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Hamiltonian Monte Carlo Methods for Subset Simulation in Reliability Analysis

Ziqi Wang, Marco Broccardo, Junho Song

arXiv:1706.01435v2stat.CO

TL;DR

Small failure probabilities make crude Monte Carlo inefficient, and Subset Simulation depends on effective sampling from nested conditional distributions. This paper develops two HMC-based Subset Simulation approaches and finds them at least as accurate as conventional Metropolis-Hastings, with efficiency advantages reported for some cases. The methods remain subject to approximation and tuning limitations.

  • Problem

    Subset Simulation requires efficient MCMC sampling of conditional distributions, but conventional random-walk methods can limit exploration and HMC had not been studied for reliability analysis.

  • Method

    The paper develops two HMC approaches for Subset Simulation: rejection sampling HMC and barrier-bouncing HMC.

  • Results

    Across reliability examples, both HMC-based Subset Simulation methods are at least as accurate as conventional Metropolis-Hastings, while RS-HMC-SS is more efficient at low probability levels.

  • Takeaways & Limitations

    HMC provides a non-random-walk sampling alternative that generally explores probability space more effectively than traditional random-walk Metropolis-Hastings.

  • Takeaways & Limitations

    BB-HMC is approximate because hitting time and direction estimates prevent rigorous detailed balance, while tuning partial momentum refreshment remains open.

Abstract

from arXiv · show

This paper studies a non-random-walk Markov Chain Monte Carlo method, namely the Hamiltonian Monte Carlo (HMC) method in the context of Subset Simulation used for structural reliability analysis. The HMC method relies on a deterministic mechanism inspired by Hamiltonian dynamics to propose samples following a target probability distribution. The method alleviates the random walk behavior to achieve a more effective and consistent exploration of the probability space compared to standard Gibbs or Metropolis-Hastings techniques. After a brief review of the basic concepts of the HMC method and its computational details, two algorithms are proposed to facilitate the application of the HMC method to Subset Simulation in structural reliability analysis. Next, the behavior of the two HMC algorithms is illustrated using simple probability distribution models. Finally, the accuracy and efficiency of Subset Simulation employing the two HMC algorithms are tested using various reliability examples. The supporting source code and data are available for download at (the URL that will become available once the paper is accepted).

1. Introduction

General reliability solutions are usually unavailable, while crude Monte Carlo becomes inefficient for small failure probabilities. The paper therefore studies Hamiltonian Monte Carlo as a non-random-walk alternative within Subset Simulation.

  • Motivation: Subset Simulation reduces computational cost by expressing failure as nested intermediate domains and multiplying their conditional probabilities.The intermediate conditional probabilities are substantially larger than the target failure probability.
  • Research problem: Subset Simulation accuracy and efficiency depend directly on the MCMC method used to sample each conditional distribution.Existing practice primarily uses random-walk MCMC methods.
  • Research problem: The paper investigates HMC, a non-random-walk MCMC method, for generating samples in Subset Simulation reliability analysis.HMC uses a deterministic mechanism inspired by Hamiltonian dynamics.
  • Novelty: The study addresses an application gap because HMC had not previously been studied for reliability analysis, and compares it with random-walk Metropolis-Hastings.The comparison examines both accuracy and efficiency.
  • Paper scope: The paper reviews HMC, develops computational procedures for HMC-based Subset Simulation, and evaluates them on simple distributions and reliability examples.The numerical studies include standard normal and non-Gaussian spaces, analytical limit states, and structural reliability examples.

2. Principles of Subset Simulation

Subset Simulation estimates small failure probabilities by decomposing the failure event into nested intermediate domains. Conditional samples from each truncated distribution are generated with MCMC and combined into the final probability estimate.

  • Reliability formulation: Reliability probabilities are formulated in standard normal space after transforming the original variables, allowing Subset Simulation to use a truncated multivariate standard normal density.The transformation expresses x in terms of independent standard normal variables u.
  • Nested failure domains: Subset Simulation represents the target failure domain as a sequence of nested intermediate domains and expresses its probability as a product of conditional probabilities.The domains satisfy F1 ⊃ F2 ⊃ . . . ⊃ FM, with F = FM.
  • Conditional sampling: MCMC generates samples from each conditional or truncated distribution so the intermediate probabilities can be estimated by Monte Carlo simulation.The samples at level j follow ϕ(u|Fj−1).
  • Adaptive construction: Intermediate domains are selected adaptively so their conditional probabilities approximately equal a prescribed percentile p0.This construction continues through the intermediate levels before estimating the final failure probability.
  • Estimator uncertainty: The failure-probability estimator has bias from sample correlation and adaptive subsets, but its O(N−1) order is considered negligible relative to the estimate’s coefficient of variation.The coefficient of variation depends on subset-level variation and correlation in the generated Markov chains.

