Monte Carlo Integration Research Articles

Risk of Adverse Perinatal Outcomes Among African-born Black Women in California, 2011-2020.

African-born women have a lower risk of preterm birth and small for gestational age (SGA) birth compared with United States-born Black women, however variation by country of origin is overlooked. Additionally, the extent that nativity disparities in adverse perinatal outcomes to Black women are explained by individual-level factors remains unclear. We conducted a population-based study of nonanomalous singleton live births to United States- and African-born Black women in California from 2011 to 2020 (n = 194,320). We used age-adjusted Poisson regression models to estimate the risk of preterm birth and SGA and reported risk ratios (RR) and 95% confidence intervals (CI). Decomposition using Monte Carlo integration of the g-formula computed the percentage of disparities in adverse outcomes between United States- and African-born women explained by individual-level factors. Eritrean women (RR = 0.4; 95% CI = 0.3, 0.5) had the largest differences in risk of preterm birth and Cameroonian women (RR = 0.5; 95% CI = 0.3, 0.6) in SGA birth, compared with United States-born Black women. Ghanaian women had smaller differences in risk of preterm birth (RR = 0.8; 95% CI = 0.7, 1.0) and SGA (RR = 0.9; 95% CI = 0.8, 1.1) compared with United States-born women. Overall, we estimate that absolute differences in socio-demographic and clinical factors contributed to 32% of nativity-based disparities in the risk of preterm birth and 26% of disparities in SGA. We observed heterogeneity in risk of adverse perinatal outcomes for African- compared with United States-born Black women, suggesting that nativity disparities in adverse perinatal outcomes were not fully explained by differences in individual-level factors.

Joint Retrieval of Multiple Species of Ice Hydrometeor Parameters from Millimeter and Submillimeter Wave Brightness Temperature Based on Convolutional Neural Networks

Submillimeter wave radiometers are promising remote sensing tools for sounding ice cloud parameters. The Ice Cloud Imager (ICI) aboard the second generation of the EUMETSAT Polar System (EPS−SG) is the first operational submillimeter wave radiometer used for ice cloud remote sensing. Ice clouds simultaneously contain three species of ice hydrometeors—ice, snow, and graupel—the physical distributions and submillimeter wave radiation characteristics of which differ. Therefore, jointly retrieving the mass parameters of the three ice hydrometeors from submillimeter brightness temperatures is very challenging. In this paper, we propose a multiple species of ice hydrometeor parameters retrieval algorithm based on convolutional neural networks (CNNs) that can jointly retrieve the total content and vertical profiles of ice, snow, and graupel particles from submillimeter brightness temperatures. The training dataset is generated by a numerical weather prediction (NWP) model and a submillimeter wave radiative transfer (RT) model. In this study, an end to end ICI simulation experiment involving forward modeling of the brightness temperature and retrieval of ice cloud parameters was conducted to verify the effectiveness of the proposed CNN retrieval algorithm. Compared with the classical Unet, the average relative errors of the improved RCNN–ResUnet are reduced by 11%, 25%, and 18% in GWP, IWP, and SWP retrieval, respectively. Compared with Bayesian Monte Carlo integration algorithm, the average relative error of the total content retrieved by RCNN–ResUnet is reduced by 71%. Compared with BP neural network algorithm, the average relative error of the vertical profiles retrieved by RCNN–ResUnet is reduced by 69%. In addition, this algorithm was applied to actual Advanced Technology Microwave Sounder (ATMS) 183 GHz observed brightness temperatures to retrieve graupel particle parameters with a relative error in the total content of less than 25% and a relative error in the profile of less than 35%. The results show that the proposed CNN algorithm can be applied to future space borne submillimeter wave radiometers to jointly retrieve mass parameters of ice, snow, and graupel.

Variance reduction via simultaneous importance sampling and control variates techniques using vegas

Monte Carlo (MC) integration is an important calculational technique in the physical sciences. Practical considerations require that the calculations are performed as accurately as possible for a given set of computational resources. To improve the accuracy of MC integration, a number of useful variance reduction algorithms have been developed, including importance sampling and control variates. In this work, we demonstrate how these two methods can be applied simultaneously, thus combining their benefits. We provide a python wrapper, named CoVVVR, which implements our approach in the VEGAS program. The improvements are quantified with several benchmark examples from the literature.

Virial equation of state for a granular system.

