Showing papers on "Monte Carlo method published in 2019"

Rank-normalization, folding, and localization: An improved $\widehat{R}$ for assessing convergence of MCMC

Abstract: Markov chain Monte Carlo is a key computational tool in Bayesian statistics, but it can be challenging to monitor the convergence of an iterative stochastic algorithm. In this paper we show that the convergence diagnostic $\widehat{R}$ of Gelman and Rubin (1992) has serious flaws. Traditional $\widehat{R}$ will fail to correctly diagnose convergence failures when the chain has a heavy tail or when the variance varies across the chains. In this paper we propose an alternative rank-based diagnostic that fixes these problems. We also introduce a collection of quantile-based local efficiency measures, along with a practical approach for computing Monte Carlo error estimates for quantiles. We suggest that common trace plots should be replaced with rank plots from multiple chains. Finally, we give recommendations for how these methods should be used in practice.

Event Generation with Sherpa 2.2

Abstract: Sherpa is a general-purpose Monte Carlo event generator for the simulation of particle collisions in high-energy collider experiments. We summarize essential features and improvements of the Sherpa 2.2 release series, which is heavily used for event generation in the analysis and interpretation of LHC Run 1 and Run 2 data. We highlight a decade of developments towards ever higher precision in the simulation of particle-collision events.

GetDist: a Python package for analysing Monte Carlo samples

Abstract: Monte Carlo techniques, including MCMC and other methods, are widely used and generate sets of samples from a parameter space of interest that can be used to infer or plot quantities of interest. This note outlines methods used the Python GetDist package to calculate marginalized one and two dimensional densities using Kernel Density Estimation (KDE). Many Monte Carlo methods produce correlated and/or weighted samples, for example produced by MCMC, nested, or importance sampling, and there can be hard boundary priors. GetDist's baseline method consists of applying a linear boundary kernel, and then using multiplicative bias correction. The smoothing bandwidth is selected automatically following Botev et al., based on a mixture of heuristics and optimization results using the expected scaling with an effective number of samples (defined to account for MCMC correlations and weights). Two-dimensional KDE use an automatically-determined elliptical Gaussian kernel for correlated distributions. The package includes tools for producing a variety of publication-quality figures using a simple named-parameter interface, as well as a graphical user interface that can be used for interactive exploration. It can also calculate convergence diagnostics, produce tables of limits, and output in latex.

Statistical power in two-level models: A tutorial based on Monte Carlo simulation.

Abstract: The estimation of power in two-level models used to analyze data that are hierarchically structured is particularly complex because the outcome contains variance at two levels that is regressed on predictors at two levels. Methods for the estimation of power in two-level models have been based on formulas and Monte Carlo simulation. We provide a hands-on tutorial illustrating how a priori and post hoc power analyses for the most frequently used two-level models are conducted. We describe how a population model for the power analysis can be specified by using standardized input parameters and how the power analysis is implemented in SIMR, a very flexible power estimation method based on Monte Carlo simulation. Finally, we provide case-sensitive rules of thumb for deriving sufficient sample sizes as well as minimum detectable effect sizes that yield a power ≥ .80 for the effects and input parameters most frequently analyzed by psychologists. For medium variance components, the results indicate that with lower level (L1) sample sizes up to 30 and higher level (L2) sample sizes up to 200, medium and large fixed effects can be detected. However, small L2 direct- or cross-level interaction effects cannot be detected with up to 200 clusters. The tutorial and guidelines should be of help to researchers dealing with multilevel study designs such as individuals clustered within groups or repeated measurements clustered within individuals. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Multivariate output analysis for Markov chain Monte Carlo