3. General concepts of Hamiltonian Monte Carlo method

HMC augments target positions with momentum and evolves them deterministically along Hamiltonian trajectories. Reversibility, energy-related invariance, and volume preservation support valid MCMC transitions that can explore the target distribution efficiently.

  • Hamiltonian mechanics: The Hamiltonian combines potential energy V(q), determined by position, with kinetic energy K(p), determined by momentum.Given initial position and momentum, this defines the system’s energy level.
  • Fundamental properties: Hamiltonian dynamics provides reversible transitions, enabling Markov chains that preserve the stationary target distribution.Reversibility supplies a unique inverse mapping for the deterministic evolution.
  • Fundamental properties: Volume preservation means Hamiltonian proposals do not require a Jacobian correction, while symplecticness motivates suitable numerical integrators.The unit Jacobian follows from the volume-preserving phase-space transformation.
  • Target-distribution connection: HMC is formulated here for continuous distributions with differentiable PDFs by connecting the target probability space to a position-momentum Hamiltonian system.The position is identified with the outcome and momentum variables expand the state space.
  • HMC construction: HMC samples a target distribution by introducing auxiliary momentum and evolving position and momentum together on an equi-Hamiltonian trajectory.The position follows the target distribution after the momentum component is projected out.
  • Sampling procedure: The HMC procedure first samples momentum, then integrates Hamilton’s equations for a specified time, with momentum negation supporting reversibility and detailed balance.The joint position-momentum distribution remains invariant under these steps.

4. Hamiltonian Monte Carlo method for Subset Simulation

HMC-SS uses Hamiltonian dynamics to generate samples from conditional distributions in Subset Simulation, with rejection-sampling and barrier-bouncing variants that address the failure-domain boundary differently. The method includes tuning and momentum-refreshment strategies to control acceptance, exploration, and random-walk behavior.

  • Framework: HMC-SS samples conditional distributions in Subset Simulation using Hamiltonian dynamics, with formulations for truncated normal and generic target PDFs.The framework applies to standard normal space and, when the target-density gradient exists, directly to non-Gaussian probability spaces.
  • Hamiltonian formulation: The Hamiltonian combines the conditional-density potential with kinetic energy, while the indicator term creates an infinite potential barrier outside the intermediate failure domain.With M^-1 set to the identity, momentum follows N(0, I); states outside Fj cannot be reached by the Hamiltonian system.
  • Barrier bouncing based HMC (BB-HMC): Barrier-bouncing HMC instead reflects the Hamiltonian trajectory at the limit-state surface, while approximate numerical hitting-time and direction calculations can break strict proposal symmetry.The algorithm preserves Hamiltonian energy, but its accuracy depends on the approximations used for the hitting time and direction vector.
  • Rejection sampling based HMC (RS-HMC): Rejection-sampling HMC proposes states along analytical Hamiltonian trajectories and accepts them using the failure-domain criterion.The method starts from a seed position and random momentum, evolves to a specified tf, then applies an acceptance step; momentum negation returns rejected states to their initial position.
  • Adaptation and limitations: The standard-normal trajectory is periodic, motivating tf choices within [−π/2, π/2] and randomization or avoidance of tf = 2π to prevent periodic chains.The mass matrix plays the role of a cross-correlation matrix and must be symmetric and positive definite.
  • Adaptation and limitations: The trajectory time tf is tuned using acceptance-rate feedback, but low probability levels may require small tf values that increase random-walk behavior.Partial momentum refreshment is proposed to suppress this effect; its tuning parameter α remains an open question beyond the study’s scope.

5.1. Behavior of HMC

Across simple Gaussian, constrained, and non-Gaussian examples, HMC generally explores target distributions with less random-walk behavior than Metropolis-based methods, while parameter choices and barrier handling affect performance and validity.

