Lasso Monte Carlo, a Novel Method for High Dimensional Uncertainty Quantification

Uncertainty quantification (UQ) is an active area of research, and an essential technique used in all fields of science and engineering. The most common methods for UQ are Monte Carlo and surrogate-modelling. The former method is dimensionality independent but has slow convergence, while the latter method has been shown to yield large computational speedups with respect to Monte Carlo. However, surrogate models suffer from the so-called curse of dimensionality, and become costly to train for high-dimensional problems, where UQ might become computationally prohibitive. In this paper we present a new technique, Lasso Monte Carlo (LMC), which combines surrogate models and the multilevel Monte Carlo technique, in order to perform UQ in high-dimensional settings, at a reduced computational cost. We provide mathematical guarantees for the unbiasedness of the method, and show that LMC can converge faster than simple Monte Carlo. The theory is numerically tested with benchmarks on toy problems, as well as on a real example of UQ from the field of nuclear engineering. In all presented examples LMC converges faster than simple Monte Carlo, and computational costs are reduced by more than a factor of 5 in some cases

The Monte Carlo Methods

In applied mathematics, the name Monte Carlo is given to the method of solving problems by means of experiments with random numbers. This name, after the casino at Monaco, was first applied around 1944 to the method of solving deterministic problems by reformulating them in terms of a problem with random elements, which could then be solved by large-scale sampling. But, by extension, the term has come to mean any simulation that uses random numbers. Monte Carlo methods have become among the most fundamental techniques of simulation in modern science. This book is an illustration of the use of Monte Carlo methods applied to solve specific problems in mathematics, engineering, physics, statistics, and science in general

Low-Variance Monte Carlo Simulation of Thermal Transport in Graphene

ue to its unique thermal properties, graphene has generated considerable interest in the context of thermal management applications. In order to correctly treat heat transfer in this material, while still reaching device-level length and time scales, a kinetic description, such as the Boltzmann transport equation, is typically required. We present a Monte Carlo method for obtaining numerical solutions of this description that dramatically outperforms traditional Monte Carlo approaches by simulating only the deviation from equilibrium. We validate the simulation method using an analytical solution of the Boltzmann equation for long graphene nanoribbons; we also use this result to characterize the error associated with previous approximate solutions of this problem.National Science Foundation (U.S.). Graduate Research Fellowship ProgramAmerican Society for Engineering Education. National Defense Science and Engineering Graduate FellowshipSingapore-MIT Allianc

Deviational simulation of phonon transport in graphene ribbons with ab initio scattering

We present a deviational Monte Carlo method for solving the Boltzmann-Peierls equation with ab initio 3-phonon scattering, for temporally and spatially dependent thermal transport problems in arbitrary geometries. Phonon dispersion relations and transition rates for graphene are obtained from density functional theory calculations. The ab initio scattering operator is simulated by an energy-conserving stochastic algorithm embedded within a deviational, low-variance Monte Carlo formulation. The deviational formulation ensures that simulations are computationally feasible for arbitrarily small temperature differences, while the stochastic treatment of the scattering operator is both efficient and exhibits no timestep error. The proposed method, in which geometry and phonon-boundary scattering are explicitly treated, is extensively validated by comparison to analytical results, previous numerical solutions and experiments. It is subsequently used to generate solutions for heat transport in graphene ribbons of various geometries and evaluate the validity of some common approximations found in the literature. Our results show that modeling transport in long ribbons of finite width using the homogeneous Boltzmann equation and approximating phonon-boundary scattering using an additional homogeneous scattering rate introduces an error on the order of 10% at room temperature, with the maximum deviation reaching 30% in the middle of the transition regime.Singapore-MIT Alliance for Research and TechnologyAmerican Society for Engineering Education. National Defense Science and Engineering Graduate FellowshipNational Science Foundation (U.S.). Graduate Research Fellowshi