Abstract: Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating expectations with respect to a target distribution. A fundamental question is when should sampling stop so that we have good estimates of the desired quantities? The key to answering this question lies in assessing the Monte Carlo error through a multivariate Markov chain central limit theorem (CLT). The multivariate nature of this Monte Carlo error largely has been ignored in the MCMC literature. We present a multivariate framework for terminating simulation in MCMC. We define a multivariate effective sample size, estimating which requires strongly consistent estimators of the covariance matrix in the Markov chain CLT; a property we show for the multivariate batch means estimator. We then provide a lower bound on the number of minimum effective samples required for a desired level of precision. This lower bound depends on the problem only in the dimension of the expectation being estimated, and not on the underlying stochastic process. This result is obtained by drawing a connection between terminating simulation via effective sample size and terminating simulation using a relative standard deviation fixed-volume sequential stopping rule; which we demonstrate is an asymptotically valid procedure. The finite sample properties of the proposed method are demonstrated in a variety of examples.

Importance Nested Sampling and the MultiNest Algorithm

Abstract: Bayesian inference involves two main computational challenges. First, in estimating the parameters of some model for the data, the posterior distribution may well be highly multi-modal: a regime in which the convergence to stationarity of traditional Markov Chain Monte Carlo (MCMC) techniques becomes incredibly slow. Second, in selecting between a set of competing models the necessary estimation of the Bayesian evidence for each is, by definition, a (possibly high-dimensional) integration over the entire parameter space; again this can be a daunting computational task, although new Monte Carlo (MC) integration algorithms offer solutions of ever increasing efficiency. Nested sampling (NS) is one such contemporary MC strategy targeted at calculation of the Bayesian evidence, but which also enables posterior inference as a by-product, thereby allowing simultaneous parameter estimation and model selection. The widely-used MultiNest algorithm presents a particularly efficient implementation of the NS technique for multi-modal posteriors. In this paper we discuss importance nested sampling (INS), an alternative summation of the MultiNest draws, which can calculate the Bayesian evidence at up to an order of magnitude higher accuracy than `vanilla' NS with no change in the way MultiNest explores the parameter space. This is accomplished by treating as a (pseudo-)importance sample the totality of points collected by MultiNest, including those previously discarded under the constrained likelihood sampling of the NS algorithm. We apply this technique to several challenging test problems and compare the accuracy of Bayesian evidences obtained with INS against those from vanilla NS.

Neural Importance Sampling

Abstract: We propose to use deep neural networks for generating samples in Monte Carlo integration. Our work is based on non-linear independent components estimation (NICE), which we extend in numerous ways to improve performance and enable its application to integration problems. First, we introduce piecewise-polynomial coupling transforms that greatly increase the modeling power of individual coupling layers. Second, we propose to preprocess the inputs of neural networks using one-blob encoding, which stimulates localization of computation and improves inference. Third, we derive a gradient-descent-based optimization for the Kullback-Leibler and the χ2 divergence for the specific application of Monte Carlo integration with unnormalized stochastic estimates of the target distribution. Our approach enables fast and accurate inference and efficient sample generation independently of the dimensionality of the integration domain. We show its benefits on generating natural images and in two applications to light-transport simulation: first, we demonstrate learning of joint path-sampling densities in the primary sample space and importance sampling of multi-dimensional path prefixes thereof. Second, we use our technique to extract conditional directional densities driven by the product of incident illumination and the BSDF in the rendering equation, and we leverage the densities for path guiding. In all applications, our approach yields on-par or higher performance than competing techniques at equal sample count.

A Monte Carlo global analysis of the Standard Model Effective Field Theory : the top quark sector

Abstract: We present a novel framework for carrying out global analyses of the Standard Model Effective Field Theory (SMEFT) at dimension-six: SMEFiT. This approach is based on the Monte Carlo replica method for deriving a faithful estimate of the experimental and theoretical uncertainties and enables one to construct the probability distribution in the space of the SMEFT degrees of freedom. As a proof of concept of the SMEFiT methodology, we present a first study of the constraints on the SMEFT provided by top quark production measurements from the LHC. Our analysis includes more than 30 independent measurements from 10 different processes at $$ \sqrt{s} $$ = 8 and 13 TeV such as inclusive $$ t\overline{t} $$ and single-top production and the associated production of top quarks with weak vector bosons and the Higgs boson. State-of-the-art theoretical calculations are adopted both for the Standard Model and for the SMEFT contributions, where in the latter case NLO QCD corrections are included for the majority of processes. We derive bounds for the 34 degrees of freedom relevant for the interpretation of the LHC top quark data and compare these bounds with previously reported constraints. Our study illustrates the significant potential of LHC precision measurements to constrain physics beyond the Standard Model in a model-independent way, and paves the way towards a global analysis of the SMEFT.

