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arxiv: 2511.23156 · v3 · submitted 2025-11-28 · 📊 stat.AP

Design loads for wave impacts -- introducing the Probabilistic Adaptive Screening (PAS) method for predicting extreme non-linear loads on maritime structures

Sanne M. van Essen , Harleigh C. Seyffert This is my paper

Pith reviewed 2026-05-17 04:24 UTC · model grok-4.3

classification 📊 stat.AP
keywords wave impact loadsextreme value predictionprobabilistic screeningcopula modelingmulti-fidelity methodsmaritime structuresnonlinear loadsadaptive sampling
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The pith

The Probabilistic Adaptive Screening method predicts extreme wave impact loads on maritime structures by mapping cheap linear indicators to nonlinear results with 2-15% accuracy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Wave impact loads on ships and offshore platforms are rare and complex, making it expensive to run enough high-fidelity simulations to capture their extremes reliably. The paper introduces the Probabilistic Adaptive Screening method, which combines copula-based dependence modeling with multi-fidelity screening and adaptive sampling. Low-fidelity linear potential flow calculations identify candidate events, after which the statistical model predicts the corresponding high-fidelity nonlinear loads. Validation on four test cases shows the method reproduces short-term distributions and most probable maximum values within 2-15% of full Monte Carlo results while using only 1-3% of the usual high-fidelity computation time.

Core claim

The Probabilistic Adaptive Screening (PAS) method integrates copula-based statistical dependence modelling with multi-fidelity screening and adaptive sampling to predict extreme non-linear wave impact loads. This framework enables efficient extreme value prediction by statistically mapping low-fidelity indicator variables to high-fidelity impact loads. The method allows efficient linear potential flow indicators to be used in the low-fidelity stage even for strongly non-linear cases. Validation against four non-linear test cases concludes that PAS with optimal settings accurately estimates both short-term distributions and extreme values, with most probable maximum values within 2-15% of the

What carries the argument

The Probabilistic Adaptive Screening (PAS) method, which uses copula-based statistical dependence modelling together with multi-fidelity screening and adaptive sampling to map low-fidelity linear potential flow indicators onto high-fidelity nonlinear impact loads.

If this is right

  • Extreme load statistics for wave impacts can be obtained at roughly one to three percent of the computational cost of conventional Monte Carlo simulation.
  • Linear potential flow indicators remain useful for screening even when the final loads are strongly nonlinear.
  • Short-term distributions of impact loads are reproduced well enough to support design load selection.
  • The same framework applies across weakly and strongly nonlinear regimes including non-linear waves, ship bending moments, green water, and slamming.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the copula mapping generalizes to other vessel types and wave environments, the method could reduce the computational barrier to including realistic impact loads in classification society rules.
  • The adaptive screening idea may transfer to other rare-event fluid-structure problems such as extreme run-up on coastal structures or fatigue from repeated slamming.
  • Combining PAS with longer low-fidelity time series could further improve tail estimates without extra high-fidelity runs.

Load-bearing premise

That copula-based statistical dependence modelling can reliably map low-fidelity linear potential flow indicators to high-fidelity non-linear impact loads even for strongly non-linear phenomena.

What would settle it

Running the PAS procedure on an additional slamming or green-water case with a long brute-force Monte Carlo reference and finding that the predicted most probable maximum lies outside the 2-15% error band reported for the original test cases.

Figures

Figures reproduced from arXiv: 2511.23156 by Harleigh C. Seyffert, Sanne M. van Essen.

