  Investigating uncertainties in computer simulations can be prohibitive in
terms of computational costs, since the simulator needs to be run over a large
number of input values. Building an emulator, i.e. a statistical surrogate
model of the simulator, using a small design of experiments, greatly alleviates
the computational burden to carry out such investigations. Nevertheless, this
can still be above the computational budget for many studies. Two major
approaches have been used are to reduce the budget needed to build the
emulator: efficient design of experiments, such as sequential designs, and
combining training data of different degrees of sophistication in a so-called
multi-fidelity method, or multilevel in case these fidelities are ordered
typically for increasing resolutions. We present here a novel method that
combines both approaches, the multilevel adaptive sequential design of computer
experiments (MLASCE) in the framework of Gaussian process (GP) emulators. We
make use of reproducing kernel Hilbert spaces as a new tool for our GP
approximations of the increments across levels. This dual strategy allows us to
allocate efficiently limited computational resource over simulations of
different levels of fidelity and build the GP emulator. The allocation of
computational resources is shown to be the solution of a simple optimization
problem in a special case where we theoretically prove the validity of our
approach. Our proposed method is compared with other existing models of
multi-fidelity Gaussian process emulation. Gains of orders of magnitudes in
accuracy for medium-size computing budgets are demonstrated in numerical
examples.