Deep Optimal Stopping

Abstract: In this paper we develop a deep learning method for optimal stopping problems which directly learns the optimal stopping rule from Monte Carlo samples. As such, it is broadly applicable in situations where the underlying randomness can efficiently be simulated. We test the approach on three problems: the pricing of a Bermudan max-call option, the pricing of a callable multi barrier reverse convertible and the problem of optimally stopping a fractional Brownian motion. In all three cases it produces very accurate results in high-dimensional situations with short computing times.

The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data

Abstract: Standard MCMC methods can scale poorly to big data settings due to the need to evaluate the likelihood at each iteration. There have been a number of approximate MCMC algorithms that use sub-sampling ideas to reduce this computational burden, but with the drawback that these algorithms no longer target the true posterior distribution. We introduce a new family of Monte Carlo methods based upon a multidimensional version of the Zig-Zag process of [Ann. Appl. Probab. 27 (2017) 846–882], a continuous-time piecewise deterministic Markov process. While traditional MCMC methods are reversible by construction (a property which is known to inhibit rapid convergence) the Zig-Zag process offers a flexible nonreversible alternative which we observe to often have favourable convergence properties. We show how the Zig-Zag process can be simulated without discretisation error, and give conditions for the process to be ergodic. Most importantly, we introduce a sub-sampling version of the Zig-Zag process that is an example of an exact approximate scheme, that is, the resulting approximate process still has the posterior as its stationary distribution. Furthermore, if we use a control-variate idea to reduce the variance of our unbiased estimator, then the Zig-Zag process can be super-efficient: after an initial preprocessing step, essentially independent samples from the posterior distribution are obtained at a computational cost which does not depend on the size of the data.

Flow-based generative models for Markov chain Monte Carlo in lattice field theory

Abstract: A Markov chain update scheme using a machine-learned flow-based generative model is proposed for Monte Carlo sampling in lattice field theories. The generative model may be optimized (trained) to produce samples from a distribution approximating the desired Boltzmann distribution determined by the lattice action of the theory being studied. Training the model systematically improves autocorrelation times in the Markov chain, even in regions of parameter space where standard Markov chain Monte Carlo algorithms exhibit critical slowing down in producing decorrelated updates. Moreover, the model may be trained without existing samples from the desired distribution. The algorithm is compared with HMC and local Metropolis sampling for ${\ensuremath{\phi}}^{4}$ theory in two dimensions.

Lithium-Ion Battery Remaining Useful Life Prediction With Box–Cox Transformation and Monte Carlo Simulation

Abstract: The current lithium-ion battery remaining useful life (RUL) prediction techniques are mainly developed dependent on offline training data. The loaded current, temperature, and state of charge of lithium-ion batteries used for electric vehicles (EVs) change dramatically under the working conditions. Therefore, it is difficult to design acceleration aging tests of lithium-ion batteries under similar working conditions as those for EVs and to collect effective offline training data. To address this problem, this paper developed an RUL prediction method based on the Box–Cox transformation (BCT) and Monte Carlo (MC) simulation. This method can be implemented independent of offline training data. In the method, the BCT was used to transform the available capacity data and to construct a linear model between the transformed capacities and cycles. The constructed linear model using the BCT was extrapolated to predict the battery RUL, and the RUL prediction uncertainties were generated using the MC simulation. Experimental results showed that accurate and precise RULs were predicted with errors and standard deviations within, respectively, [-20, 10] cycles and [1.8, 7] cycles. If some offline training data are available, the method can reduce the required online training data and, thus, the acceleration aging test time of lithium-ion batteries. Experimental results showed that the acceleration time of the tested cells can be reduced by 70%–85% based on the developed method, which saved one to three months’ acceleration test time compared to the particle filter method.

