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DOE OSTI2023

Efficient Subset Simulation using Hamiltonian Neural Network enhanced Markov Chain Monte Carlo Methods

The Monte Carlo method delivers an unbiased estimate of the probability of failure. However, the variance of the estimate depends on the number of evaluated samples. This number must be very large for estimations of a low probability of failure. If the evaluation of each sample is computationally expensive, the crude Monte Carlo simulation strategy is impracticable. Therefore, subset simulations are used to reduce the required number of evaluations. Subset simulations require a Markov Chain Monte Carlo sampler, such as the random walk Metropolis-Hastings algorithm. The algorithm, however, struggles with sampling in low-probability regions, especially if they are narrow. As a consequence, advanced Markov Chain Monte Carlo simulations have been developed. In particular, the Hamiltonian Monte Carlo method explores the target distribution rapidly. Driven by the idea of Hamiltonian dynamics, this sampler provides a non-random walk through the target distribution. The incorporation of subset simulation and Hamiltonian Monte Carlo methods has shown promising results for reliability analysis. One downside of the Hamiltonian Monte Carlo method is that gradient evaluations are computationally expensive, especially when dealing with high-dimensional problems and evaluating long trajectories. We show that integrating Hamiltonian neural networks in Hamiltonian Monte Carlo simulations significantly speeds up the sampling task. Furthermore, the enhancement of adaptive trajectory length within the Hamiltonian Monte Carlo results in the efficient proposal of the following states. Based on this recent enhancement, we provide a fast sampling strategy for subset simulations using Hamiltonian neural networks to replace the evaluation of the gradient and significantly speed up the Hamiltonian Monte Carlo simulation.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Optimized structure and electronic band gap of monolayer GeSe from quantum Monte Carlo methods

Here, we have used highly accurate quantum Monte Carlo methods to determine the chemical structure and electronic band gaps of monolayer GeSe. Two-dimensional (2D) monolayer GeSe has received a great deal of attention due to its unique thermoelectric, electronic, and optoelectronic properties with a wide range of potential applications. Density functional theory (DFT) methods have usually been applied to obtain optical and structural properties of bulk and 2D GeSe. For the monolayer, DFT typically yields a larger band-gap energy than for bulk GeSe but cannot conclusively determine if the monolayer has a direct or indirect gap. Moreover, the DFT-optimized lattice parameters and atomic coordinates for monolayer GeSe depend strongly on the choice of approximation for the exchange-correlation functional, which makes the ideal structure-and its electronic properties-unclear. In order to obtain accurate lattice parameters and atomic coordinates for the monolayer, we use a surrogate Hessian-based parallel line search within diffusion Monte Carlo to fully optimize the GeSe monolayer structure. The DMC-optimized structure is different from those obtained using DFT, as are calculated band gaps. The potential energy surface has a shallow minimum at the optimal structure. This, combined with the sensitivity of the electronic structure to strain, suggests that the optical properties of monolayer GeSe are highly tunable by strain.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Magnetic structure of A ≤ 10 nuclei using the Norfolk nuclear models with quantum Monte Carlo methods

Here we present quantum Monte Carlo calculations of magnetic moments, form factors, and densities of A ≤ 10 nuclei within a chiral effective field theory approach. We use the Norfolk two- and three-body chiral potentials and their consistent electromagnetic one- and two-nucleon current operators. We find that two-body contributions to the magnetic moment can be large (up to ≈ 33% in A = 9 systems). We study the model dependence of these observables and place particular emphasis on investigating their sensitivity to using different cutoffs to regulate the many-nucleon operators. Calculations of elastic magnetic form factors for A ≤ 10 nuclei show excellent agreement with the data out to momentum transfers q ≈ 3 fm -1 .