The equation of state for an ideal gas is simple, which is . In the case of imperfect gases where mutual interactions among the constituents are important, pressure P can be expressed as the series expansion of density n with appropriate coefficients, known as virial coefficients . In this paper, we have obtained the first four virial coefficients for a model interaction potential using multidimensional Monte-Carlo integration and importance sampling methods. Next, we perform molecular dynamics simulations with the same for a many-particle system to obtain P as a function of T and n. We compare our numerical data with the virial equation of state.

Sampling and Change of Measure for Monte Carlo Integration on Simplices

Simplices are the fundamental domain when integrating over convex polytopes. The aim of this work is to establish a novel framework of Monte Carlo integration over simplices, throughout from sampling to variance reduction. Namely, we develop a uniform sampling method on the standard simplex consisting of two independent procedures and construct theories on change of measure on each of the two independent elements in the developed sampling technique with a view towards variance reduction by importance sampling. We provide illustrative figures and numerical results to support our theoretical findings and demonstrate the strong potential of the developed framework for effective implementation and acceleration of Monte Carlo integration over simplices.

On the one-point quadrature discretization in peridynamics: A novel perspective from Monte Carlo integration

One-point quadrature discretization is one of the most popular discretizations for peridynamic simulation, but it suffers from poor accuracy. Employing smoothly decaying influence functions or precise volume correction can address this, but the former applies only to specific materials, and the latter may be computationally expensive and limited to square or cubic meshes. This paper discusses one-point quadrature discretization in the context of Monte Carlo integration and presents the quadrature rule, considering the smoothed acceptance-rejection method. In this context, the volume correction factor is explained as the probability of two particles interacting with each other. We summarize several volume correction algorithms from the literature and then present a new one based on the proposed quadrature rule. Numerical tests were conducted in both 2D and 3D. According to the comparison, the proposed algorithm only costs approximately 0.2% to 40% of the time that precise volume correction needs to compute the quadrature weights, while its integration accuracy is still at the same level. The results suggest that the continuity of the acceptance-rejection rule and the precision of the expectation of the horizon volume primarily matter in the accuracy of quadrature rules.

Towards Quantum Ray Tracing.

Rendering on conventional computers is capable of generating realistic imagery, but the computational complexity of these light transport algorithms is a limiting factor of image synthesis. Quantum computers have the potential to significantly improve rendering performance through reducing the underlying complexity of the algorithms behind light transport. This paper investigates hybrid quantum-classical algorithms for ray tracing, a core component of most rendering techniques. Through a practical implementation of quantum ray tracing in a 3D environment, we show quantum approaches provide a quadratic improvement in query complexity compared to the equivalent classical approach. Based on domain specific knowledge, we then propose algorithms to significantly reduce the computation required for quantum ray tracing through exploiting image space coherence and a principled termination criteria for quantum searching. We show results obtained using a simulator for both Whitted style ray tracing, and for accelerating ray tracing operations when performing classical Monte Carlo integration for area lights and indirect illumination.

Probabilistic evaluation of dynamic positioning operability with a Quasi-Monte Carlo approach

During the design phase of an offshore unit, estimating the station-keeping capabilities of the dynamic positioning (DP) system is mandatory. This means, in conventional offshore applications, to determine the maximum sustainable wind speed as a function of the encounter heading, which the unit may counteract by employing the onboard actuators or mooring lines only. Besides the deterministic estimation of DP capability, it is possible to assess the operability of the DP system following a non-deterministic probabilistic process by employing the site-specific joint wind-wave distributions to model the environment. In such a case, the operability results from a Monte Carlo integration process. Here it is proposed to enhance the applicability of the probabilistic analysis of DP operability, investigating the application of a Quasi-Monte Carlo method. In this sense, the procedure uses quasi-random samplings following a Sobol sequence instead of employing random samples of the joint distributions. In this paper, the Quasi-Monte Carlo process is tested and compared on a reference ship, highlighting the improvements to the established probabilistic DP prediction concerning the number of calculations needed to estimate operability. The significant reduction of computational time makes the newly implemented method suitable for the early design stage applications.