Analysis of Langevin Monte Carlo via Convex Optimization

Abstract: In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as a first order optimization algorithm of an objective functional defined on the Wasserstein space of order $2$. Using this interpretation and techniques borrowed from convex optimization, we give a non-asymptotic analysis of this method to sample from logconcave smooth target distribution on $\mathbb{R}^d$. Based on this interpretation, we propose two new methods for sampling from a non-smooth target distribution, which we analyze as well. Besides, these new algorithms are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm, which is a popular extension of the Unadjusted Langevin Algorithm. Similar to SGLD, they only rely on approximations of the gradient of the target log density and can be used for large-scale Bayesian inference.

Learning New Physics from a Machine

Abstract: We propose using neural networks to detect data departures from a given reference model, with no prior bias on the nature of the new physics responsible for the discrepancy. The virtues of neural networks as unbiased function approximants make them particularly suited for this task. An algorithm that implements this idea is constructed, as a straightforward application of the likelihood-ratio hypothesis test. The algorithm compares observations with an auxiliary set of reference-distributed events, possibly obtained with a Monte Carlo event generator. It returns a $p$ value, which measures the compatibility of the reference model with the data. It also identifies the most discrepant phase-space region of the data set, to be selected for further investigation. The most interesting potential applications are model-independent new physics searches, although our approach could also be used to compare the theoretical predictions of different Monte Carlo event generators, or for data validation algorithms. In this work we study the performance of our algorithm on a few simple examples. The results confirm the model independence of the approach, namely that it displays good sensitivity to a variety of putative signals. Furthermore, we show that the reach does not depend much on whether a favorable signal region is selected based on prior expectations. We identify directions for improvement towards applications to real experimental data sets.

Atomistic Simulation: A Unique and Powerful Computational Tool for Corrosion Inhibition Research

Abstract: It is difficult to understand the atomistic information on the interaction at the metal/corrosion inhibitor interface experimentally which is a key to understanding the mechanism by which inhibitors prevent the corrosion of metals. Atomistic simulations (molecular dynamics and Monte Carlo) are mostly performed in corrosion inhibition research to give deeper insights into the mechanism of inhibition of corrosion inhibitors on metal surfaces at the atomic and molecular time scales. A lot of works on the use of molecular dynamics and Monte Carlo simulation to investigate corrosion inhibition phenomenon have appeared in the literature in recent times. However, there is still a lack of comprehensive review on the understanding of corrosion inhibition mechanism using these atomistic simulation methodologies. In this review paper, we first of all introduce briefly some important molecular modeling simulations methods. Thereafter, the basic theories of molecular dynamics and Monte Carlo simulations are highlighted. Several studies on the use of atomistic simulations as a modern tool in corrosion inhibition research are presented. Some mechanistic and energetic information on how organic corrosion inhibitors interact with iron and copper metals are provided. This atomic and molecular level information could aid in the design, synthesis and development of new and novel corrosion inhibitors for industrial applications.

How to determine the number of factors to retain in exploratory factor analysis: A comparison of extraction methods under realistic conditions.