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

High Accuracy Transition Metal Effective Cores for the Many-Body Diffusion Monte Carlo Method

Practical applications of the real-space diffusion Monte Carlo (DMC) method require the removal of core electrons, where currently localization approximations of semilocal potentials are generally used in the projector. Accurate calculations of complex solids and large molecules demand minimizing the impact of approximated atomic cores. Prior works have shown that the errors from such approximations can be sizable in both finite and periodic systems. In this work, we show that a class of differential pseudopotentials, known as pseudo-Hamiltonians, can be constructed for the 3d transition metal atoms, entirely removing the need for any localization scheme in the DMC projector. As a proof of principle, we demonstrate the approach for the case of Co. In order to minimize errors in the pseudo-Hamiltonian at the many-body level, we generalize the recently proposed correlation-consistent pseudopotential generation scheme to successively close semilocal representations of the differential potentials. Our generation scheme successfully produces potentials tailored specifically for real space projector quantum Monte Carlo methods with low error at the many-body level, i.e., with many-body scattering properties very close to relativistic all-electron results. In particular, we show that the agreement with respect to atomic and molecular quantities reach chemical accuracy in many cases-on par with the most accurate semilocal pseudopotentials available. Further, our pseudo-Hamiltonian generation scheme utilizes standard quantum chemistry codes designed only to work with semilocal pseudopotentials, enabling straightforward generation of pseudo-Hamiltonians for additional elements in future works.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Quantum Monte Carlo Methods for Astrophysical Applications

In recent years, new astrophysical observations have provided a wealth of exciting input for nuclear physics. For example, the observations of two-solar-mass neutron stars put strong constraints on possible phase transitions to exotic phases in strongly interacting matter at high densities. Furthermore, the recent observation of a neutron-star merger in both the electromagnetic spectrum and gravitational waves has provided compelling evidence that neutron-star mergers are an important site for the production of extremely neutron-rich nuclei within the r-process. In the coming years, an abundance of new observations is expected, which will continue to provide crucial constraints on the nuclear physics of these events. To reliably analyze such astrophysical observations and extract information on nuclear physics, it is very important that a consistent approach to nuclear systems is used. Such an approach consists of a precise and accurate method to solve the nuclear many-body problem in nuclei and nuclear matter, combined with modern nuclear Hamiltonians that allow to estimate the theoretical uncertainties. Quantum Monte Carlo methods are ideally suited for such an approach and have been successfully used to describe atomic nuclei and nuclear matter. In this contribution, I will present a detailed description of Quantum Monte Carlo methods focusing on the application of these methods to astrophysical problems. In particular, I will discuss how to use Quantum Monte Carlo methods to describe nuclear matter of relevance to the physics of neutron stars.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Adaptive, Active Learning, and Multifidelity Monte Carlo Methods in the MOOSE Stochastic Tools Module

MOOSE is an open-source computational platform for constructing multi-physics models and executing them in a massively parallel fashion. It has a stochastic tools module (STM) for forward/inverse uncertainty quantification (UQ) and surrogate modeling. This presentation details some recent developments to the STM with respect to the implementation of adaptive, active learning, and multifidelity Monte Carlo methods for forward UQ of computational models. Specifically, the adaptive Monte Carlo methods include Markov Chain Monte Carlo (MCMC)-driven algorithms like adaptive importance sampling and parallelized subset simulation for statistical QoI estimation, rare events analysis, and stochastic gradient-free optimization. The active learning methods include Gaussian Process (GP) surrogates and their training via Adam optimization, design of acquisition functions, and integration with samplers like Monte Carlo, adaptive importance, and parallelized subset simulation. These active learning methods are also designed to work in a batch mode, wherein, the required calls to the full computational model are executed in parallel whenever a user-specified batch size is met. The multifidelity methods in STM are broadly divided into two categories: hierarchical, where a defined hierarchy exists among the low-fidelity models, and peer, where all the low-fidelity models are treated equally. A GP surrogate is used to learn the differences between the low- and high-fidelity models in both multifidelity categories, and acquisition functions from the active learning classes are used to decide whether to rely on a low-fidelity model or call the expensive high-fidelity model. Alongside the software description and usage, applications are also presented to nuclear engineering computational models including a TRISO nuclear fuel particle, a reactor pressure vessel, and a heat-pipe microreactor.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Advanced Multilevel Monte Carlo Methods

