Calibration of expensive computer models using emulators for high-dimensional output fields can become increasingly intractable with the size of the field(s) being compared to observational data. In these settings, dimension reduction is attractive, reducing the number of emulators required to mimic the field(s) by orders of magnitude. By comparing to popular independent emulation approaches that fit univariate emulators to each grid cell in the output field, we demonstrate that using a basis structure for emulation, aside from the clear computational benefits, is essential for obtaining coherent draws that can be compared with data or used in prediction. We show that calibrating on the subspace spanned by the basis is not generally equivalent to calibrating on the full field (the latter being generally infeasible owing to the large number of matrix inversions required for calibration and the size of the matrices on the full field). We then present a projection that allows accurate calibration on the field for exactly the cost of calibrating in the subspace, by projecting in the norm induced by our uncertainties in observations and model discrepancy and given a one-off inversion of a large matrix. We illustrate the benefits of our approach and compare with standard univariate approaches for emulating and calibrating the high-dimensional ice sheet model Glimmer.
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