Abstract: Exploratory factor analyses are commonly used to determine the underlying factors of multiple observed variables. Many criteria have been suggested to determine how many factors should be retained. In this study, we present an extensive Monte Carlo simulation to investigate the performance of extraction criteria under varying sample sizes, numbers of indicators per factor, loading magnitudes, underlying multivariate distributions of observed variables, as well as how the performance of the extraction criteria are influenced by the presence of cross-loadings and minor factors for unidimensional, orthogonal, and correlated factor models. We compared several variants of traditional parallel analysis (PA), the Kaiser-Guttman Criterion, and sequential χ2 model tests (SMT) with 4 recently suggested methods: revised PA, comparison data (CD), the Hull method, and the Empirical Kaiser Criterion (EKC). No single extraction criterion performed best for every factor model. In unidimensional and orthogonal models, traditional PA, EKC, and Hull consistently displayed high hit rates even in small samples. Models with correlated factors were more challenging, where CD and SMT outperformed other methods, especially for shorter scales. Whereas the presence of cross-loadings generally increased accuracy, non-normality had virtually no effect on most criteria. We suggest researchers use a combination of SMT and either Hull, the EKC, or traditional PA, because the number of factors was almost always correctly retrieved if those methods converged. When the results of this combination rule are inconclusive, traditional PA, CD, and the EKC performed comparatively well. However, disagreement also suggests that factors will be harder to detect, increasing sample size requirements to N ≥ 500. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Inference with Arbitrary Clustering

Abstract: Analyses of spatial or network data are now very common. Nevertheless, statistical inference is challenging since unobserved heterogeneity can be correlated across neighboring observational units. We develop an estimator for the variance-covariance matrix (VCV) of OLS and 2SLS that allows for arbitrary dependence of the errors across observations in space or network structure and across time periods. As a proof of concept, we conduct Monte Carlo simulations in a geospatial setting based on U.S. metropolitan areas. Tests based on our estimator of the VCV asymptotically correctly reject the null hypothesis, whereas conventional inference methods, e.g., those without clusters or with clusters based on administrative units, reject the null hypothesis too often. We also provide simulations in a network setting based on the IDEAS structure of coauthorship and real-life data on scientific performance. The Monte Carlo results again show that our estimator yields inference at the correct significance level even in moderately sized samples and that it dominates other commonly used approaches to inference in networks. We provide guidance to the applied researcher with respect to (i) whether or not to include potentially correlated regressors and (ii) the choice of cluster bandwidth. Finally, we provide a companion statistical package (acreg) enabling users to adjust the OLS and 2SLS coefficient's standard errors to account for arbitrary dependence.

DijetGAN: A Generative-Adversarial Network Approach for the Simulation of QCD Dijet Events at the LHC

Abstract: A Generative-Adversarial Network (GAN) based on convolutional neural networks is used to simulate the production of pairs of jets at the LHC. The GAN is trained on events generated using MadGraph5, Pythia8, and Delphes3 fast detector simulation. We demonstrate that a number of kinematic distributions both at Monte Carlo truth level and after the detector simulation can be reproduced by the generator network. The code can be checked out or forked from the publicly accessible online repository https://gitlab.cern.ch/disipio/DiJetGAN .

Monte Carlo and Reconstruction Membership Inference Attacks against Generative Models

Abstract: Abstract We present two information leakage attacks that outperform previous work on membership inference against generative models. The first attack allows membership inference without assumptions on the type of the generative model. Contrary to previous evaluation metrics for generative models, like Kernel Density Estimation, it only considers samples of the model which are close to training data records. The second attack specifically targets Variational Autoencoders, achieving high membership inference accuracy. Furthermore, previous work mostly considers membership inference adversaries who perform single record membership inference. We argue for considering regulatory actors who perform set membership inference to identify the use of specific datasets for training. The attacks are evaluated on two generative model architectures, Generative Adversarial Networks (GANs) and Variational Autoen-coders (VAEs), trained on standard image datasets. Our results show that the two attacks yield success rates superior to previous work on most data sets while at the same time having only very mild assumptions. We envision the two attacks in combination with the membership inference attack type formalization as especially useful. For example, to enforce data privacy standards and automatically assessing model quality in machine learning as a service setups. In practice, our work motivates the use of GANs since they prove less vulnerable against information leakage attacks while producing detailed samples.

