan option pricing via Regression Monte Carlo) show
that adaptive batching brings significant computational speed-ups with minimal
loss of modeling fidelity.
