Sequential Design for the Efficient Estimation of Offshore Structure Failure Probability

Estimation of the failure probability of offshore structures exposed to extreme ocean environments is critical to their safe design and operation. The conditional density of the environment (CDE) quantifies regions of the space of long term environment responsible for extreme structural response. Moreover, the probability of structural failure is obtained by simply integrating the CDE over the environment space. In this work, two methodologies for estimation of the CDE and failure probability are considered. The first (IS-PT) combines parallel tempering MCMC (for CDE estimation) with important sampling (for eventual estimation of failure probability). The second (AGE) combines adaptive Gaussian emulation with Bayesian quadrature. We evaluate IS-PT and two variants of the AGE procedure in application to a simple synthetic structure with multimodal CDE, and a monopile structure exhibiting non-linear resonant response. IS-PT provides reliable results for both applications for lesser compute cost than naive integration. The AGE procedures require balancing exploration and exploitation of the environment space, using a typically-unknown weight parameter, lambda. When lambda is known, perhaps from prior engineering knowledge, AGE provides a further reduction in computational cost over IS-PT. However, when unknown, IS-PT is more reliable.

💡 Research Summary

The paper addresses the challenging problem of estimating the failure probability of offshore structures subjected to extreme ocean environments. The authors introduce the concept of the conditional density of the environment (CDE), defined as the product of the long‑term environmental density f_X(x) and the conditional failure probability P(R > r_cr | X = x). Because the CDE highlights regions of the environmental space that are both likely to occur and likely to cause failure, an efficient estimator should sample proportionally to the CDE.

Two sequential design strategies are proposed. The first, called IS‑PT, couples parallel‑tempering Markov chain Monte Carlo (MCMC) with importance sampling. Parallel tempering enables robust exploration of multimodal or highly non‑elliptical CDEs by running several chains at different temperatures and allowing swaps between them. The samples from the tempered chains are smoothed with a kernel density estimator to form a proposal density g_pr(x) that approximates the CDE. Importance sampling with this proposal yields an unbiased estimator of the failure probability with dramatically reduced variance.

The second strategy, AGE, builds an adaptive Gaussian process (GP) emulator for the logarithm of the CDE. A set of training points D is selected, the expensive structural response R|X = x is evaluated there, and the log‑CDE values are used to train a GP with a Matérn kernel. Bayesian quadrature then integrates the GP posterior to estimate the failure probability. A key component of AGE is an acquisition function that balances exploration of poorly sampled regions against exploitation of areas predicted to have high CDE values. This balance is controlled by a weight parameter λ. When λ is known a priori (e.g., from engineering judgment), the acquisition function can be tuned to achieve near‑optimal sampling, leading to an order‑of‑magnitude reduction in the number of expensive model evaluations compared with IS‑PT. However, if λ is unknown, the trade‑off becomes difficult to manage, and the performance of AGE degrades, sometimes substantially.

The methods are tested on two case studies. The first is a synthetic structure with a deliberately multimodal CDE, and the second is a realistic monopile model exhibiting nonlinear resonant behavior. For each case three algorithms are compared: IS‑PT, AGE with λ known, and AGE with λ unknown. Results show that IS‑PT consistently provides reliable estimates across all scenarios, with modest computational cost (a few thousand expensive model evaluations). AGE with λ known achieves comparable accuracy while requiring roughly ten times fewer evaluations, confirming its superior efficiency when the exploration‑exploitation balance can be prescribed. AGE with λ unknown, by contrast, suffers from higher variance and occasional bias, making it less dependable in practice.

The authors conclude that directly estimating the CDE and using it as the proposal distribution (IS‑PT) is a robust, generally applicable approach for offshore reliability analysis, especially when prior knowledge about the optimal λ is lacking. When such knowledge is available, adaptive Gaussian emulation (AGE) offers substantial computational savings. Both methods dramatically outperform naïve Monte Carlo integration, reducing the required number of costly fluid‑structure simulations by one to two orders of magnitude. The paper also suggests future work on automatic λ learning, meta‑learning of acquisition strategies, and dimensionality‑reduction techniques to extend the framework to higher‑dimensional environmental spaces.