Complexity analysis of quasi continuous level Monte Carlo

TL;DR

The paper analyzes a quasi-random variant of continuous level Monte Carlo (QCLMC) as an unbiased estimator for magnetic quantities of interest, replacing i.i.d. tail samples with a deterministic quasi-random sequence to reduce variance. It proves a complexity theorem showing that QCLMC attains the same optimal complexity as CLMC and MLMC up to constants while potentially achieving lower variance. The authors validate the theory on a 2D elliptic PDE with a log-Gauss coefficient, showing that QCLMC can outperform CLMC in time-to-error, especially when the variance-to-bias ratio favors the quasi-random tail sampling. The work demonstrates that quasi-random tail sampling provides variance reductions with negligible extra cost and is robust to problem dimensionality, making QCLMC a favorable alternative to CLMC in uncertainty quantification tasks with adaptive meshes.

Abstract

Continuous level Monte Carlo is an unbiased, continuous version of the celebrated multilevel Monte Carlo method. The approximation level is assumed to be continuous resulting in a stochastic process describing the quantity of interest. Continuous level Monte Carlo methods allow naturally for samplewise adaptive mesh refinements, which are indicated by goal-oriented error estimators. The samplewise refinement levels are drawn in the estimator from an exponentially-distributed random variable. Unfortunately in practical examples this results in higher costs due to high variance in the samples. In this paper we propose a variant of continuous level Monte Carlo, where a quasi Monte Carlo sequence is utilized to "sample" the exponential random variable. We provide a complexity theorem for this novel estimator and show that this results theoretically and practically in a variance reduction of the whole estimator.

Figures (9)

• Figure 2.1: Demonstration of the convergence result of Lemma \ref{['lma:inverse_transform_and_convergence']} for $r = 1.3$ and four independent runs (different seeds) of quasi-random Sobol numbers Sobol1967_Sobol with Owen scrambling, cf. Owen1995_ScramblingOwen1998_Scrambling generated through the scipy library Scipy2020, and pseudo-random numbers generated with the numpy library Numpy2020. The quasi-random Sobol numbers have their optimal discrepancy for powers of two (location of downward spikes in the light blue lines), but we observe that for values in between powers of two, the discrepancy still converges with rate one, i.e. $\kappa = 0$.
• Figure 4.1: Visualization of samples of the log-Gauss random coefficient for $\nu=1.5$, $\lambda = 0.1$, $v=0.5$ (upper left), $\nu=1.5$, $\lambda = 0.1$, $v=1$ (upper right), $\nu=1.5$, $\lambda = 0.2$, $v=0.5$ (lower left), $\nu=1$, $\lambda = 0.1$, $v=0.5$ (lower right). The KL-expansion \ref{['eq:log_gauss_coefficient']} was truncated after $R=36$ terms in each case.
• Figure 4.2: Single sample of the log-Gauss random coefficient for $\nu=1.5$, $\lambda = 0.1$, $v=0.5$ (left) on adaptive mesh (middle) generated by $5$ iterative refinement steps, see Remark \ref{['rem:refinement']}, and corresponding PDE solution (right). The KL-expansion was truncated after $R=36$ terms.
• Figure 4.3: Mean of maximal levels $\bar{L}$ generated by quasi-random Sobol sequence for $M$ samples (left) and corresponding mean of the upper bound to the bias of QCLMC (right) over $100$ independent runs realized via Owen Scrambling.
• Figure 4.4: Upper bounds to the variance (left) and MSE (right) for CLMC and QCLMC estimated over $100$ independent runs. Hyperparameters for log-Gauss field \ref{['eq:matern_kernel']}: $\nu=1$, $\lambda = 0.1$, $v=0.5$.

Theorems & Definitions (13)

• Remark 1.1
• Remark 1.2
• Lemma 2.1
• Remark 2.2
• Remark 2.3
• Proposition 3.1
• proof
• Remark 3.2
• Theorem 3.3: QCLMC - complexity theorem
• proof