Adaptive Sampling Strategy

• Adaptive sampling strategy is a dynamic method that selects new evaluation points based on real-time analysis of previous data to enhance accuracy and convergence.
• It uses radial basis function interpolation with derivative-based selection to effectively capture localized variations and maintain global grid uniformity.
• Applied in uncertainty quantification and expensive simulations, this approach can reduce evaluations by up to 50% while achieving stringent error thresholds.

Adaptive sampling strategy refers to any method that dynamically selects new data points for evaluation or measurement based on online analysis of previously acquired information. The central objective is to improve efficiency—often measured by accuracy, convergence, or information gain per sample—when compared to uniform or predetermined sampling schemes. Adaptive sampling is broadly applied across fields such as uncertainty quantification, stochastic optimization, model order reduction, system identification, and scientific computing, where the cost per sample can be substantial and the system under investigation may have heterogeneous or localized complexity.

1. Adaptive Sampling in Uncertainty Quantification

In the context of uncertainty quantification (UQ), adaptive sampling addresses the problem of estimating probabilistic characteristics—such as the cumulative distribution function (cdf) of an output $Y = g(X)$, where $g$ is a computationally expensive nonlinear map and $X$ is a random variable with known density—using a minimal number of evaluations of $g$ (Camporeale et al., 2016). The adaptive strategy proceeds as follows:

• Initialization: Begin with a coarse set of sampling points, e.g., a sparse grid in the domain of $X$.
• Incremental Refinement: At each step, candidate midpoints between existing sample points are proposed.
• Derivative-based Selection: For each candidate, the derivative of the interpolant (constructed using all prior samples) is estimated. The candidate with the maximum or minimum derivative magnitude is selected next, alternating between these to capture regions of rapid variation and near-flatness in $g(x)$.
• Grid Uniformity Control: To avoid excessive local refinement, a ratio condition $r = \max\{d_i\} / \min\{d_i\}$ (where $d_i$ are distances between adjacent samples) is imposed, maintaining global coverage.

The interpolant $\tilde{g}(x)$ is constructed using a radial basis function (RBF) approach:

$$\tilde{g}(x) = \sum_{i=1}^N \lambda_i \Phi_i(x, c_i)$$

where $\Phi_i(x, c_i) = \sqrt{(x-x_i)^2 + c_i^2}$ and $c_i$ scales with the local grid spacing, e.g., $c_i = 0.85 \cdot \min\{d_{i-1}, d_i\}$. Interpolant derivatives:

$$\frac{d}{dx} \tilde{g}(x) = \sum_{i=1}^N \lambda_i \frac{(x-x_i)}{\sqrt{(x-x_i)^2 + c_i^2}}$$

inform subsequent sampling decisions.

This method typically yields faster convergence of the estimated cdf $C(y)$ than both Clenshaw–Curtis collocation and adaptive hierarchical surplus techniques, achieving target cdf errors (e.g., $\varepsilon_C \approx 10^{-5}$) with up to half as many expensive $g(x)$ evaluations.

2. Mathematical and Algorithmic Structure

The adaptive sampling procedure in UQ can be summarized as an iterative, information-driven refinement process characterized by the following components:

• Global Interpolant: Maintain and update a surrogate model $\tilde{g}(x)$ encompassing all collected $(x_i, g(x_i))$ pairs.
• Local Error/Derivative Metric: For each candidate new sample, compute an explicit quantity (e.g., interpolant derivative, curvature, or bias) to quantify prospective information gain.
• Selection Rule: Apply an algorithmic criterion—typically, maximize or minimize the error metric or alternate between extremes—to guide sampling location.
• Uniformity/Quality Control: Impose grid ratio or spacing constraints to prevent over-concentration, enhancing global estimation properties (not just local accuracy).
• Adaptive Interpolation Properties: For non-uniform grids, interpolate using RBF or other mesh-free methods that are robust to irregular sample locations and heterogeneity in $g(x)$.

