Mathematical Modelling with Multidisciplinary Applications (M3A)
http://hdl.handle.net/20.500.11824/13
2021-09-11T20:22:51ZUnveiling Interfacial Li-Ion Dynamics in Li7La3Zr2O12/PEO(LiTFSI) Composite Polymer-Ceramic Solid Electrolytes for All-Solid-State Lithium Batteries
http://hdl.handle.net/20.500.11824/1334
Unveiling Interfacial Li-Ion Dynamics in Li7La3Zr2O12/PEO(LiTFSI) Composite Polymer-Ceramic Solid Electrolytes for All-Solid-State Lithium Batteries
Rincón, M.; García Daza, F.A.; Ranque, P.; Aguesse, F.; Carrasco, J.; Akhmatskaya, E.
Unlocking the full potential of solid-state electrolytes (SSEs) is key to enabling safer and more-energy dense technologies than today’s Li-ion batteries. In particular, composite materials comprising a conductive, flexible polymer matrix embedding ceramic filler particles are emerging as a good strategy to provide the combination of conductivity and mechanical and chemical stability demanded from SSEs. However, the electrochemical activity of these materials strongly depends on their polymer/ceramic interfacial Li-ion dynamics at the molecular scale, whose fundamental understanding remains elusive. While this interface has been explored for nonconductive ceramic fillers, atomistic modeling of interfaces involving a potentially more promising conductive ceramic filler is still lacking. We address this shortfall by employing molecular dynamics and enhanced Monte Carlo techniques to gain unprecedented insights into the interfacial Li-ion dynamics in a composite polymer-ceramic electrolyte, which integrates polyethylene oxide plus LiN(CF3SO2)2 lithium imide salt (LiTFSI), and Li-ion conductive cubic Li7La3Zr2O12 (LLZO) inclusions. Our simulations automatically produce the interfacial Li-ion distribution assumed in space-charge models and, for the first time, a long-range impact of the garnet surface on the Li-ion diffusivity is unveiled. Based on our calculations and experimental measurements of tensile strength and ionic conductivity, we are able to explain a previously reported drop in conductivity at a critical filler fraction well below the theoretical percolation threshold. Our results pave the way for the computational modeling of other conductive filler/polymer combinations and the rational design of composite SSEs.
2021-06-23T00:00:00ZA Tool for Custom Construction of QMC and RQMC Point Sets
http://hdl.handle.net/20.500.11824/1331
A Tool for Custom Construction of QMC and RQMC Point Sets
Godin, M.; Marion, P.; L'Ecuyer, P.; Puchhammer, F.
We present LatNet Builder, a software tool to find good parameters for lattice rules, polynomial lattice rules, and digital nets in base 2, for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) sampling over the s-dimensional unit hypercube. The selection criteria are figures of merit that give different weights to different subsets of coordinates. They are upper bounds on the worst-case error (for QMC) or variance (for RQMC) for integrands rescaled to have a norm of at most one in certain Hilbert spaces of functions. We summarize what are the various Hilbert spaces, discrepancies, types of weights, figures of merit, types of constructions, and search methods supported by LatNet Builder. We briefly discuss its organization and we provide simple illustrations of what it can do.
2021-01-01T00:00:00ZDensity Estimation by Monte Carlo and Quasi-Monte Carlo
http://hdl.handle.net/20.500.11824/1330
Density Estimation by Monte Carlo and Quasi-Monte Carlo
L'Ecuyer, P.; Puchhammer, F.
Estimating the density of a continuous random variable $X$ has been studied extensively in statistics, in the setting where $n$ independent observations of $X$ are given a priori and one wishes to estimate the density from that. Popular methods include histograms and kernel density estimators. In this review paper, we are interested instead in the situation where the observations are generated by Monte Carlo simulation from a model. Then, one can take advantage of variance reduction methods such as stratification, conditional Monte Carlo, and randomized quasi-Monte Carlo (RQMC), and obtain a more accurate density estimator than with standard Monte Carlo for a given computing budget. We discuss several ways of doing this, proposed in recent papers, with a focus on methods that exploit RQMC. A first idea is to directly combine RQMC with a standard kernel density estimator. Another one is to adapt a simulation-based derivative estimation method such as smoothed perturbation analysis or the likelihood ratio method to obtain a continuous estimator of the cumulative density function (CDF), whose derivative is an unbiased estimator of the density. This can then be combined with RQMC. We summarize recent theoretical results with these approaches and give numerical illustrations of how they improve the convergence of the mean square integrated error.
2021-01-01T00:00:00ZMonte Carlo and Quasi-Monte Carlo Density Estimation via Conditioning
http://hdl.handle.net/20.500.11824/1327
Monte Carlo and Quasi-Monte Carlo Density Estimation via Conditioning
L'Ecuyer, P.; Puchhammer, F.; Ben Abdellah, A.
Estimating the unknown density from which a given independent sample originates is more difficult than estimating the mean, in the sense that for the best popular non-parametric density estimators, the mean integrated square error converges more slowly than at the canonical rate of $\mathcal{O}(1/n)$. When the sample is generated from a simulation model and we have control over how this is done, we can do better. We examine an approach in which conditional Monte Carlo yields, under certain conditions, a random conditional density which is an unbiased estimator of the true density at any point. By averaging independent replications, we obtain a density estimator that converges at a faster rate than the usual ones. Moreover, combining this new type of estimator with randomized quasi-Monte Carlo to generate the samples typically brings a larger improvement on the error and convergence rate than for the usual estimators, because the new estimator is smoother as a function of the underlying uniform random numbers.
2021-01-01T00:00:00Z