Adaptive Numerical Designs for the Calibration of Computer Codes

Abstract

Making good predictions of a physical system using a computer code requires the inputs to be carefully specified. Some of these inputs, called control variables, reproduce physical conditions whereas other inputs, called parameters, are specific to the computer code and most often uncertain. The goal of statistical calibration consists in reducing their uncertainty with the help of a statistical model which links the code outputs with the field measurements. In a Bayesian setting, the posterior distribution of these parameters is typically sampled using MCMC methods. However, they are impractical when the code runs are highly timeconsuming. A way to circumvent this issue consists of replacing the computer code with a Gaussian process emulator, then sampling a surrogate posterior distribution based on it. Doing so, calibration is subject to an error which strongly depends on the numerical design of experiments used to fit the emulator. Under the assumption that there is no code discrepancy, we aim to reduce this error by constructing a sequential design by means of the Expected Improvement criterion. Numerical illustrations in several dimensions assess the efficiency of such sequential strategies.

Iterative method for tuning complex simulation code

“…Some authors, including Cailliez, Bourasseau, and Pernot (2014), and Beck and Guillas (2016), explored this topic. We agree that a sequential tuning approach is practically useful, as in Pratola et al (2013), Kumar (2015), and Damblin et al (2018). Further research on relevant designs under the sequential tuning approach will be helpful.…”

“…We refer to this method as the full MLE. It can be done for each GP model defined in (6). The −2 times concentrated log likelihood function (except for constants) of all parameters for the combined data withβ B andσ 2 B plugged in is…”

“…It is thus computationally infeasible to run the code as many times as needed for an iterative nonlinear optimizer to find τ . The ANLS method first fits the GP Model 1 or Model 2 as defined in equation (6), with MLE using computer data alone. Then, upon treating the fitted prediction model as if it were the true model, the method attempts to determine the τ that minimizes the residual sum of squares with predictors:…”

“…This makes the problem computationally feasible. It is applied for each GP model (Model 1 and Model 2) stated in (6). The advantages of this method are that it is reasonably economical and easy to implement, and the computer and experimental data are uncoupled.…”

“…Step 2 (initial solution): set iteration i = 1, and find τ by minimizing RSS p (τ ) in (11) using surrogate (6).…”