A high-level schematic is:

Step	Action	Purpose
1	Evaluate $g(x)$ at initial points	Coarse model initialization
2	Build/update interpolant/RBF	Surrogate for $g(x)$ over domain
3	Propose candidate midpoints	Identify where to refine
4	Compute derivative at candidates	Quantify potential information gain
5	Select new $x^*$ following rule	Adaptive placement
6	Check grid uniformity (enforce $r$)	Avoid local over-refinement
7	Iterate until global target achieved	Efficient UQ

3. Comparison with Classical and Alternative Methods

Fixed-grid or non-adaptive UQ methods such as stochastic collocation (e.g., Clenshaw–Curtis) and adaptive hierarchical surplus sampling have the following limitations:

• Stochastic Collocation (CC): Employs nested, boundary-concentrated grids, well-suited for moments but unresponsive to observed $g(x)$ structure. Sampling locations are not updated based on function evaluations.
• Hierarchical Surplus: Adds points where the difference between coarse- and fine-level interpolants is large. However, it may prematurely halt in regions where accidental matches occur between the interpolant and $g(x)$, overlooking subtler features critical to the cdf estimation.

In contrast, the adaptive derivative-based approach continuously re-evaluates sampling locations in light of all available data and directly targets rapid cdf convergence.

Empirical results show that, for given cdf error thresholds, adaptive sampling can reduce the number of required $g(x)$ evaluations by factors of 2× or more relative to CC and offers greater reliability than the surplus method, particularly in the presence of local sharp features or unanticipated function variability.

4. Applications and Computational Implications

This class of adaptive sampling is particularly applicable in UQ contexts where:

• Computationally Expensive Simulations: $g(x)$ evaluation entails solving PDEs, running complex multi-physics models, or executing high-fidelity simulations.
• UQ for Reliability/Risk: Accurate tail probability and cdf estimation is critical, e.g., for rare event prediction, design margins, or risk assessment.
• Smooth but Heterogeneous Mappings: Global smoothness in $g(x)$ does not guarantee global cdf smoothness; localized adaptation is crucial.

Examples given include the Lotka–Volterra predator–prey model and the Van der Pol oscillator, where adaptive strategies target regions responsible for rapid cdf changes.

Resource and scaling considerations include the computational cost of surrogate model updates (for moderate $N$, RBF interpolation is tractable), derivative evaluations, and numerical robustness as the number of points increases. Care in basis selection (including adaptive or optimal shape parameters $c_i$ for RBFs) is recommended, especially as the interpolation matrix may become ill-conditioned.

5. Limitations and Extensions

The derivative-driven adaptive method described is currently developed for one-dimensional input spaces. Extending to higher dimensions is nontrivial due to:

• Curse of Dimensionality: The required sampling density for uniform direct sampling scales exponentially in $d$.
• Sampling in Non-rectangular Domains: The mesh-free nature of RBFs offers promise for generalization, but high-dimensional derivative estimation and candidate selection become computationally intensive.

Potential future directions include:

• Multi-dimensional Adaptive Sampling: Leveraging the mesh-free properties of RBFs or combining with quasi Monte Carlo schemes.
• Optimization of Basis Parameters: Adaptive or theoretically optimal selection of shape parameters $c_i$ to manage conditioning and accuracy, especially in the presence of steep gradients or discontinuities.
• Integration with Monte Carlo or Quasi-Monte Carlo Sampling: Using adaptive selection to guide point placement in QMC frameworks for extremely high-dimensional UQ problems.

6. Impact and Significance

Adaptive sampling strategies that learn from previous evaluations and optimize the placement of new samples are a significant advancement for efficient, accurate quantification of uncertainty in computational models. By specifically targeting rapid convergence of distributional functionals of $g(X)$, rather than just interpolant error, these methods provide a practical solution for computationally intensive UQ applications. The combination of RBF-based interpolation, derivative-based adaptivity, and explicit grid quality control decisively outperforms classical collocation-based and surplus-driven approaches in both accuracy and computational efficiency.

The ongoing development and extension of these strategies to higher-dimensional, nontrivial domains, and more complex interpolation bases remain vital research directions to further broaden their applicability.