KL-Cov Technique in Uncertainty Quantification

• KL-Cov is a technique utilizing conditional Karhunen-Loève expansions to represent spatially varying random fields and reduce uncertainty in SPDEs.
• Its methodology integrates Gaussian process conditioning, eigenfunction truncation, and active learning for adaptive experimental design with sharper variance reduction.
• Compared to alternative conditioning strategies, KL-Cov offers improved computational efficiency and robust moment approximations, enhancing data-driven uncertainty quantification.

The KL-Cov technique, in the context of uncertainty quantification for stochastic partial differential equations (SPDEs), refers to the use of conditional Karhunen-Loève (KL) expansions to represent and actively reduce uncertainty in random spatial coefficients, leveraging observed data and targeted experimental design. The approach enables dimension reduction and efficient moment computation for solutions of PDEs involving partially-known, stochastic input fields. KL-Cov operates at the intersection of Gaussian process modeling, eigenfunction expansions, conditioning on observations, and active learning in computational science.

1. Mathematical Foundation: Conditional KL Expansion

The core of KL-Cov lies in representing a spatially varying coefficient $Y(x)$ as a Gaussian process, typically parameterized by its mean and covariance kernel. The unconditional KL expansion of a zero-mean process $g(x)$ (e.g., $g(x) = \log k(x)$ for log-normal coefficients) is given by: $$g(x, \omega) = \sum_{i=1}^{\infty} \sqrt{\lambda_i}\,\phi_i(x)\,\xi_i(\omega)$$ where $\lambda_i$ and $\phi_i(x)$ are, respectively, the eigenvalues and eigenfunctions of the covariance operator, and the $\xi_i$ are independent standard Gaussian variables.

Upon observing noisy or noiseless measurements $y$ of $Y(x)$ at locations $X$, the process is conditioned using Gaussian process regression, yielding a conditional mean $\mu^c_g(x)$ and a conditional covariance $C^c_g(x, x')$: $$\mu^c_g(x) = C_g(x, X)[C_s + \sigma_\varepsilon^2I]^{-1} y$$

$$C^c_g(x, x') = C_g(x, x') - C_g(x, X)[C_s + \sigma_\varepsilon^2I]^{-1} C_g(X, x')$$

The conditional field can be expressed by its own KL expansion: $$g^c(x, \omega) = \mu^c_g(x) + \sum_{i=1}^{d^c} \sqrt{\lambda^c_i}\,\phi^c_i(x)\,\xi_i$$ where $\{\lambda^c_i, \phi^c_i(x)\}$ are obtained from the eigenproblem for $C^c_g$.

This formulation directly embeds observation data into the stochastic representation, ensuring exact uncertainty collapse to zero at the measurement points, and enables dimension reduction by selecting only significant modes.

2. Conditioning and Truncation Strategies

Two non-commutative approaches are distinguished for incorporating measurements into the KL expansion:

• Approach 1 (“Condition then Truncate”) conditions the infinite-dimensional GP to obtain $C_g^c$, then computes and truncates its KL expansion. This propagation of conditioning through the full space before truncation ensures maximal fidelity to the observed data and sharp variance reduction at observed locations.
• Approach 2 (“Truncate then Condition”) first truncates the unconditional expansion to $d$ modes, then conditions the finite set of $\xi = \{\xi_1,\ldots,\xi_d\}$. Conditioning modifies the joint law of these variables and, given $N_s$ measurements, reduces the effective dimension: the number of independent conditional random variables becomes $d-N_s$.

Empirical results demonstrate that for a fixed random dimension, Approach 1 outperforms Approach 2 in approximating both the conditional coefficient and the solution of the SPDE. Approach 2, however, remains valuable for approximate rank estimation and computational efficiency, providing a lower bound on necessary KL terms after conditioning.

3. Active Learning for Uncertainty Reduction

Beyond static use of measurements, KL-Cov facilitates active selection of further observation locations to reduce uncertainty via two acquisition criteria:

• Method 1 (Classical, Parameter-Driven): Selects the new measurement location $x^*$ to minimize the integrated conditional variance of $g$,

$$x^* = \operatorname{argmin}_{x'} \int_D C_g^c(x, x\,|\,\{X, x'\})\,dx$$

targeting maximal decrease in uncertainty in the parameter field itself.

• Method 2 (Forward, Solution-Driven): Selects $x^*$ to minimize the integrated conditional variance of the solution $u$,

$$x^* = \operatorname{argmin}_{x'} \int_D \left\{C_u^c(x, x\,|\,X) - C_{ug}^c(x, x')[C_g^c(x', x')]^{-1}C_{gu}^c(x', x)\right\} dx$$

where $C_u^c$ is the conditional covariance of the solution, and $C_{ug}^c$ is the cross-covariance between solution and coefficient. This criterion can require joint GP regression for both $u$ and $g$.

Empirically, Method 2 induces more rapid and targeted variance reduction in the SPDE solution compared to Method 1, particularly in cases with high parameter variability.

4. Dimension Reduction and Computational Advantages

Conditioning on even a small number of measurements reduces the effective random dimension of the KL representation, leading to a more parsimonious, computationally tractable model. This reduction manifests as a lower number of nonzero eigenvalues $\lambda^c_i$ after conditioning, with the magnitude and number of retained modes directly reflecting the information content of the observations.

The collapse of uncertainty at observed points and its propagation to nearby regions under the covariance kernel ensures that computational resources prioritize the directions and zones of genuine uncertainty, suppressing spurious fluctuations and supporting scalable UQ and adaptive experimental design.

5. Comparative Impact of Conditioning and Active Learning

Comparative analysis of conditioning strategies confirms:

• Superiority of Conditioning Then Truncation: For fixed random dimension, Approach 1 provides sharper conditional fields and solution statistics, as variance is fully “annealed” around measurement locations and propagated globally.
• Dimension Suitability of Truncate Then Condition: Approach 2, though potentially less accurate, is computationally lighter and accurately forecasts the effective KL rank needed post-conditioning.

Regarding active learning:

• Solution-Variance-Based Selection (Method 2) is more effective for UQ in quantities of interest (such as SPDE states) than parameter-variance-based criteria (Method 1), as demonstrated by greater reductions in output variance at each iteration. This improvement becomes increasingly marked with growing coefficient uncertainty.

A plausible implication is that for high-dimensional or computationally intensive UQ workflows, iterative application of Approach 2 can guide rank selection, while combining Approach 1 and Method 2 delivers the greatest predictive improvement per new measurement.

6. Broader Implications for UQ in SPDEs

The findings associated with KL-Cov extend the state-of-the-art in UQ for SPDEs by:

• Enabling rigorous and scalable reduction in both epistemic and parametric uncertainty through measurement integration;
• Providing a systematic framework for joint dimension reduction and adaptive sampling;
• Clarifying the tradeoff between accuracy and computational efficiency under different conditioning and truncation regimes;
• Laying the groundwork for advanced UQ strategies focusing on state quantities of interest, not just input fields.

This suggests that KL-Cov techniques are particularly well-suited for large-scale, data-sensitive, and high-fidelity modeling scenarios—such as digital twins, subsurface flows, and material property characterization—where measurement-informed reduced-order models are essential.