Surrogate-assisted reliability-based design optimization: a survey and a unified modular framework

Abstract: Reliability-based design optimization (RBDO) is an active field of research with an ever increasing number of contributions. Numerous methods have been proposed for the solution of RBDO, a complex problem that combines optimization and reliability analysis. Classical approaches are based on approximation methods and have been classified in review papers. In this paper, we first review classical approaches based on approximation methods such as FORM, and also more recent methods that rely upon surrogate modelling and Monte Carlo simulation. We then propose a generalization of the existing surrogate-assisted and simulation-based RBDO techniques using a unified framework that includes three independent blocks, namely adaptive surrogate modelling, reliability analysis, and optimization. These blocks are non-intrusive with respect to each other and can be plugged independently in the framework. After a discussion on numerical considerations that require attention for the framework to yield robust solutions to various types of problems, the latter is applied to three examples (using two analytical functions and a finite element model). Kriging and support vector machines regression together with their own active learning schemes are considered in the surrogate model block. In terms of reliability analysis, the proposed framework is illustrated using both crude Monte Carlo and subset simulation. Finally, the covariance matrix adaptation-evolution scheme (CMA-ES), a global search algorithm, or sequential quadratic programming (SQP), a local gradient-based method, is used in the optimization block. The comparison of the results to benchmark studies shows the effectiveness and efficiency of the proposed framework.

Best Practices for Quantification of Uncertainty and Sampling Quality in Molecular Simulations [Article v1.0].

Abstract: The quantitative assessment of uncertainty and sampling quality is essential in molecular simulation. Many systems of interest are highly complex, often at the edge of current computational capabilities. Modelers must therefore analyze and communicate statistical uncertainties so that "consumers" of simulated data understand its significance and limitations. This article covers key analyses appropriate for trajectory data generated by conventional simulation methods such as molecular dynamics and (single Markov chain) Monte Carlo. It also provides guidance for analyzing some 'enhanced' sampling approaches. We do not discuss systematic errors arising, e.g., from inaccuracy in the chosen model or force field.

Quantum risk analysis

Abstract: We present a quantum algorithm that analyzes risk more efficiently than Monte Carlo simulations traditionally used on classical computers. We employ quantum amplitude estimation to price securities and evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer. Additionally, we show how to implement this algorithm and how to trade-off the convergence rate of the algorithm and the circuit depth. The shortest possible circuit depth—growing polynomially in the number of qubits representing the uncertainty—leads to a convergence rate of O(M−2/3), where M is the number of samples. This is already faster than classical Monte Carlo simulations which converge at a rate of O(M−1/2). If we allow the circuit depth to grow faster, but still polynomially, the convergence rate quickly approaches the optimum of O(M−1). Thus, for slowly increasing circuit depths our algorithm provides a near quadratic speed-up compared to Monte Carlo methods. We demonstrate our algorithm using two toy models. In the first model we use real hardware, such as the IBM Q Experience, to price a Treasury-bill (T-bill) faced by a possible interest rate increase. In the second model, we simulate our algorithm to illustrate how a quantum computer can determine financial risk for a two-asset portfolio made up of government debt with different maturity dates. Both models confirm the improved convergence rate over Monte Carlo methods. Using simulations, we also evaluate the impact of cross-talk and energy relaxation errors.

$\texttt{HEPfit}$: a Code for the Combination of Indirect and Direct Constraints on High Energy Physics Models

Abstract: $\texttt{HEPfit}$ is a flexible open-source tool which, given the Standard Model or any of its extensions, allows to $\textit{i)}$ fit the model parameters to a given set of experimental observables; $\textit{ii)}$ obtain predictions for observables. $\texttt{HEPfit}$ can be used either in Monte Carlo mode, to perform a Bayesian Markov Chain Monte Carlo analysis of a given model, or as a library, to obtain predictions of observables for a given point in the parameter space of the model, allowing $\texttt{HEPfit}$ to be used in any statistical framework. In the present version, around a thousand observables have been implemented in the Standard Model and in several new physics scenarios. In this paper, we describe the general structure of the code as well as models and observables implemented in the current release.

