Nonlinear regression is a useful statistical tool, relating observed data and a nonlinear function of unknown parameters. When the parameter-dependent nonlinear function is computationally intensive, a straightforward regression analysis by maximum likelihood is not feasible. The method presented in this paper proposes to construct a faster running surrogate for such a computationally intensive nonlinear function, and to use it in a related nonlinear statistical model that accounts for the uncertainty associated with this surrogate. A pivotal quantity in the Earth's climate system is the climate sensitivity: the change in global temperature due to doubling of atmospheric CO2 concentrations. This, along with other climate parameters, are estimated by applying the statistical method developed in this paper, where the computationally intensive nonlinear function is the MIT 2D climate model.

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Computer experiments are increasingly used in scientific investigations as substitutes for physical experiments in cases where the latter are difficult or impossible to perform. A computer experiment consists of several runs of a computer model or "code" for the purpose of better understanding the input → output relationship. One practical difficulty in the use of these models is that a single run may require a prohibitive amount of computational resources in some situations. A recent approach uses statistical approximations as less expensive surrogates for such computer codes; these provide both point predictors and uncertainty characterization of the outputs. A widely used class of computer codes is the finite-difference solvers of differential equations, which produce multivariate output (e.g., time series). The finitedifference relationship underpins the statistical model proposed here, and we show that this strategy has clear computational and accuracy advantages over a direct multivariate extension of the popular scalar modeling methodology.

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