Known Boundary Emulation of Complex Computer Models

Abstract

Computer models are now widely used across a range of scientific disciplines to describe various complex physical systems, however to perform full uncertainty quantification we often need to employ emulators. An emulator is a fast statistical construct that mimics the complex computer model, and greatly aids the vastly more computationally intensive uncertainty quantification calculations that a serious scientific analysis often requires. In some cases, the complex model can be solved far more efficiently for certain parameter settings, leading to boundaries or hyperplanes in the input parameter space where the model is essentially known. We show that for a large class of Gaussian process style emulators, multiple boundaries can be formally incorporated into the emulation process, by Bayesian updating of the emulators with respect to the boundaries, for trivial computational cost. The resulting updated emulator equations are given analytically. This leads to emulators that possess increased accuracy across large portions of the input parameter space. We also describe how a user can incorporate such boundaries within standard black box GP emulation packages that are currently available, without altering the core code. Appropriate designs of model runs in the presence of known boundaries are then analysed, with two kinds of general purpose designs proposed. We then apply the improved emulation and design methodology to an important systems biology model of hormonal crosstalk in Arabidopsis Thaliana.

Efficient Emulation of Computer Models Utilising Multiple Known Boundaries of Differing Dimensions

“…, y (m) , which can be analysed using the standard Bayes linear update. Having said this, the calculations are structured so that they can be generalised to continuous model evaluations on K [39]. Denote the corresponding length m vector of model evaluations K. Plugging these m runs into the Bayes Linear update equations (3), ( 4) and ( 5) by replacing D with K may be infeasible due to the size of the m × m matrix inversion Var[K] −1 (m may need to be extremely large to capture all the information available from K).…”

“…A direct update of the emulator is therefore non-trivial, hence we show from first principles that this update can be performed analytically for a wide class of emulators. This is done by exploiting a sufficiency argument briefly described in the supplementary material of [23], and in [31], though only utilised for the first time in the context of known boundary emulation in [39]. The emulation problem is further compounded when we have both a set of evaluations K on the boundary, and a set of evaluations D in the bulk of the input space.…”

“…Emulation has been successfully applied across a variety of scientific disciplines, such as astrophysics [19,36,22], climate science [4,40,5], engineering [11] and volcanology [2,16]. Vernon et al [39] describe an advance in emulation strategy that can lead to substantial improvements in emulator performance when applicable. The strategy discussed exploits the fact that, for some simulators, there exist input parameter settings where the simulator can be solved far more efficiently (whether this be analytically or just significantly faster using a simpler numerical solver), for example, by allowing various modules to decouple from more complex parts of the model.…”

“…The remainder of this article is organised as follows. In Section 2, we review and extend the work of [39] to the case of a single known boundary of any dimension (as opposed to p−1, where p is the number of input components to the computer model). Section 3 extends the theory to multiple boundaries of various dimensions.…”

“…As discussed in [39], since the results rely on the product correlation structure of the emulator, expansion of these methods to more general emulator forms, such as [8,9,36,37]:…”