This article reviews the application of some advanced Monte Carlo techniques in the context of multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations, which can be biassed in some sense, for instance, by using the discretization of an associated probability law. The MLMC approach works with a hierarchy of biassed approximations, which become progressively more accurate and more expensive. Additionally, using a telescoping representation of the most accurate approximation, the method is able to reduce the computational cost for a given level of error versus i.i.d. sampling from this latter approximation. All of these ideas originated for cases where exact sampling from couples in the hierarchy is possible. This article considers the case where such exact sampling is not currently possible. We consider some Markov chain Monte Carlo and sequential Monte Carlo methods, which have been introduced in the literature, and we describe different strategies that facilitate the application of MLMC within these methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Uncertainty quantification (UQ) includes the characterization, integration, and propagation of uncertainties that result from stochastic variations and a lack of knowledge or data in the natural world. Monte Carlo (MC) method is a sampling-based approach that has widely used for quantification and propagation of uncertainties. However, the standard MC method is often time-consuming if the simulation-based model is computationally intensive. This article gives an overview of modern MC methods to address the existing challenges of the standard MC in the context of UQ. Specifically, multilevel Monte Carlo (MLMC) extending the concept of control variates achieves a significant reduction of the computational cost by performing most evaluations with low accuracy and corresponding low cost, and relatively few evaluations at high accuracy and corresponding high cost. Multifidelity Monte Carlo (MFMC) accelerates the convergence of standard Monte Carlo by generalizing the control variates with different models having varying fidelities and varying computational costs. Multimodel Monte Carlo method (MMMC), having a different setting of MLMC and MFMC, aims to address the issue of UQ and propagation when data for characterizing probability distributions are limited. Multimodel inference combined with importance sampling is proposed for quantifying and efficiently propagating the uncertainties resulting from small data sets. All of these three modern MC methods achieve a significant improvement of computational efficiency for probabilistic UQ, particularly uncertainty propagation. Finally, an algorithm summary and the corresponding code implementation are provided for each of the modern MC methods. The extension and application of these methods are discussed in detail.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Quantum Monte Carlo Method in the Steady State

We present a numerically exact steady-state inchworm Monte Carlo method for nonequilibrium quantum impurity models. Rather than propagating an initial state to long times, the method is directly formulated in the steady state. This eliminates any need to traverse the transient dynamics and grants access to a much larger range of parameter regimes at vastly reduced computational costs. We benchmark the method on equilibrium Green’s functions of quantum dots in the noninteracting limit and in the unitary limit of the Kondo regime. We then consider correlated materials described with dynamical mean field theory and driven away from equilibrium by a bias voltage. We show that the response of a correlated material to a bias voltage differs qualitatively from the splitting of the Kondo resonance observed in bias-driven quantum dots.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2020

Simulating sticky particles: A Monte Carlo method to sample a stratification

Many problems in materials science and biology involve particles interacting with strong, short-ranged bonds that can break and form on experimental timescales. Treating such bonds as constraints can significantly speed up sampling their equilibrium distribution, and there are several methods to sample probability distributions subject to fixed constraints. We introduce a Monte Carlo method to handle the case when constraints can break and form. More generally, the method samples a probability distribution on a stratification: a collection of manifolds of different dimensions, where the lower-dimensional manifolds lie on the boundaries of the higher-dimensional manifolds. We show several applications of the method in polymer physics, self-assembly of colloids, and volume calculation in high dimensions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Monte Carlo method for constructing confidence intervals with unconstrained and constrained nuisance parameters in the NOvA experiment

Measuring observables to constrain models using maximum-likelihood estimation is fundamental to many physics experiments. Wilks' theorem provides a simple way to construct confidence intervals on model parameters, but it only applies under certain conditions. These conditions, such as nested hypotheses and unbounded parameters, are often violated in neutrino oscillation measurements and other experimental scenarios. Monte Carlo methods can address these issues, albeit at increased computational cost. In the presence of nuisance parameters, however, the best way to implement a Monte Carlo method is ambiguous. Furthermore, this paper documents the method selected by the NOvA experiment, the profile construction. It presents the toy studies that informed the choice of method, details of its implementation, and tests performed to validate it. It also includes some practical considerations which may be of use to others choosing to use the profile construction.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Introduction to the Monte Carlo Method [Slides]

The goal here is to introduce you to the Monte Carlo (MC) method. MC is a very powerful simulation technique. Very simple examples will be used. Potential MC users have diverse backgrounds, but you should see from the examples how the method could be applied to your field of study.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

A Monte-Carlo method for Coulomb collisions in moderately coupled plasmas

Here, we introduce a new approach for coupling cumulative and single-scattering distributions, extending Monte Carlo methods for Coulomb collisions in weakly coupled plasmas into the moderately coupled regime. Using time steps typical of cumulative-scattering methods, we show that this approach not only recovers leading-order corrections proportional to the inverse Coulomb logarithm but also solves the full Boltzmann collision integral. A key feature of the method is its simplicity: it can be integrated into cumulative-scattering methods for weakly coupled plasmas with only ~ 10 additional lines of code.