Simulation and the Monte Carlo Method

This accessible new edition explores the major topics in Monte Carlo simulation Simulation and the Monte Carlo Method, Second Edition reflects the latest developments in the field and presents a fully updated and comprehensive account of the major topics that have emerged in Monte Carlo simulation since the publication of the classic First Edition over twenty-five years ago. While maintaining its accessible and intuitive approach, this revised edition features a wealth of up-to-date information that facilitates a deeper understanding of problem solving across a wide array of subject areas, such as engineering, statistics, computer science, mathematics, and the physical and life sciences. The book begins with a modernized introduction that addresses the basic concepts of probability, Markov processes, and convex optimization. Subsequent chapters discuss the dramatic changes that have occurred in the field of the Monte Carlo method, with coverage of many modern topics including: Markov Chain Monte Carlo Variance reduction techniques such as the transform likelihood ratio method and the screening method The score function method for sensitivity analysis The stochastic approximation method and the stochastic counter-part method for Monte Carlo optimization The cross-entropy method to rare events estimation and combinatorial optimization Application of Monte Carlo techniques for counting problems, with an emphasis on the parametric minimum cross-entropy method An extensive range of exercises is provided at the end of each chapter, with more difficult sections and exercises marked accordingly for advanced readers. A generous sampling of applied examples is positioned throughout the book, emphasizing various areas of application, and a detailed appendix presents an introduction to exponential families, a discussion of the computational complexity of stochastic programming problems, and sample MATLAB programs. Requiring only a basic, introductory knowledge of probability and statistics, Simulation and the Monte Carlo Method, Second Edition is an excellent text for upper-undergraduate and beginning graduate courses in simulation and Monte Carlo techniques. The book also serves as a valuable reference for professionals who would like to achieve a more formal understanding of the Monte Carlo method

Monte-Carlo-Simulation-Based, Product-Quality-Focused Analysis Of Nanocoating Curing And Post Curing Process

ABSTRACT MONTE CARLO SIMULATION BASED PRODUCT QUALITY ANALYSIS OF POLYMER COATING CURING AND POST CURING By Jianming Zhao May 2017 Advisor：Dr. Yinlun Huang Major: Chemical Engineering Degree: Master of Science To achieve better property of polymer coatings, different categories of nanoparticles are applied before coating’s curing process. However, one of the adverse effects is the change of final property, which leads the difficulty of product quality control. Using mathematical modeling method can actually improve the cost and time to get a prediction of product quality. Still now, different series of models are developed for various purposes. For example, Monte-Carlo simulation suits short-term usage prediction; kinetics simulation suits long-term usage prediction; potential energy simulation suits to model microcosmic particle’s energy change. To solve the difficulty of quality control, Monte-Carlo simulation is used to provide relatively accurate data under given conditions. A series of models are redesigned, based on those developed by Xiao et al. (2009, 2010). From the simulation data, a visualized conversion and final Young’s modulus change can be clearly seen. In this thesis, a linear plot of the general curing process is gained. Monte Carlo simulation methodology is a new method to describe the conversion change. The post-curing process is also simulated, with the contrast of the real data from Yari (2014), the post-curing process and principle can be explained. The effect of the nanoparticle can also be gained in this work. Additionally, with the work of this thesis, people can control the product quality more easily

A Monte Carlo packing algorithm for poly-ellipsoids and its comparison with packing generation using Discrete Element Model

Granular material is showing very often in geotechnical engineering, petroleum engineering, material science and physics. The packings of the granular material play a very important role in their mechanical behaviors, such as stress-strain response, stability, permeability and so on. Although packing is such an important research topic that its generation has been attracted lots of attentions for a long time in theoretical, experimental, and numerical aspects, packing of granular material is still a difficult and active research topic, especially the generation of random packing of non-spherical particles. To this end, we will generate packings of same particles with same shapes, numbers, and same size distribution using geometry method and dynamic method, separately. Specifically, we will extend one of Monte Carlo models for spheres to ellipsoids and poly-ellipsoids

Simulation of Heat Transport in Graphene Nanoribbons Using the Ab-Initio Scattering Operator

We present a deviational Monte Carlo method for simulating phonon transport in graphene using the ab initio 3-phonon scattering operator. This operator replaces the commonly used relaxation-time approximation, which is known to neglect, among other things, coupling between out of equilibrium states that are particularly important in graphene. Phonon dispersion relations and transition rates are obtained from density functional theory calculations. The proposed method provides, for the first time, means for obtaining solutions of the Boltzmann transport equation with ab initio scattering for time- and spatially-dependent problems. The deviational formulation ensures that simulations are computationally feasible for arbitrarily small temperature differences; within this formulation, the ab initio scattering operator is treated using an efficient stochastic algorithm which, in the limit of large number of states, outperforms the more traditional deterministic methods used in solutions of the homogeneous Boltzmann equation. We use the proposed method to study heat transport in graphene ribbons.National Science Foundation (U.S.). Graduate Research Fellowship ProgramAmerican Society for Engineering Education. National Defense Science and Engineering Graduate FellowshipSingapore-MIT Allianc