Sequential Design with Mutual Information for Computer Experiments (MICE): Emulation of a Tsunami Model
Joakim Beck, Serge Guillas
SIAM/ASA Journal on Uncertainty Quantification
Computer simulators can be computationally intensive to run over a large number of input values, as required for optimization and various uncertainty quantification tasks. The standard paradigm for the design and analysis of computer experiments is to employ Gaussian random fields to model computer simulators. Gaussian process models are trained on input-output data obtained from simulation runs at various input values. Following this approach, we propose a sequential design algorithm MICE
... al information for computer experiments) that adaptively selects the input values at which to run the computer simulator in order to maximize the expected information gain (mutual information) over the input space. The superior computational efficiency of the MICE algorithm compared to other algorithms is demonstrated by test functions and by a tsunami simulator with overall gains of up to 20% in that case. Introduction. Computer experiments are widely employed to study physical processes [31, 36] and involve running a computer simulator which mimics the physical process at various input values. When the computer simulator is computationally expensive to run, say, minutes, hours, or even days, often on a high performance cluster, only a limited number of simulation runs can be afforded, making the planning of such experiments even more important. Surrogate models, also known as emulators, are often used as means for designing and analyzing computer experiments  . Emulators are statistical models that have been used to approximate the input-output behavior of computer simulators for making probabilistic predictions. In this setting, we want to find a design of computer experiments that with minimal computational effort leads to a surrogate model with a good overall fit. We restrict our attention to deterministic computer simulators with a scalar output. In design of experiments it is customary to use space-filling designs  such as uniform designs, multilayer designs, maximin (Mm)-and minimax (mM)-distance designs, and Latin hypercube designs (LHD). Space-filling designs treat all regions of the design space as equally important, but are "one shot" designs that may waste computations over some unnecessary regions of the input space. A variety of adaptive designs have been proposed which can take advantage of information collected during the experimental design process [21, 31] , typically in the form of input-output * Downloaded 06/28/16 to 22.214.171.124. Redistribution subject to CCBY license 740 JOAKIM BECK AND SERGE GUILLAS data from simulation runs. An algorithm is called adaptive if it updates its behavior to new data. Some classical adaptive design criteria are the maximum mean squared prediction error (MMSPE), the integrated MSPE (IMSPE), and the entropy criterion (see, e.g.,  ). We adopt the design and analysis of computer experiments (DACE) framework proposed in the seminal paper of Sacks et al.  , within which the computer simulator output is modeled as a realization of a random field, typically assumed Gaussian. When given a set of input-output data, the best linear unbiased predictor (BLUP) and the associated MSPE for the random field can be expressed in closed forms [30, 31] . Moreover, when the random field is Gaussian, the resulting BLUP is a so-called Gaussian process (GP) emulator. GP emulators are routinely applied to handle computationally intensive computer simulators in the fields of simulation , global optimization  , and uncertainty quantification [3, 32], among others. Applications include CFD simulation of a rocket booster  and climate simulation  . By using the GP approach, a range of statistical design criteria can be estimated directly [5, 34]. Finding an optimal design is usually computationally very intensive, except for relatively small designs. A way to circumvent this issue is to consider sequential designs [5, 13, 21] . In a sequential design, points are systematically chosen, often one at a time. Sequential designs are generally not optimal, but often very effective in practice. Two popular sequential designs are active learning MacKay (ALM), and active learning Cohn (ALC). ALC tends to have better overall predictive performance but involves a higher computational cost  . In this work we propose a new sequential algorithm, called MICE (mutual information for computer experiments), which is based on the information theoretic mutual information measure given in  , where the objective of maximizing the information that a design provides about the other input values, as suggested by Caselton and Zidek . Mutual information is a measure of the information contained in one random variable about another. Krause, Singh, and Guestrin  proposed a sequential MI algorithm for sensor placement, which sequentially maximizes the mutual information between a GP over the chosen sensor locations and another GP over the locations which have not yet been selected. The MICE criterion is a modified version of the MI criterion in  , where an extra parameter is introduced to improve robustness. This modification is critical when high-dimensional spaces are considered. We demonstrate by numerical examples that MICE balances well prediction accuracy and computational complexity. We are particularly interested in deterministic computer simulation experiments with more than just a few input variables. This paper is organized as follows. Section 2 reviews GP modeling for prediction and presents some popular sequential design algorithms within the DACE framework. In section 2.2, an MI-based design criterion is proposed for computer experiments. The MI algorithm is described in section 3.1, and a practical limitation is shown in section 3.1.2. Section 3.2 presents the MICE algorithm and some theoretical results. Section 4 details the computational costs associated with the different sequential design algorithms. A numerical comparison of MICE with other methods is provided for a few standard test functions, in lieu of computer simulators, in section 5, and for a tsunami simulator that solves nonlinear shallow water equations in section 6. Critically, we examine accuracy versus computational cost, as some algorithms can be quite time consuming. Section 7 summarizes our conclusions. Proofs of the theorems are given in Appendix A.