Figure 1
Figure 1. Figure 1: Two examples of wave impacts on marine structures: a wind turbine foundation near Fécamp in 2023 (left, photo: K. King) and research vessel Discoverer on the Bering Sea in 1979 (right, photo: R. Behn / NOAA). 𝑥 ′ / 𝑥 ′′ LF / HF version of variable 𝑥 RWE Relative Wave Elevation 𝑛𝑤 # wave encounters in MCS ̂𝑥 MPM of variable 𝑥 VBM Vertical Bending Moment 𝑁 # seeds in MCS 𝑃exp PoE level corr. to 𝑇exp and 𝑇𝑝,𝑒… view at source ↗
Figure 2
Figure 2. Figure 2: Some of the possible statistical levels where multi-fidelity methods can derive or learn the relation between a LF indicator variables (black) and HF non-linear loads (red), including the location of AS and PAS. Modified from [18]. 1.4. Paper objectives & contribution As discussed above and in [18], we need a new EVPM for wave impact loads. Serious incidents with maritime structures still occur at sea, and… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustration of PAS, where the numbers roughly correspond to the method steps in Section 2.1. The left plot only shows a small part of the MCS time traces, and only a few HF samples are included in the middle and right distributions to illustrate the principle. explained that AS was developed at the cumulative distribution level, but in complex cases this approach may have discarded too much info… view at source ↗
Figure 4
Figure 4. Figure 4: Example copula-fitting procedure (here shown for case 3a of the present paper at 40 iterations), with from left to right: scatter diagram of the LF-HF data, the corresponding pseudo-observations in uniform U,V-space, the fitted copula model in U,V-space, and the transformed data from the copula model in the ‘real’ data space. the maximum response over the target exposure duration exceeds the MPM is 62.3%. … view at source ↗
Figure 5
Figure 5. Figure 5: MARIN ferry 2 and the relevant instrumentation around the bow before the CRS SCREAM experiments (left), example green water impact (middle) and example bow-flare slamming impact (right). 3.4. Case 4 (strongly non-linear): bow-flare slamming forces Case 4 studies another strongly non-linear problem, based on the same experiments as case 3. Here we predict extreme values of the bow-flare slamming forces on t… view at source ↗
Figure 6
Figure 6. Figure 6: Case 1 - waves: convergence of one-hour MPM as a function of number of samples (top) and final converged distributions (bottom) from AS and PAS. The copula in the name of the PAS results is the used model in the last iteration. 0 20 40 60 80 100 Number of samples j [-] 0.0 0.5 1.0 1.5 2.0 3 0-min V 00hog [N m] 1e9 True MCS converged PAS: +2.5% converged AS: -4.4% Indicator: V 0hog PAS, Gaussian, MD PAS sam… view at source ↗
Figure 7
Figure 7. Figure 7: Case 2 - VBM: convergence of one-hour MPM as a function of number of samples (top) and final converged distributions (bottom) from AS and PAS. The copula in the names of the PAS results is the used model in the last iteration. van Essen, Seyffert: Preprint submitted to Elsevier Page 14 of 27 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Case 3 - green water: convergence of one-hour MPM as a function of number of samples (top) and final converged distributions (bottom) from AS and PAS with indicator 𝑅′ 19𝑝 , where GP fit = Generalised Pareto fit to 30% highest true validation peaks. The copula in the name of the PAS results is the used model in the last iteration. 0 50 100 150 200 250 300 Number of samples j [-] 0 100 200 300 400 500 600 7… view at source ↗
Figure 9
Figure 9. Figure 9: Case 4 - slamming: convergence of one-hour MPM as a function of number of samples (top) and final converged distributions (bottom) from AS and PAS with indicator 𝑅′ 19𝑝 , where GP fit = Generalised Pareto fit to 30% highest true validation peaks. The copula in the name of the PAS results is the used model in the last iteration. van Essen, Seyffert: Preprint submitted to Elsevier Page 15 of 27 [PITH_FULL_I… view at source ↗
Figure 10
Figure 10. Figure 10: Performance metrics 𝑀1 , 𝑀2 and 𝑀3 defined in Section 4 for all considered cases. shows that both methods produce MPM results with accuracy well within 5% for these two cases, which is acceptable for design purposes. However, the results for cases 3 and 4 clearly highlight the advantage of PAS over AS combined with linear potential flow. The inclusion of probabilistic copula fitting in PAS significantly r… view at source ↗
Figure 11
Figure 11. Figure 11: Fitted copula models (and the histogram of the associated marginals) at different iteration numbers for green water case 3a from the full PAS procedure, where it. = iteration. This case was converged with 73 HF samples. 0 500 1000 1500 2000 2500 3000 3500 4000 f [kN] 10 4 10 3 10 2 10 1 10 0 P(F 00X f) Pexp (Texp = 1800 s) It. 10 (19 HF samp.) It. 20 (29 HF samp.) It. 30 (39 HF samp.) It. 40 (49 HF samp.)… view at source ↗
Figure 12
Figure 12. Figure 12: Predicted distributions at different iteration numbers for green water case 3a from the full PAS procedure. 20 s event corresponds to two wave encounters. The computational time required for these simulations (in CPU hours) also depends on the CFD tool used, as well as the chosen grid size, domain, and time-step settings for each case. Because the results of PAS partly depend on the fitted copula, and bec… view at source ↗
Figure 13
Figure 13. Figure 13: Scatter plots of LF indicator peaks and matched HF validation peaks for all cases. 𝑆(𝑗) = { stop if ( 𝐸50(𝑗) < 𝜖1 ) ∩ ( 𝐶50(𝑗) < 𝜖2 ) continue otherwise (27) Compared to the stopping criterion used for AS in [18], we removed the rejection criterion for distribution shapes that violate distribution assumptions (as drawing from the copula model in Step 8 of PAS always produces a proper distribution), we add… view at source ↗
Figure 14
Figure 14. Figure 14: Log-likelihood of the four copula candidate models as a function of the number of HF samples, for green water case 3a (which is converged at 73 HF samples). 0 50 100 150 200 250 Number of samples j [-] 1a 1b 2a 2b 3a 3b 4a 4b Cases Gaussian Gumbel Clayton Frank [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Selected copula per case and iteration, plotted until convergence in each case. 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 lsel 0 500 1000 1500 2000 2500 3000 3500 h s el (a) Gaussian 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 lsel 0 500 1000 1500 2000 2500 3000 3500 h s el (b) Gumbel 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 lsel 0 500 1000 1500 2000 2500 3000 3500 h s el (c) Clayton 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 lsel… view at source ↗
Figure 16
Figure 16. Figure 16: Different fitted copula models (and the histogram of the associated marginals) to the HF samples generated with the full PAS procedure at 20 iterations (29 HF samples, not converged yet), for green water load case 3a. provides an overview of the most likely copula models for each case and each iteration (up to a number of iterations past convergence). E.2. Influence of selected copula model on results For… view at source ↗
Figure 17
Figure 17. Figure 17: Exceedance probability distributions (mean and 95th percentile) resulting from 20 draws of each of the four copulas fitted to the 29 HF samples from iteration 20 and 100, for green water load case 3a. Declaration of generative AI / AI-assisted technology in the manuscript preparation process During the preparation of this work the authors used ChatGPT in order to rephrase a few individual sentences. After… view at source ↗
read the original abstract

