Gaussian mixture Taylor approximations of risk measures constrained by PDEs with Gaussian random field inputs

TL;DR

This paper addresses the computational challenge of estimating risk measures for PDE-driven QoIs with Gaussian random-field inputs. It introduces a Gaussian mixture Taylor framework that replaces the input Gaussian measure with a mixture of lower-variance components and performs Taylor expansions about each component’s mean, enabling accurate mean, variance, and CVaR estimates with far fewer PDE solves than Monte Carlo. The authors derive analytical expressions and bounds for mixture-based moments and CVaR, develop multiple directions (KL and Hessian-informed) to construct the mixture, and provide error analyses that separate mixture- and Taylor-induced errors. Numerical experiments on an ADR PDE with log-normal conductivity and a Helmholtz scattering problem demonstrate substantial computational savings and sub-percent CVaR errors with tens of PDE solves, highlighting the practical impact for risk-informed PDE analysis under limited budgets.

Abstract

This work considers the computation of risk measures for quantities of interest governed by PDEs with Gaussian random field parameters using Taylor approximations. While efficient, Taylor approximations are local to the point of expansion, and hence may degrade in accuracy when the variances of the input parameters are large. To address this challenge, we approximate the underlying Gaussian measure by a mixture of Gaussians with reduced variance in a dominant direction of parameter space. Taylor approximations are constructed at the means of each Gaussian mixture component, which are then combined to approximate the risk measures. The formulation is presented in the setting of infinite-dimensional Gaussian random parameters for risk measures including the mean, variance, and conditional value-at-risk. We also provide detailed analysis of the approximations errors arising from two sources: the Gaussian mixture approximation and the Taylor approximations. Numerical experiments are conducted for a semilinear advection-diffusion-reaction equation with a random diffusion coefficient field and for the Helmholtz equation with a random wave speed field. For these examples, the proposed approximation strategy can achieve less than $1\%$ relative error in estimating CVaR with only $\mathcal{O}(10)$ state PDE solves, which is comparable to a standard Monte Carlo estimate with $\mathcal{O}(10^4)$ samples, thus achieving significant reduction in computational cost. The proposed method can therefore serve as a way to rapidly and accurately estimate risk measures under limited computational budgets.

Figures (13)

• Figure 1: Probability density functions of Gaussian mixture approximations of the standard normal distribution using $N_{\mathrm{mix}} = 3, 5, 7$ mixture components obtained from the univariate splitting library of VittaldevRussell16. The mixture components have standard deviations $\sigma_i = N_{\mathrm{mix}}^{-p}$ with $p = 1/2$.
• Figure 1: Left: diagram of the PDE domain and set. Center: a sample of the parameter $m \sim \mathcal{N}(\bar{m}, \mathcal{C})$. Right: corresponding sample PDE solution $u = u(m)$. Samples are obtained for correlation length and pointwise variances equal to 1.
• Figure 2: Gaussian mixture approximations of a 2D Gaussian distribution by 1D mixture decompositions along the dominant KLE direction, an arbitrary direction, and recursively by further decomposing a mixture component along a second direction. Top row shows the PDFs of the individual mixture components, where the opacity corresponds to the component weight and the dashed lines are the decomposition directions.. The bottom row shows the total mixture PDFs.
• Figure 2: Magnitudes of the first 100 eigenvalues (as a ratio to the largest eigenvalue) for the covariance operator (left) and the covariance-preconditioned Hessian of the QoIs at $\bar{m}$ (left to right: $L^2$, $L^3$, and energy). Spectra are shown for correlation lengths of $0.1, 0.25, 0.5, 1, 2$ and a pointwise variance of $1$.
• Figure 3: Dominant eigenvector of the covariance (left) and covariance-preconditioned Hessian at $\bar{m}$ for the $L^2$ QoI (right) with correlation length and pointwise variance both equal to one.

Theorems & Definitions (16)

• Proposition 5.1
• Proposition 5.2: Dimension-independent mixture approximation error
• Proof 1
• Lemma 5.3
• Proof 2
• Lemma 5.4
• Proof 3
• Proposition 5.5
• Proof 4
• Proposition 5.6