Understanding step selection analysis through numerical integration

Abstract Step selection functions (SSFs) are flexible statistical models used to jointly describe animals' movement and habitat preferences. The popularity of SSFs has grown rapidly, and various extensions have been developed to increase their utility, including the ability to use multiple statistical distributions to describe movement constraints, interactions to allow movements to depend on local environmental features, and random effects and latent states to account for within‐ and among‐individual variability. Although the SSF is a relatively simple statistical model, its presentation has not been consistent in the literature, leading to confusion about model flexibility and interpretation. We believe that part of the confusion has arisen from the conflation of the SSF model with the methods used for statistical inference, and in particular, parameter estimation. Notably, conditional logistic regression (CLR) can be used to fit SSFs in exponential form, and this model fitting approach is often presented interchangeably with the actual model (the SSF itself). However, reliance on CLR reduces model flexibility, and suggests a misleading interpretation of step selection analysis as being equivalent to a case–control study. In this review, we explicitly distinguish between model formulation and inference technique, presenting a coherent framework to fit SSFs based on numerical integration and maximum likelihood estimation. We provide an overview of common numerical integration techniques (including Monte Carlo integration, importance sampling and quadrature), and explain how they relate to popular methods used in step selection analyses. This general framework unifies different model fitting techniques for SSFs, and opens the way for improved inferential methods. In this approach, it is straightforward to model movement with distributions outside the exponential family, and to apply different SSF model formulations to the same data set and compare them with AIC. By separating the model formulation from the inference technique, we hope to clarify many important concepts in step selection analysis.

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. Non-local nonlinear PDE models can accurately capture these phenomena, but traditional numerical approximation methods are infeasible when the considered non-local PDE is high-dimensional. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions. In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman–Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the non-local term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

A Note on Monte Carlo Integration in High Dimensions

Monte Carlo integration is a commonly used technique to compute intractable integrals and is typically thought to perform poorly for very high-dimensional integrals. To show that this is not always the case, we examine Monte Carlo integration using techniques from the high-dimensional statistics literature by allowing the dimension of the integral to increase. In doing so, we derive nonasymptotic bounds for the relative and absolute error of the approximation for some general classes of functions through concentration inequalities. We provide concrete examples in which the magnitude of the number of points sampled needed to guarantee a consistent estimate varies between polynomial to exponential, and show that in theory arbitrarily fast or slow rates are possible. This demonstrates that the behavior of Monte Carlo integration in high dimensions is not uniform. Through our methods we also obtain nonasymptotic confidence intervals which are valid regardless of the number of points sampled.

Numerical Integration of locally Peaked Bivariate Functions

The aim of this paper is to compare the relative accuracies between deterministic and stochastic methods for solving bounded integrals numerically to observe which methods tend to function well and converge to a small amount of error based on computational resources. For the deterministic method, the Gauss-Legendre quadrature method has been selected and for the stochastic method, the Monte Carlo integration has been selected. For each case, the number of variables will be adjusted to observe the effect on error. For the Gauss-Legendre quadrature method the permutations increased with the inaccuracy of 9% when the number of nodes increased to 3 but was reduced by 90% and later on the error depicted a drop as the number of nodes raised further. For the stochastic method, that was chosen from large sample size, the inaccuracy was found to be inversely proportional to the sample size. This concluded that the monte-carlo approach was not affected by the impact of dimensionality moreover, deterministic method also seemed to overcome the dimensionality constraint.

IBL‐NeRF: Image‐Based Lighting Formulation of Neural Radiance Fields

AbstractWe propose IBL‐NeRF, which decomposes the neural radiance fields (NeRF) of large‐scale indoor scenes into intrinsic components. Recent approaches further decompose the baked radiance of the implicit volume into intrinsic components such that one can partially approximate the rendering equation. However, they are limited to representing isolated objects with a shared environment lighting, and suffer from computational burden to aggregate rays with Monte Carlo integration. In contrast, our prefiltered radiance field extends the original NeRF formulation to capture the spatial variation of lighting within the scene volume, in addition to surface properties. Specifically, the scenes of diverse materials are decomposed into intrinsic components for rendering, namely, albedo, roughness, surface normal, irradiance, and prefiltered radiance. All of the components are inferred as neural images from MLP, which can model large‐scale general scenes. Especially the prefiltered radiance effectively models the volumetric light field, and captures spatial variation beyond a single environment light. The prefiltering aggregates rays in a set of predefined neighborhood sizes such that we can replace the costly Monte Carlo integration of global illumination with a simple query from a neural image. By adopting NeRF, our approach inherits superior visual quality and multi‐view consistency for synthesized images as well as the intrinsic components. We demonstrate the performance on scenes with complex object layouts and light configurations, which could not be processed in any of the previous works.

