Smoothed Circulant Embedding with Applications to Multilevel Monte Carlo Methods for PDEs with Random Coefficients

TL;DR

This work tackles efficient uncertainty quantification for PDEs with random coefficients by integrating circulant embedding (CE) sampling of Gaussian random fields with smoothing (CES) into Multilevel Monte Carlo (MLMC). By discarding high-frequency components on coarse MLMC levels and using smooth periodisation to maintain positive definiteness, the authors show that the coarsest level can be chosen independently of the covariance length, reducing the overall computational cost by substantial factors (reported as 5–10x in experiments). They derive error bounds for the smoothing procedure, establish level coupling, and provide MLMC complexity results that mirror standard theory while featuring improved constants. Numerical experiments for groundwater-flow-type PDEs with exponential and Matérn covariances demonstrate major practical gains, especially for small correlation lengths, and compare CES to KL-based approaches, highlighting FFT-based efficiency advantages. The methods offer a flexible, general framework for fast sampling of highly oscillatory random fields in MLMC and related uncertainty quantification tasks.

Abstract

We consider the computational efficiency of Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods applied to partial differential equations with random coefficients. These arise, for example, in groundwater flow modelling, where a commonly used model for the unknown parameter is a random field. We make use of the circulant embedding procedure for sampling from the aforementioned coefficient. To improve the computational complexity of the MLMC estimator in the case of highly oscillatory random fields, we devise and implement a smoothing technique integrated into the circulant embedding method. This allows to choose the coarsest mesh on the first level of MLMC independently of the correlation length of the covariance function of the random field, leading to considerable savings in computational cost. We illustrate this with numerical experiments, where we see a saving of factor 5-10 in computational cost for accuracies of practical interest.

Figures (11)

• Figure 1: Plots of Gaussian field samples $\mathbf{Z}(\mathbf{x}, \omega)$ and $\tilde{\mathbf{Z}}(\mathbf{x}, \omega)$ obtained using circulant embedding for the exponential covariance \ref{['eq: p-norm-cov-function']} with $m_1 = m_2 = 32$, $p=1$, $\sigma^2=1$ and $\lambda=0.1$.
• Figure 2: Plot of variance decay for the quantity of interest $Q = {u(\mathbf{x}^*)}$ using the Matérn covariance function for the random coefficient with $\sigma^2=1$, $\nu=1.5$ and $\lambda=0.03$.
• Figure 3: Decay rate of $\lvert \mathbb{E}[Q_{h} - \tilde{Q}_h] \rvert$ and $\mathbb{V}[Q_{h} - \tilde{Q}_h]$ with $h$ on a log scale, for the exponential covariance \ref{['eq: p-norm-cov-function']} with $\lambda = 0.3$ and $\lambda = 0.1$, using ${\tau = \sqrt{s}}$. The quantity of interest is $Q = u(\mathbf{x}^*, \mathbf{y}^*)$.
• Figure 4: Decay rate of $\lvert \mathbb{E}[Q_{h} - \tilde{Q}_h] \rvert$ and $\mathbb{V}[Q_{h} - \tilde{Q}_h]$ with $h$ on a log scale, for the Matérn covariance \ref{['eq: matern-cov-function']} with $\lambda = 0.1$ and $\lambda = 0.03$. Here, ${\tau}$ is computed using Eq. \ref{['eq: tau-definition']}. The quantity of interest is $Q = \|u(\cdot, \omega)\|_{L^2(D)}$.
• Figure 5: Plots of $\alpha$$\beta$, and $\gamma$ and their best linear fit, and of the number of samples $N_\ell$ on a log scale computed with and without smoothing for $\lambda = 0.3$ and $\sigma^2 = 1$ using MLMC with $\ell = 0, \dots, 6$. The quantity of interest is the mean value of the pressure at $\mathbf{x}^*=\left(\frac{7}{15},\frac{7}{15}\right)$.

Theorems & Definitions (18)

• Lemma 1
• proof
• Lemma 2
• proof
• Remark 1
• Theorem 1
• proof
• Proposition 1
• Lemma 3
• Theorem 2