Boltzmann collision integral↗

DOE OSTI2023

Metals and Quantum Materials with Spin-orbit Interactions by Quantum Monte Carlo methods

The key goals of this project were as follows: 1) Analysis and benchmarks of electron correlation effects recovered in the fixed-node approximation that is inherent to quantum Monte Carlo (QMC) method as applied to metallic states; 2) development of new algorithms for electron spin-degrees of freedom to be treated as explicit quantum variables; 3) designing electronic structure QMC algorithm for efficient evaluation of spin-orbit effects in systems with heavy atoms; 4) adapting the algorithm to complex wave functions and developing corresponding fixed-phase approximation; 5) design and testing of algorithm for valence-only non-local spin-orbit operators; 6) analysis of fixed-node vs fixed-phase errors and their comparisons. The key accomplishments: i) We carried out a systematic study of Li systems by the fixed-node diffusion Monte Carlo method. This involved Li atom, molecule, cluster and solid calculated by the full range of QMC methods including fixed-node QMC.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2026

Renovating Monte Carlo Methods and Codebases with Generative Models

The code will implement a standardized interface for Monte Carlo sampling methods, including conventional techniques, and going beyond current available packages to also incorporate generative model-enabled Monte Carlo sampling to provide a unified framewor

Garcia-Cardona, Cristina↗

DOE OSTI2025

Efficient analysis of small-angle scattering curves for large biomolecular assemblies using Monte Carlo methods

Structure elucidation from small-angle scattering curves of large biomolecular assemblies is notoriously challenging. This is because the simulation of high-resolution features in the structure of large macromolecular assemblies, such as de novo protein assemblies, is computationally demanding when it needs to cover a broad range of length scales. Conventional methods, such as the numerical approximation to the Debye equation or the use of spherical harmonics, do not scale well as the size of the assembly increases, which limits their application to small structures (e.g. individual proteins). This work explores the effectiveness of a Monte Carlo method to simulate and fit scattering curves for large biomolecular assemblies spanning over ranges covering atomic and molecular detail (e.g. spacing and orientation of proteins in an assembly) as well as large-scale (hundreds of nanometres) features. Owing to its speed and scalability, it can be combined with a fitting algorithm to extract structural features from experimental small-angle scattering curves in biomolecular assemblies that are otherwise intractable for interpretation. This work first demonstrates the effectiveness of the tool using experimental small-angle X-ray scattering (SAXS) data from tile-like proteins that assemble into 1D tube-like macromolecular structures. Here, the diameter distribution of tubes is extracted from SAXS fits, and this is quantitatively compared with distributions from electron microscopy. SAXS data are also obtained from 2D sheet-like protein assemblies, and the proposed method is used to quantify structural features such as the separation distance between protein building blocks and the flexing of the sheet. An open-source implementation of the methodology is provided for use in a broad range of biological systems involving multi-scale scattering analysis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad-Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costs by performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductions typically on the order of 60 or more and possibly as large as 200. Furthermore, adaptive griding results in more accurate computation of geometric quantities such as x-points associated with the model.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Chiral Effective Field Theory’s Impact on Advancing Quantum Monte Carlo Methods

Thirty years ago, Steven Weinberg published his seminal paper on “Nuclear Forces from chiral Lagrangians” which has revolutionized the field of theoretical nuclear physics. Nowadays, interactions derived from chiral effective field theory are routinely used to describe nuclear systems ranging from atomic nuclei to the dense matter explored in the core of neutron stars with theoretical uncertainty estimates. Here, in our contribution to the special issue “Celebrating 30 years of Steven Weinberg’s paper Nuclear Forces from Chiral Lagrangians”, we focus on the impact that chiral effective field theory interactions have played in advancing microscopic studies of atomic nuclei and the nuclear-matter equation of state using quantum Monte Carlo methods.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