Miroslav Vořechovský

Miroslav Vořechovský, Jan Mašek

Space-filling experimental designs are widely used in engineering computer experiments, where only a limited number of expensive model evaluations can be afforded. Distance-based designs such as Maximin or Minimax ensure global space-filling, while Latin hypercube sampling enforces uniform one-dimensional projections, yet neither guarantees uniformity in lowdimensional subspaces. Maximum Projection (MaxPro) designs were introduced to improve uniformity in low-dimensional subspaces, yet their original formulation relies on the Euclidean distance and may induce systematic density distortions in bounded domains. We demonstrate that the standard MaxPro criterion leads to statistically non-uniform sampling, resulting in undersampling of corner regions and biased Monte Carlo estimates. To remedy this issue, we introduce a periodic variant of the criterion, termed Uniform Maximum Projection (uMaxPro), in which the Euclidean metric is replaced by a periodic distance based on the minimum image convention. The proposed uMaxPro designs preserve the projection-aware structure of MaxPro while achieving statistical uniformity of the design-generation mechanism. Numerical experiments show unbiased Monte Carlo integration with reduced variance, excellent subspace projection performance, and competitive discrepancy properties. The methodology is further validated on benchmark engineering problems, including a meso-scale finite element model of concrete, demonstrating improved accuracy in surrogate modeling and probabilistic estimation. The resulting criterion provides a simple and computationally efficient modification of MaxPro that enhances its robustness for nonadaptive computer experiments. The construction algorithm, open-source implementation, and reproducible optimized designs are provided to facilitate practical adoption of the method.

Lukáš Novák, Michael D. Shields, Václav Sadílek et al.

The paper presents a novel methodology to build surrogate models of complicated functions by an active learning-based sequential decomposition of the input random space and construction of localized polynomial chaos expansions, referred to as domain adaptive localized polynomial chaos expansion (DAL-PCE). The approach utilizes sequential decomposition of the input random space into smaller sub-domains approximated by low-order polynomial expansions. This allows approximation of functions with strong nonlinearties, discontinuities, and/or singularities. Decomposition of the input random space and local approximations alleviates the Gibbs phenomenon for these types of problems and confines error to a very small vicinity near the non-linearity. The global behavior of the surrogate model is therefore significantly better than existing methods as shown in numerical examples. The whole process is driven by an active learning routine that uses the recently proposed $Θ$ criterion to assess local variance contributions. The proposed approach balances both \emph{exploitation} of the surrogate model and \emph{exploration} of the input random space and thus leads to efficient and accurate approximation of the original mathematical model. The numerical results show the superiority of the DAL-PCE in comparison to (i) a single global polynomial chaos expansion and (ii) the recently proposed stochastic spectral embedding (SSE) method developed as an accurate surrogate model and which is based on a similar domain decomposition process. This method represents general framework upon which further extensions and refinements can be based, and which can be combined with any technique for non-intrusive polynomial chaos expansion construction.

Miroslav Vořechovský

The paper presents a new efficient and robust method for rare event probability estimation for computational models of an engineering product or a process returning categorical information only, for example, either success or failure. For such models, most of the methods designed for the estimation of failure probability, which use the numerical value of the outcome to compute gradients or to estimate the proximity to the failure surface, cannot be applied. Even if the performance function provides more than just binary output, the state of the system may be a non-smooth or even a discontinuous function defined in the domain of continuous input variables. In these cases, the classical gradient-based methods usually fail. We propose a simple yet efficient algorithm, which performs a sequential adaptive selection of points from the input domain of random variables to extend and refine a simple distance-based surrogate model. Two different tasks can be accomplished at any stage of sequential sampling: (i) estimation of the failure probability, and (ii) selection of the best possible candidate for the subsequent model evaluation if further improvement is necessary. The proposed criterion for selecting the next point for model evaluation maximizes the expected probability classified by using the candidate. Therefore, the perfect balance between global exploration and local exploitation is maintained automatically. The method can estimate the probabilities of multiple failure types. Moreover, when the numerical value of model evaluation can be used to build a smooth surrogate, the algorithm can accommodate this information to increase the accuracy of the estimated probabilities. Lastly, we define a new simple yet general geometrical measure of the global sensitivity of the rare-event probability to individual variables, which is obtained as a by-product of the proposed algorithm.