Consistency without Inference: Instrumental Variables in Practical Application

Abstract: I use Monte Carlo simulations, the jackknife and multiple forms of the bootstrap to study a comprehensive sample of 1359 instrumental variables regressions in 31 papers published in the journals of the American Economic Association. Monte Carlo simulations based upon published regressions show that non-iid error processes adversely affect the size and power of IV estimates, while increasing the bias of IV relative to OLS, producing a very low ratio of power to size and mean squared error that is almost always larger than biased OLS. Weak instrument pre-tests based upon F-statistics are found to be largely uninformative of both size and bias. In published papers, statistically significant IV results generally depend upon only one or two observations or clusters, IV confidence intervals almost always include OLS point estimates, there is little statistical evidence that OLS is biased, and excluded instruments often appear to be irrelevant. *I am grateful to David Broadstone, Brian Finley and anonymous referees for valuable suggestions, and to Ruoqi Zhou for excellent research assistance.

Reflection probability in wireless networks with metasurface-coated environmental objects: an approach based on random spatial processes

Abstract: An emerging and promising vision of wireless networks consists of coating the environmental objects with reconfigurable metasurfaces that are capable of modifying the radio waves impinging upon them according to the generalized law of reflection. By relying on tools from point processes, stochastic geometry, and random spatial processes, we model the environmental objects with a modified random line process of fixed length and with random orientations and locations. Based on the proposed modeling approach, we develop the first analytical framework that provides one with the probability that a randomly distributed object that is coated with a reconfigurable metasurface acts as a reflector for a given pair of transmitter and receiver. In contrast to the conventional network setup where the environmental objects are not coated with reconfigurable metasurfaces, we prove that the probability that the typical random object acts as a reflector is independent of the length of the object itself. The proposed analytical approach is validated against Monte Carlo simulations, and numerical illustrations are given and discussed.

Sampling can be faster than optimization.

Abstract: Optimization algorithms and Monte Carlo sampling algorithms have provided the computational foundations for the rapid growth in applications of statistical machine learning in recent years. There is, however, limited theoretical understanding of the relationships between these 2 kinds of methodology, and limited understanding of relative strengths and weaknesses. Moreover, existing results have been obtained primarily in the setting of convex functions (for optimization) and log-concave functions (for sampling). In this setting, where local properties determine global properties, optimization algorithms are unsurprisingly more efficient computationally than sampling algorithms. We instead examine a class of nonconvex objective functions that arise in mixture modeling and multistable systems. In this nonconvex setting, we find that the computational complexity of sampling algorithms scales linearly with the model dimension while that of optimization algorithms scales exponentially.

Spirit : Multifunctional framework for atomistic spin simulations

Abstract: The Spirit framework is designed for atomic-scale spin simulations of magnetic systems with arbitrary geometry and magnetic structure, providing a graphical user interface with powerful visualizations and an easy-to-use scripting interface. An extended Heisenberg-type spin-lattice Hamiltonian including competing exchange interactions between neighbors at arbitrary distances, higher-order exchange, Dzyaloshinskii-Moriya and dipole-dipole interactions is used to describe the energetics of a system of classical spins localized at atom positions. A variety of common simulation methods are implemented including Monte Carlo and various time evolution algorithms based on the Landau-Lifshitz-Gilbert (LLG) equation of motion. These methods can be used to determine static ground-state and metastable spin configurations, sample equilibrium and finite-temperature thermodynamical properties of magnetic materials and nanostructures, or calculate dynamical trajectories including spin torques induced by stochastic temperature or electric current. Methods for finding the mechanism and rate of thermally assisted transitions include the geodesic nudged elastic band method, which can be applied when both initial and final states are specified, and the minimum mode-following method when only the initial state is given. The lifetimes of magnetic states and rates of transitions can be evaluated within the harmonic approximation of transition-state theory. The framework offers performant central processing unit (CPU) and graphics processing unit (GPU) parallelizations. All methods are verified and applications to several systems, such as vortices, domain walls, skyrmions, and bobbers are described.