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Computing conservative probabilities of rare events with surrogates

Nicolas Bousquet

TL;DR

This article provides a critical review of the main methods used to produce conservative estimators of probabilities of rare events, or critical failures, for reliability and certification studies in the broadest sense and suggests avenues of research to improve the guarantees currently reachable.

Abstract

This article provides a critical review of the main methods used to produce conservative estimators of probabilities of rare events, or critical failures, for reliability and certification studies in the broadest sense. These probabilities must theoretically be calculated from simulations of (certified) numerical models, but which typically suffer from prohibitive computational costs. This occurs frequently, for instance, for complex and critical industrial systems. We focus therefore in adapting the common use of surrogates to replace these numerical models, the aim being to offer a high level of confidence in the results. We suggest avenues of research to improve the guarantees currently reachable.

Computing conservative probabilities of rare events with surrogates

TL;DR

This article provides a critical review of the main methods used to produce conservative estimators of probabilities of rare events, or critical failures, for reliability and certification studies in the broadest sense and suggests avenues of research to improve the guarantees currently reachable.

Abstract

This article provides a critical review of the main methods used to produce conservative estimators of probabilities of rare events, or critical failures, for reliability and certification studies in the broadest sense. These probabilities must theoretically be calculated from simulations of (certified) numerical models, but which typically suffer from prohibitive computational costs. This occurs frequently, for instance, for complex and critical industrial systems. We focus therefore in adapting the common use of surrogates to replace these numerical models, the aim being to offer a high level of confidence in the results. We suggest avenues of research to improve the guarantees currently reachable.
Paper Structure (13 sections, 3 theorems, 37 equations, 7 figures, 4 tables, 1 algorithm)

This paper contains 13 sections, 3 theorems, 37 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Surrogate-based methods
  3. Ensuring conservatism
  4. Obtaining deterministic bounds on $p$ from geometrical properties
  5. Lipschitz smoothness
  6. Monotonicity
  7. Convexity and quasi-convexity.
  8. Biasing surrogates to get conservative failure probabilities
  9. Principle
  10. Using Bernstein concentration inequalities
  11. Learning minimal bias
  12. Discussion
  13. Appendix: Semi-adaptive sampling and computing bounds within staircase subspaces

Key Result

Theorem 1

Under the following assumptions: then where with, for $d\geq 2$, this latter result being the optimal convergence rate, in the sense it cannot be improved by any other algorithm defined under the sole general assumptions.

Figures (7)

  • Figure 1: An illustration of various levels of partitioning $[0,1]^2$ by a family of dyadic cubes.
  • Figure 2: An illustration in dimension 2 of the bounding principle due to monotonicity. On the left is a set $A$ of $m$ points included in $[0,1]^d$. On the right, we plot the dominated spaces $\mathbb{U}^-(A)$ and $\mathbb{U}^+(A)$ surrounding $\Gamma$. It is clear that $\mathbb{U}^-(A)\preceq_P \Gamma \preceq_P \mathbb{U}^+(A)$. The volume of these subspaces allows to compute deterministic bounds for $p$: $\lambda(\mathbb{U}^-(A))\leq p \leq 1 - \lambda(\mathbb{U}^+(A))$, with $\lambda$ the Lebesgue measure.
  • Figure 3: Average calculation time (in seconds) as a function of size. "No MCMC" means "brute force Monte Carlo." Figure extracted from bergere2021.
  • Figure 4: Values of $\lambda(n,p)$ with $C=6$, in function of $\log(n)$, for some values of $p$.
  • Figure 5: Values of $log(n)$ such that $\lambda(n,p)\simeq p$, for some typical values of $p$.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1: Dyadic cubes.
  • Theorem 1: Derived from bernard2021recursive.
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 1
  • Theorem 2: Inspired from Ducoffe2020, Corollary 1
  • Corollary 1