Multi-level informed optimization via decomposed Kriging for large design problems under uncertainty

TL;DR

The paper tackles the challenge of design under uncertainty for large, resource‑intensive engineering models by introducing MLIO, a tri‑level, non‑intrusive framework that maps the full uncertainty COST function $COST(u,p)$ using an ensemble of decomposed Kriging surrogates. By organizing surrogates into symmetric, separable, and assumption‑free layers, MLIO achieves scalable accuracy and sample efficiency, outperforming the state‑of‑the‑art two‑step baseline PCE+GA on analytical benchmarks up to 200 dimensions. The approach leverages three iterative levels—Solve, Explore, and Exploit—and Bayesian-inspired acquisition to refine the uncertainty map while keeping evaluations cheap. Results show MLIO attains subpercent uncertainty metrics with roughly $10^3$ samples, offering orders‑of‑magnitude improvements in efficiency and broad applicability to robust and stochastic optimization, reliability analysis, and risk assessment; future work includes parallelization, multi‑fidelity, and gradient‑based enhancements.

Abstract

Engineering design involves demanding models encompassing many decision variables and uncontrollable parameters. In addition, unavoidable aleatoric and epistemic uncertainties can be very impactful and add further complexity. The state-of-the-art adopts two steps, uncertainty quantification and design optimization, to optimize systems under uncertainty by means of robust or stochastic metrics. However, conventional scenario-based, surrogate-assisted, and mathematical programming methods are not sufficiently scalable to be affordable and precise in large and complex cases. Here, a multi-level approach is proposed to accurately optimize resource-intensive, high-dimensional, and complex engineering problems under uncertainty with minimal resources. A non-intrusive, fast-scaling, Kriging-based surrogate is developed to map the combined design/parameter domain efficiently. Multiple surrogates are adaptively updated by hierarchical and orthogonal decomposition to leverage the fewer and most uncertainty-informed data. The proposed method is statistically compared to the state-of-the-art via an analytical testbed and is shown to be concurrently faster and more accurate by orders of magnitude.

Figures (7)

• Figure 1: Graphical representation of the uncertainty map, projected from a multi-variate energy system on parameter and design spaces
• Figure 2: Logical flow chart of MLIO approach with the three levels, solution, exploration, and exploitation, and the iterative loops of the adaptive algorithm
• Figure 3: Decomposed Kriging algorithm within the MLIO scheme and its three iteration layers.
• Figure 4: Aggregated median errors over the testbed for tuned PCE+GA (setting #2 over 6) and MLIO (setting #1 over 2) vs. 1e6 Halton set. a) shows MLIO vs. PCE+GA inaccuracy and suboptimality for robust and stochastic optimizations. b) shows MLIO vs. PCE+GA statistical error merging all results.
• Figure 5: Complexity of MLIO and PCE+GA methods for 1% accuracy as a function of $f$ dimensionality ($D$) and evaluation cost.