Root Finding and Metamodeling for Rapid and Robust Computer Model Calibration

Abstract

We concern computer model calibration problem where the goal is to find the parameters that minimize the discrepancy between the multivariate real-world and computer model outputs. We propose to solve an approximation using signed residuals that enables a root finding approach and an accelerated search. We characterize the distance of the solutions to the approximation from the solutions of the original problem for the strongly-convex objective functions, showing that it depends on variability of the signed residuals across output dimensions, as wells as their variance and covariance. We develop a metamodel-based root finding framework under kriging and stochastic kriging that is augmented with a sequential search space reduction. We derive three new acquisition functions for finding roots of the approximate problem along with their derivatives usable by first-order solvers. Compared to kriging, stochastic kriging accounts for observational noise, promoting more robust solutions. We also analyze the case where a root may not exist. Our analysis of the asymptotic behavior in this context show that, since existence of roots in the approximation problem may not be known a priori, using new acquisition functions will not compromise the outcome. Numerical experiments on data-driven and physics-based examples demonstrate significant computational gains over standard calibration approaches.

Figures (15)

• Figure 1: Left: a metamodel fitted to the signed discrepancy $\tilde{f}_n(\theta)$, where opposite signs at $\theta_a$ and $\theta_b$ guarantees the existence of a root within $[\theta_a, \theta_b]$ by continuity. Right: a metamodel fitted to the squared discrepancy $f_n(\theta)$, where additional evaluations may be needed to locate lower estimates within $[\theta_a, \theta_b]$.
• Figure 2: Probability of improvement under the root finding (left) and minimization framework (right).
• Figure 3: Computed hypervolume from three evaluated configurations in a two-dimensional input space ($m_\theta = 2$), among which two pairs exhibit opposite signs. These pairs define two candidate subregions, and the subregion with the smallest hypervolume is highlighted in green.
• Figure 4: RSS in a stochastic setting in a two-dimensional input space ($m_\theta = 2$). Under the probabilistic framework, the hypervolume accounts for sampling variability, resulting in a different selected subregion compared to \ref{['siam:fig:rss']}.
• Figure 5: Calibration results of the 2D synthetic example. Objective function value at each best observed solution is obtained from 1,000 post replications and reported with 95% confidence intervals.

Theorems & Definitions (23)

• Definition 1: Spatial Variability
• Theorem 1: Lebesgue's Dominated Convergence Theorem
• Corollary 2: Strong Convexity of the Calibration Objective
• Proposition 3: Uniform Law of Large Numbers
• Theorem 4: Convergence Rate of Sample Average Approximation
• Corollary 5: Sample Average Convergence Rates in Expectation
• proof
• Theorem 6: Solution Accuracy Bound
• Corollary 7: Convergence Rate of Solution Accuracy
• proof