Wave impact loads on maritime structures can cause casualties, damage, pollution and operational delays. Consequently, their extreme values should be accounted for in the design of these structures. However, this is challenging, as wave impact events are both rare and highly complex, requiring both high-fidelity simulations and long analysis durations to reliably quantify the associated design loads. Moreover, existing extreme value prediction methods are neither specifically developed nor adequately validated for wave impact phenomena. We therefore introduce the new Probabilistic Adaptive Screening (PAS) method for predicting extreme non-linear loads on maritime structures. The method integrates copula-based statistical dependence modelling with multi-fidelity screening and adaptive sampling. This framework enables efficient extreme value prediction by statistically mapping low-fidelity indicator variables to high-fidelity impact loads. The method allows for efficient linear potential flow indicators to be used in the low-fidelity stage, even for strongly non-linear cases. Its statistical framework is validated against four non-linear test cases, including non-linear waves, ship vertical bending moments, green water impact loads, and slamming loads. It is concluded that PAS with optimal settings accurately estimates both the short-term distributions and extreme values in these test cases, with most probable maximum (MPM) values within 2-15% of the reference brute-force Monte-Carlo Simulation (MCS) results. In addition, PAS achieves this performance very efficiently, requiring in the order of 1-3% of the high-fidelity simulation time needed for conventional MCS. These results demonstrate that PAS can reliably reproduce the statistics of both weakly and strongly non-linear extreme load problems, while significantly reducing the associated computational cost compared to MCS.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces the Probabilistic Adaptive Screening (PAS) method, which combines copula-based statistical dependence modelling, multi-fidelity screening, and adaptive sampling to predict extreme non-linear wave impact loads on maritime structures. Low-fidelity linear potential flow indicators are mapped to high-fidelity non-linear loads, with validation on four test cases (non-linear waves, ship vertical bending moments, green water impacts, and slamming loads) reporting most probable maximum (MPM) values within 2-15% of brute-force Monte Carlo simulation (MCS) results and computational cost reduced to 1-3% of conventional MCS.

