Uniform Maximum Projection Designs for Computer Experiments

1. Problem Statement

Computer experiments often rely on space-filling designs to explore high-dimensional input spaces with a limited number of expensive model evaluations. While Latin Hypercube Sampling (LHS) ensures uniformity in one-dimensional projections and distance-based criteria (like Maximin) ensure global space-filling, neither guarantees uniformity in low-dimensional subspaces.

To address subspace uniformity, Maximum Projection (MaxPro) designs were introduced. However, the authors identify a critical flaw in the standard MaxPro formulation:

• Systematic Boundary Bias: The standard MaxPro criterion relies on the Euclidean distance metric. This leads to statistical non-uniformity in the design-generation mechanism. Specifically, it systematically undersamples corner regions of the bounded domain and oversamples regions near the boundaries.
• Consequences: This bias results in:
  • Biased Monte Carlo integration estimates.
  • Poor performance when the active input variables are unknown (i.e., when the design is projected onto lower-dimensional subspaces).
  • Inability to provide unbiased uncertainty quantification without complex, often impractical, importance-sampling corrections.

2. Methodology

The authors propose a modification to the MaxPro criterion to eliminate boundary bias while preserving its superior projection properties.

The Proposed Solution: uMaxPro

The paper introduces the Uniform Maximum Projection (uMaxPro) criterion. The core innovation is replacing the standard Euclidean distance with a periodic distance metric based on the minimum image convention.

• Metric Modification: Instead of calculating the direct distance $\Delta_{ij,v} = |x_{i,v} - x_{j,v}|$, the periodic distance is defined as:
  $$\Delta_{ij,v} = \min(\Delta_{ij,v}, 1 - \Delta_{ij,v})$$
  This effectively treats the design domain $[0,1]^{N_{var}}$ as a torus (periodic boundary conditions), where the distance between points wraps around the boundaries.
• The Criterion: The uMaxPro objective function minimizes the inverse product of these periodic distances across all dimensions, similar to the original MaxPro but using the periodic metric.
• Statistical Uniformity: Because the periodic metric is translation-invariant, the resulting optimization process does not favor any specific region (corners or boundaries). Repeated independent realizations of the design generation mechanism yield a statistically uniform distribution where every subregion is visited with probability proportional to its volume.
• Algorithm: The designs are constructed using Simulated Annealing (SA) within the class of Latin Hypercube Samples (LHS) to preserve marginal stratification. The implementation is provided as an open-source Python package.

3. Key Contributions

1. Identification of Bias: The paper rigorously demonstrates that standard MaxPro designs are statistically non-uniform, leading to systematic undersampling of corners and biased integration results.
2. Novel Criterion (uMaxPro): Introduction of a periodic variant of MaxPro that achieves dual uniformity:
   • Geometric Uniformity: Excellent space-filling and projection properties in every deterministic realization.
   • Statistical Uniformity: An unbiased design-generation mechanism where point density is uniform across the domain over multiple realizations.
3. Theoretical Justification: Proof that the periodic metric ensures translation invariance, removing the boundary effects inherent in Euclidean-based space-filling designs.
4. Open Source Implementation: Provision of the uMaxPro Python package and reproducible optimized designs to facilitate adoption.

4. Results and Numerical Studies

The authors validate uMaxPro through extensive numerical experiments comparing it against standard MaxPro, LHS, Halton sequences, and random sampling.

• Statistical Uniformity (Histograms):
  • Standard MaxPro histograms show significant dips in frequency near domain corners (undersampling).
  • uMaxPro histograms are flat (value $\approx$ 1.0), confirming perfect statistical uniformity across all dimensions and subspaces.
• Discrepancy:
  • uMaxPro achieves Wrap-around Discrepancy (WD2) values comparable to the Halton sequence (a gold standard for low-discrepancy sequences), maintaining $O(1/N_{sim})$ convergence rates in 2D.
• Monte Carlo Integration:
  • Bias: Standard MaxPro yields biased estimates for functions with strong variable interactions. uMaxPro provides unbiased estimates.
  • Variance: uMaxPro significantly reduces the Root Mean Square Error (RMSE) compared to MaxPro and often outperforms the Halton sequence, particularly in higher dimensions.
• Subspace Projection:
  • When projecting 5D designs onto 2D, 3D, or 4D subspaces, standard MaxPro performance degrades due to accumulated boundary bias.
  • uMaxPro maintains superior performance in all subspaces, effectively outperforming Halton sequences in low-sample regimes.
• Engineering Applications:
  • Short Column & Cantilever Beam: In surrogate modeling (Gaussian Process regression) for structural mechanics problems, uMaxPro designs yielded lower prediction errors (RMSE) than MaxPro and LHS, especially when the number of active variables was fewer than the design dimension.
  • Meso-scale Concrete FEM: In a complex finite element model involving correlated material properties, uMaxPro provided more accurate estimation of stress quantiles with fewer samples, offering a better precision-to-computational-cost ratio.

5. Significance

The paper addresses a fundamental gap in the design of computer experiments: the trade-off between projection quality and statistical unbiasedness.

• Robustness for Unknown Inputs: In real-world engineering, the "active" variables are often unknown. uMaxPro ensures that even if the problem reduces to a lower-dimensional subspace, the sampling remains uniform and unbiased.
• Unbiased Integration: For uncertainty propagation and reliability analysis, unbiasedness is a non-negotiable requirement. uMaxPro enables this without requiring complex post-hoc corrections.
• Practical Efficiency: The method offers a computationally efficient modification to existing MaxPro codes, making it immediately applicable for improving surrogate modeling and probabilistic estimation in engineering contexts.

In conclusion, uMaxPro represents a significant advancement over standard space-filling designs, offering a robust, unbiased, and highly efficient solution for high-dimensional computer experiments where subspace uniformity is critical.