Multilevel Monte Carlo Methods for Stochastic Convection-Diffusion Eigenvalue Problems.

We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection-diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element (FE) discretizations of the eigenvalue problem on a hierarchy of increasingly finer meshes. For the discretized, algebraic eigenproblems we use both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA), providing an analysis of the cost in each case. By studying the variance on each level and adapting classical FE error bounds to the stochastic setting, we are able to bound the total error of our MLMC estimator and provide a complexity analysis. As expected, the complexity bound for our MLMC estimator is superior to plain Monte Carlo. To improve the efficiency of the MLMC further, we exploit the hierarchy of meshes and use coarser approximations as starting values for the eigensolvers on finer ones. To improve the stability of the MLMC method for convection-dominated problems, we employ two additional strategies. First, we consider the streamline upwind Petrov-Galerkin formulation of the discrete eigenvalue problem, which allows us to start the MLMC method on coarser meshes than is possible with standard FEs. Second, we apply a homotopy method to add stability to the eigensolver for each sample. Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster convergence of QMC, this improves the overall complexity. We provide detailed numerical results comparing our different strategies to demonstrate the practical feasibility of the MLMC method in different use cases. The results support our complexity analysis and further demonstrate the superiority over plain Monte Carlo in all cases.

Separating internal and externally forced contributions to global temperature variability using a Bayesian stochastic energy balance framework.

Earth's temperature variability can be partitioned into internal and externally forced components. Yet, underlying mechanisms and their relative contributions remain insufficiently understood, especially on decadal to centennial timescales. Important reasons for this are difficulties in isolating internal and externally forced variability. Here, we provide a physically motivated emulation of global mean surface temperature (GMST) variability, which allows for the separation of internal and external variations. To this end, we introduce the "ClimBayes" software package, which infers climate parameters from a stochastic energy balance model (EBM) with a Bayesian approach. We apply our method to GMST data from temperature observations and 20 last millennium simulations from climate models of intermediate to high complexity. This yields the best estimates of the EBM's forced and forced + internal response, which we refer to as emulated variability. The timescale-dependent variance is obtained from spectral analysis. In particular, we contrast the emulated forced and forced + internal variance on interannual to centennial timescales with that of the GMST target. Our findings show that a stochastic EBM closely approximates the power spectrum and timescale-dependent variance of GMST as simulated by modern climate models. Small deviations at interannual timescales can be attributed to the simplified representation of internal variability and, in particular, the absence of (pseudo-)oscillatory modes in the stochastic EBM. Altogether, we demonstrate the potential of combining Bayesian inference with conceptual climate models to emulate statistics of climate variables across timescales.

Multilevel simulation of hard-sphere mixtures.

We present a multilevel Monte Carlo simulation method for analyzing multi-scale physical systems via a hierarchy of coarse-grained representations, to obtain numerically exact results, at the most detailed level. We apply the method to a mixture of size-asymmetric hard spheres, in the grand canonical ensemble. A three-level version of the method is compared with a previously studied two-level version. The extra level interpolates between the full mixture and a coarse-grained description where only the large particles are present-this is achieved by restricting the small particles to regions close to the large ones. The three-level method improves the performance of the estimator, at fixed computational cost. We analyze the asymptotic variance of the estimator and discuss the mechanisms for the improved performance.

Critical point for demixing of binary hard spheres.

We use a two-level simulation method to analyze the critical point associated with demixing of binary hard-sphere mixtures. The method exploits an accurate coarse-grained model with two- and three-body effective interactions. Using this model within the two-level methodology allows computation of properties of the full (fine-grained) mixture. The critical point is located by computing the probability distribution for the number of large particles in the grand canonical ensemble and matching to the universal form for the 3D Ising universality class. The results have a strong and unexpected dependence on the size ratio between large and small particles, which is related to three-body effective interactions and the geometry of the underlying hard-sphere packings.

Correction of coarse-graining errors by a two-level method: Application to the Asakura-Oosawa model.

We present a method that exploits self-consistent simulation of coarse-grained and fine-grained models in order to analyze properties of physical systems. The method uses the coarse-grained model to obtain a first estimate of the quantity of interest, before computing a correction by analyzing properties of the fine system. We illustrate the method by applying it to the Asakura-Oosawa model of colloid-polymer mixtures. We show that the liquid-vapor critical point in that system is affected by three-body interactions which are neglected in the corresponding coarse-grained model. We analyze the size of this effect and the nature of the three-body interactions. We also analyze the accuracy of the method as a function of the associated computational effort.