Significance. If the accuracy holds, PAS would offer a practical route to incorporate rare extreme non-linear loads into maritime design at far lower cost than brute-force MCS, supporting safer assessment of slamming and green-water events. A notable strength is the use of independent brute-force MCS for validation rather than fitted parameters or self-referential checks, which reduces circularity and strengthens the empirical support for the efficiency claims.

major comments (1)
  1. [Abstract and validation on four test cases] The central claim for strongly non-linear phenomena rests on the copula accurately transferring tail statistics from low-fidelity indicators to high-fidelity loads. The validation reports 2-15% MPM agreement, but the manuscript should include explicit diagnostics (e.g., comparison of conditional quantiles or upper-tail dependence coefficients for thresholds above the 95th percentile of the indicator) to confirm that the fitted copula and its parameter estimation from screening samples preserve the required conditional extreme behavior; without this, bias in the adaptive sampling step cannot be ruled out for slamming and green-water cases.
minor comments (1)
  1. [Method description] Specify the exact copula families employed, the procedure for parameter estimation from the screening samples, and any sensitivity of results to copula choice to improve reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback and recommendation for major revision. We appreciate the recognition of the PAS method's potential and the emphasis on independent brute-force validation. We address the major comment below and will incorporate the suggested diagnostics in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and validation on four test cases] The central claim for strongly non-linear phenomena rests on the copula accurately transferring tail statistics from low-fidelity indicators to high-fidelity loads. The validation reports 2-15% MPM agreement, but the manuscript should include explicit diagnostics (e.g., comparison of conditional quantiles or upper-tail dependence coefficients for thresholds above the 95th percentile of the indicator) to confirm that the fitted copula and its parameter estimation from screening samples preserve the required conditional extreme behavior; without this, bias in the adaptive sampling step cannot be ruled out for slamming and green-water cases.

    Authors: We thank the referee for highlighting this important point regarding tail behavior in strongly non-linear cases. While the reported agreement in most probable maximum (MPM) values within 2-15% and short-term distributions, validated against independent brute-force Monte Carlo simulations across all four test cases (including slamming and green water), provides empirical support for the copula mapping, we agree that explicit upper-tail diagnostics would further strengthen the validation and address potential concerns about bias in adaptive sampling. In the revised manuscript, we will add comparisons of conditional quantiles of the high-fidelity loads conditioned on low-fidelity indicators exceeding the 95th percentile, along with upper-tail dependence coefficients, with focused analysis on the slamming and green-water cases. These additions will confirm that the copula fitted from screening samples preserves the required conditional extreme statistics. revision: yes

Circularity Check

0 steps flagged

No circularity: PAS validated empirically against independent MCS references

full rationale

The paper introduces the PAS method as a combination of copula-based dependence modeling, multi-fidelity screening, and adaptive sampling to map low-fidelity linear indicators to high-fidelity nonlinear loads. All performance claims (short-term distributions, extreme values, and 2-15% MPM agreement) are presented as direct empirical comparisons to separate brute-force Monte Carlo Simulation results on four distinct test cases. No equations, parameters, or central premises reduce to self-definitions, fitted inputs renamed as predictions, or self-citation chains; the validation benchmarks are external and independent of the method's internal fitting process. This keeps the derivation self-contained without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms or invented entities; the approach relies on standard copula techniques and multi-fidelity concepts whose specific parameter choices and assumptions are not detailed.

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