  • Bivariate standard normal distribution: HMC reaches the high-density region faster, has lower sample autocorrelation, and estimates the target CDF or CCDF more effectively than CW-MH.These benefits matter for Subset Simulation because lower autocorrelation can reduce estimator variability and better conditional-probability estimates can reduce bias.
  • Bivariate standard normal distribution: Poor trajectory settings can increase HMC's random-walk behavior, but partial momentum refreshment can suppress it and improve exploration.The comparison uses tf = π/10 with α = 0 versus α = 0.5.
  • Bivariate standard normal distribution: Setting α = 1 produces an elliptical periodic orbit, trapping the chain and preventing effective exploration because the Markov chain is not ergodic.The limiting case therefore cannot be used for statistical computing purposes.
  • Truncated bivariate standard normal distribution: In the constrained distribution, RS-HMC and BB-HMC show less random-walk behavior than CW-MH, with BB-HMC especially effective but computationally more expensive.The comparison uses 1,000 iterations and includes trajectories and first-coordinate values for all three algorithms.
  • Banana-shaped bivariate non-Gaussian distribution: For the banana-shaped distribution, HMC reaches the bulk of the probability density in one iteration, whereas random-walk Metropolis-Hastings proposals are highly likely to be rejected.The example compares 100 HMC iterations with 100 Metropolis-Hastings iterations.

5.2. Reliability example in standard normal space

The study evaluates HMC-based Subset Simulation against conventional Metropolis-Hastings sampling across standard-normal reliability examples, varying probability level, dimension, curvature, and structural setting. HMC-SS is generally at least as accurate as CWMH-SS, while RS-HMC-SS often improves efficiency at low failure probabilities.

  • Linear limit-state function: HMC-SS approaches are at least as accurate as conventional CWMH-SS, while RS-HMC-SS is more efficient at low probability levels for similar evaluation costs.BB-HMC-SS can achieve the lowest coefficient of variation but requires more limit-state evaluations; Newton-Raphson uses fewer evaluations than the secant method at the cost of gradients.
  • Linear limit-state function: Using analytical hitting times makes BB-HMC-SS evaluation costs similar to RS-HMC-SS and CWMH-SS while achieving significantly smaller coefficients of variation.The analytical formulation removes the principal extra cost associated with solving the hitting time numerically.
  • Dimension study: All compared sampling methods show performance that is not sensitive to dimension over the tested range from 10 to 1000.The dimension study fixes β0 = 4 and reports results in Tables 3.a and 3.b.
  • Nonlinear limit-state function: For nonlinear limit states with positive or negative curvature, HMC-SS remains as accurate as CWMH-SS, while RS-HMC-SS is more efficient at low probability levels.The comparison uses n = 100, β0 = 4.0, and varying curvature parameter κ.
  • Structural and non-Gaussian examples: In random-vibration and elliptical-limit-state examples, RS-HMC-SS remains more efficient than conventional MH-SS, while HMC-SS explores narrow probability regions and both tails effectively.The elliptical example attributes this exploration to Hamiltonian dynamics; the random-vibration example compares first-passage estimates across several thresholds.

6. Conclusion

The paper develops two HMC approaches for Subset Simulation and finds them accurate relative to conventional Metropolis-Hastings, with rejection-sampling HMC more efficient and BB-HMC strongest in one studied case. It concludes that HMC is an attractive general alternative while identifying high-dimensional correlated spaces as a target for future improvement.

  • 6. Conclusion: The paper develops HMC for Subset Simulation using deterministic Hamiltonian dynamics to improve exploration beyond random-walk methods.Two variants are proposed: rejection sampling and barrier bouncing when trajectories encounter constraints.
  • 6. Conclusion: The two HMC approaches are at least as accurate as conventional Metropolis-Hastings, while rejection-sampling HMC is noticeably more efficient and BB-HMC outperforms the other approaches in one case.BB-HMC incurs additional computational cost to determine barrier hitting times.
  • 6. Conclusion: The study concludes that HMC provides an attractive and general alternative for MCMC sampling in Subset Simulation.
  • 6. Conclusion: Further work is suggested on Riemann manifold HMC for high-dimensional, strongly correlated spaces and on Bayesian reliability with parameter uncertainties.The latter setting involves sampling complex posterior parameter distributions for reliability updating.
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