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A new gradient-free active subspace estimation method with application to rare event probability estimation

Valentin Breaz, Miguel Munoz Zuniga, Olivier Zahm, Richard Wilkinson

Abstract

To reduce the cost of estimating the probability of a rare event involving a very large number of random parameters, we propose a new strategy for dimension reduction coupled with a surrogate model for the expensive part of the algorithm. To this end, we extend the Ordinary Kriging Active Subspace (OK-AS) method into a sequential version. Our approach consists of iteratively re-estimating the active subspace using a Kriging surrogate trained in a rotated coordinate system until the active subspace stabilises. This method allows for a reduction in prediction error and a better approximation of the active subspace on a benchmark of test problems. Furthermore, we integrate our algorithm into an efficient pre-existing approach for estimating the probability of a rare event. This approach is based on learning the active subspace associated with the random event whose probability is to be estimated. The sequential learning of an importance sampling density is necessary and corresponds to the expensive part of this strategy. To circumvent this issue, we integrate our sequential OK-AS version into the estimation of the importance sampling density. The numerical results indicate that our method allows for reducing the cost required to obtain a precise estimate of the rare event probability.

A new gradient-free active subspace estimation method with application to rare event probability estimation

Abstract

To reduce the cost of estimating the probability of a rare event involving a very large number of random parameters, we propose a new strategy for dimension reduction coupled with a surrogate model for the expensive part of the algorithm. To this end, we extend the Ordinary Kriging Active Subspace (OK-AS) method into a sequential version. Our approach consists of iteratively re-estimating the active subspace using a Kriging surrogate trained in a rotated coordinate system until the active subspace stabilises. This method allows for a reduction in prediction error and a better approximation of the active subspace on a benchmark of test problems. Furthermore, we integrate our algorithm into an efficient pre-existing approach for estimating the probability of a rare event. This approach is based on learning the active subspace associated with the random event whose probability is to be estimated. The sequential learning of an importance sampling density is necessary and corresponds to the expensive part of this strategy. To circumvent this issue, we integrate our sequential OK-AS version into the estimation of the importance sampling density. The numerical results indicate that our method allows for reducing the cost required to obtain a precise estimate of the rare event probability.

Paper Structure

This paper contains 22 sections, 21 equations, 6 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Abstract
  2. Introduction
  3. Dimension reduction for a high-dimensional Gaussian Process
  4. Summary review of existing methodologies
  5. Kernel Dimension Reduction (KDR).
  6. Embedding Learning (EL).
  7. Partial Least Squares (PLS).
  8. Other linear dimension reduction approaches.
  9. Active Subspace (AS).
  10. Gradient-free Active Subspaces.
  11. New method: Sequential Ordinary Kriging Active Subspace (seqOK-AS)
  12. Application to high-dimensional rare event probability estimation
  13. The importance sampling framework.
  14. Strategy via smooth approximation and dimension reduction.
  15. Towards an algorithm.
  16. ...and 7 more sections

Figures (6)

  • Figure 1: Validation of the seqOK-AS method on the quadratic problem (solid line: average over 10 trials)
  • Figure 2: Validation of seqOK-AS on the more realistic models (solid line: average over 10 trials)
  • Figure 3: Method comparison for GP surrogate modeling and AS estimation for the quadratic test problems
  • Figure 4: Method comparison for GP surrogates and AS estimation for the more realistic test problems
  • Figure 5: GP surrogate comparison for the quadratic test problems (seqOK-AS versus popular methods)
  • ...and 1 more figures