Global Sensitivity of Penman–Monteith Reference Evapotranspiration to Climatic Variables in Mato Grosso, Brazil

Understanding how climatic variables impact the reference evapotranspiration (ETo) is essential for water resource management, especially considering potential fluctuations due to climate change. Therefore, we used the Sobol’ method to analyze the spatiotemporal variations of Penman–Monteith ETo sensitivity to the climatic variables: downward solar radiation, relative humidity, maximum and minimum air temperature, and wind speed. The Sobol’ indices variances were estimated by Monte Carlo integration, with sample limits set to the 2.5th and 97.5th percentiles of the daily data of 33 automatic weather stations located in the state of Mato Grosso, Brazil. The results of the Sobol’ analysis indicate considerable spatiotemporal variations in the sensitivity of ETo to climatic variables and their interactions. The dominant climatic variable responsible for ETo fluctuations in Mato Grosso is incident solar radiation (53% to 93% of annual total sensitivity—Stot), which has a more significant impact in humid environments (70% to 90% of Stot), as observed in the areas of the Amazon biome in the state. Air relative humidity and wind speed have higher sensitivity indices during the dry season in the Cerrado biome (savanna) areas in Mato Grosso (20% and 30% of the Stot, respectively). Our findings show that changes in solar radiation, relative humidity, and wind speed are the main driving forces that impact the reference evapotranspiration.

Geometric Analysis of Free and Accessible Volume in Atmospheric Nanoparticles.

Computer-generated atomistic microstructures of atmospheric nanoparticles are geometrically analyzed using Delaunay tessellation followed by Monte Carlo integration to compute their free and accessible volume. The nanoparticles studied consist of cis-pinonic acid (a biogenic organic aerosol component), inorganic ions (sulfate and ammonium), and water. Results are presented for the free or unoccupied volume in different domains of the nanoparticles and its dependence on relative humidity and organic content. We also compute the accessible volume to small penetrants such as water molecules. Most of the free volume or volume accessible to a penetrant as large as a water molecule is located in the domains occupied by organics. In contrast, regions dominated by inorganics do not have any cavities with sizes larger than 1 Å. Solid inorganic domains inside the particle are practically impermeable to any small molecule, thereby offering practically infinite resistance to diffusion. A guest molecule can find diffusive channels to wander around within the nanoparticle only through the aqueous and organic-rich domains. The largest pores are observed in nanoparticles with high levels of organic mass and low relative humidity. At high relative humidity, the presence of more water molecules reduces the empty space in the inner domains of the nanoparticle, since areas rich in organic molecules (which are the only ones where appreciable pores are found) are pushed to the outer area of the particle. This, however, should not be expected to affect the diffusive process as transport through the aqueous phase inside the particle will be, by default, fast due to its fluid-like nature.

Some Theoretical and Computational Aspects of the Truncated Multivariate Skew-Normal/Independent Distributions

In this article, we derive a closed-form expression for computing the probabilities of p-dimensional rectangles by means of a multivariate skew-normal distribution. We use a stochastic representation of the multivariate skew-normal/independent distributions to derive expressions that relate their probability density functions to the expected values of positive random variables. We also obtain an analogous expression for probabilities of p-dimensional rectangles for these distributions. Based on this, we propose a procedure based on Monte Carlo integration to evaluate the probabilities of p-dimensional rectangles through multivariate skew-normal/independent distributions. We use these findings to evaluate the probability density functions of a truncated version of this class of distributions, for which we also suggest a scheme to generate random vectors by using a stochastic representation involving a truncated multivariate skew-normal random vector. Finally, we derive distributional properties involving affine transformations and marginalization. We illustrate graphically several of our methodologies and results derived in this article.

Tropical Feynman integration in the Minkowski regime

We present a new computer program, feyntrop, which uses the tropical geometric approach to evaluate Feynman integrals numerically. In order to apply this approach in the physical regime, we introduce a new parametric representation of Feynman integrals that implements the causal iε prescription concretely while retaining projective invariance. feyntrop can efficiently evaluate dimensionally regulated, quasi-finite Feynman integrals, with not too exceptional kinematics in the physical regime, with a relatively large number of propagators and with arbitrarily many kinematic scales. We give a systematic classification of all relevant kinematic regimes, review the necessary mathematical details of the tropical Monte Carlo approach, give fast algorithms to evaluate (deformed) Feynman integrands, describe the usage of feyntrop and discuss many explicit examples of evaluated Feynman integrals. Program summaryProgram title:feyntrop.CPC Library link to program files:https://doi.org/10.17632/k6r62hdgvd.1Developer's repository link:https://github.com/michibo/feyntrop.Licensing provisions: MIT License.Programming language: The tropical Monte Carlo code is written in C++. The high-level interface is written in python.Supplementary material: The repository includes installation and usage instructions (README.md), a jupyter notebook tutorial (tutorial_2L_3pt.ipynb), the collection of examples presented in section 6 (see the folder /examples), and a test suite to ensure a successful installation (see the folder /tests).Nature of problem: Sufficiently fast numerical integration of (dimensionally regularized) Feynman integrals (also in the Minkowski regime of phase space).Solution method: Tropical Monte Carlo integration of a manifestly iε-free parametric representation of Feynman integrals.Additional comments: The program feyntrop is based on previous code available at https://github.com/michibo/tropical-feynman-quadrature, which was published as a proof-of-concept with, Michael Borinsky, ‘Tropical Monte Carlo quadrature for Feynman integrals’, Ann. Inst. Henri Poincaré Comb. Phys. Interact. (in press)[1]. This previous code did not have features which are required for phenomenological studies in high-energy physics. In particular, it only allowed for phase space points in the Euclidean regime, and only computed the leading term in the ϵ expansion.Restrictions: The Feynman integral must be quasi-finite and the momentum configuration must be sufficiently generic. Numerators of Feynman integrals are not implemented. ReferencesEigen3[2]. The xoshiro256+ pseudo-random-number generator [3]. python[4]. pybind11[5]. sympy[6].

PSGen, a generator of phase space parameterizations for the multichannel Monte Carlo integration

PSGen is a new general purpose Fortran program which has been written to facilitate the Monte Carlo phase space integration of the S matrix element of any 2→n scattering process, with n=2,...,9, provided by the user. The program is written in Fortran 90/95. It uses a new very fast algorithm that automatically generates calls to Fortran subroutines containing different phase space parameterizations of the considered class of processes. The parameterizations take into account mappings of poles due to the Feynman propagators of unstable heavy particles decaying into 2 or 3 on shell final state particles according to predefined patterns, possible single or double t-channel poles and peaks due to one on shell photon or gluon radiation. The individual subroutines are organized in a single multichannel kinematics subroutine which can be easily called while computing the phase space integral of the S matrix element as a function of generated particle four momenta, in either the leading or higher orders of the perturbation series. The particle four momenta can be used in a quadruple precision version, if necessary. Program summaryProgram Title:PSGenCPC Library link to program files:https://doi.org/10.17632/y7nprysy8g.1Developer's repository link:https://www.kk.us.edu.pl/psgen.htmlLicensing provisions: GPLv3Programming language: Fortran 90/95Nature of problem: Automatic generation of Fortran routines which allow for the multichannel Monte Carlo integration of user's provided S matrix element as a function of generated particle four momenta of any 2→n, with n=2,...,9, scattering process. The S matrix can be defined in either the leading or higher orders.Solution method: A new very fast algorithm written in Fortran 90/95 generates calls to different handwritten subroutines containing phase space parameterizations which, among others, take into account poles due to the Feynman propagators of unstable heavy particles decaying into 2 or 3 on shell final state particles according to predefined patterns, possible single or double t-channel poles and peaks due to one on shell photon or gluon radiation. The generated routines are moved to a directory, where they are combined in a single properly normalized multichannel phase space parameterization. The corresponding kinematics subroutine is self-consistent and can be readily combined with any program which calculates the S matrix element in either the leading or higher orders of the perturbation series. The particle four momenta can be generated in a quadruple precision version, if necessary. The automatically generated subroutine containing calls to different kinematics subroutines can be easily supplemented with calls to user's own made kinematics subroutines.

ALGORITHM FOR RECOGNITION TECHNOLOGY OF ACOUSTIC SIGNALS

An algorithm for processing the reflected acoustic signal for the technology of object recognition against the background of noise interference is presented. The considered mathematical model takes into account that the input data (acoustic signals) have a previously unknown probability distribution. To build an acoustic signal processing technology the Markov chain based probabilistic models, the Monte Carlo integration method and the Bootstrap filter method built on it were used. Based on them, an algorithm for processing an acoustic signal reflected from obstacles was created. The signal processing technology implements the transition from the input data to the representation of the input data in the form of a grid, where a characteristic-intensity normally distributed according to the central limit theorem is assigned to each cell. A likelihood function is used to detect noise at a random point. The algorithm is ready for practical use, which is confirmed by examples. The result of the algorithm application for the recognition of two moving objects, which have a reflected acoustic signal close to the noise, is graphically presented. The dependence of the processing speed on the amount of data is noted. Several constructive approaches to solving this problem are given. It is noted that by using the methods of storing sparse matrices the estimate of the memory used is reduced from quadratic to linear with deterioration of the speed estimate by a logarithm.

Quantum Metropolis-Hastings algorithm with the target distribution calculated by quantum Monte Carlo integration

Quantum Metropolis-Hastings algorithm with the target distribution calculated by quantum Monte Carlo integration