arXiv subjects

Pieterjan Robbe

Publications and source records attributed to Pieterjan Robbe.

At least 19 recordsLinked to original sources

arXiv2026

Materials Behavior as Mechanism Ensembles: A Probabilistic Framework for Emergent Behaviors

Materials behavior is often treated as a deterministic mapping from structure to properties, yet many important phenomena emerge from the conditional activation of multiple mechanisms across scales. This is especially evident in fatigue of metals, where crack growth is typically modeled as monotonic and irreversible process, despite evidence that local microstructure, loading history, and competing unit processes can shift the balance among propagation, arrest, and self-healing. Here we present a probabilistic framework that describes materials behavior as an ensemble of constituent mechanisms whose activation, interaction, and evolution determine emergent outcomes. The framework connects mechanism activation, state evolution, and macroscopic observables in a probabilistic way. In the case of fatigue crack propagation, it reframes damage tolerance as an inference problem over mechanism competition and provides a basis for integrating multiscale simulation, multimodal characterization, and machine learning. The same logic extends to other physical and chemical systems suggesting a portable framework for any system in which emergent behavior reflects mechanism competition under changing conditions. The broader ambition of this perspective review is a shift from correlating structure and performance after the fact to identifying, in advance, the conditions that make desired emergent behavior probable.

cond-mat.mtrl-sci↗

arXiv2026

Fast multitask Gaussian processes, with application to surrogate modeling of the quark-gluon plasma

Gaussian processes (GPs) are broadly used for the surrogate modeling of computer experiments with reliable uncertainty quantification. Our motivating application comes from the study of the quark-gluon plasma (QGP), an extreme state of nuclear matter that filled the universe shortly after the Big Bang. To reliably infer properties of the QGP, multiple surrogate models need to be constructed for related particle collision simulation systems (i.e., multiple "tasks"). While there is a body of literature on multitask GPs (MTGPs), such models can be computationally expensive with large datasets: they require $\mathcal{O}(N^3)$ work and $\mathcal{O}(N^2)$ storage for model training, where $N$ is the total number of samples over all tasks. To address this, we propose a new fast MTGP approach, which pairs structured low-discrepancy design points, such as Sobol' points, with special kernel forms for efficient and exact model fitting. This pairing of a kernel and potentially different design points of different sizes across tasks provides a structured Gram matrix, e.g., a circulant block matrix, which we exploit via a novel algorithm for efficient Gram matrix inversion and determinant computation. Our algorithm reduces training costs to $\mathcal{O}(N \log N)$ work and $\mathcal{O}(N)$ storage in the case of equal sample sizes for each tasks. In the worst case of severely unbalanced sample sizes across tasks, our algorithm may require up to $\mathcal{O}(N^2)$ work and storage, with continuous interpolation between these best and worst cases depending on the balance of sample sizes across tasks. An open-source Python implementation is made available in the FastGPs package (https://alegresor.github.io/fastgps/). We demonstrate the effectiveness of our fast MTGP approach on a range of simulation experiments and on our motivating QGP application.

stat.CO↗

arXiv2026

Towards Spatio-Temporal Extrapolation of Phase-Field Simulations with Convolution-Only Neural Networks

Phase-field simulations of liquid metal dealloying (LMD) can capture complex microstructural evolutions but can be prohibitively expensive for large domains and long time horizons. In this paper, we introduce a fully convolutional, conditionally parameterized U-Net surrogate designed to extrapolate far beyond its training data in both space and time. The architecture integrates convolutional self-attention, physically informed padding, and a flood-fill corrector method to maintain accuracy under extreme extrapolation, while conditioning on simulation parameters allows for flexible time-step skipping and adaptation to varying alloy compositions. To remove the need for costly solver-based initialization, we couple the surrogate with a conditional diffusion model that generates synthetic, physically consistent initial conditions. We train our surrogate on simulations generated over small domain sizes and short time spans, but, by taking advantage of the convolutional nature of U-Nets, we are able to run and extrapolate surrogate simulations for longer time horizons than what would be achievable with classic numerical solvers. Across multiple alloy compositions, the framework is able to reproduce the LMD physics accurately. It predicts key quantities of interest and spatial statistics with relative errors typically below 5% in the training regime and under 15% during large-scale, long time-horizon extrapolations. Our framework can also deliver speed-ups of up to 36,000 times, bringing the time to run weeks-long simulations down to a few seconds. This work is a first stepping stone towards high-fidelity extrapolation in both space and time of phase-field simulation for LMD.

cs.CE↗

arXiv2025

Fast Bayesian Multilevel Quasi-Monte Carlo

Existing multilevel quasi-Monte Carlo (MLQMC) methods often rely on multiple independent randomizations of a low-discrepancy (LD) sequence to estimate statistical errors on each level. While this approach is standard, it can be less efficient than simply increasing the number of points from a single LD sequence. However, a single LD sequence does not permit statistical error estimates in the current framework. We propose to recast the MLQMC problem in a Bayesian cubature framework, which uses a single LD sequence and quantifies numerical error through the posterior variance of a Gaussian process (GP) model. When paired with certain LD sequences, GP regression and hyperparameter optimization can be carried out at only $\mathcal{O}(n \log n)$ cost, where $n$ is the number of samples. Building on the adaptive sample allocation used in traditional MLQMC, where the number of samples is doubled on the level with the greatest expected benefit, we introduce a new Bayesian utility function that balances the computational cost of doubling against the anticipated reduction in posterior uncertainty. We also propose a new digitally-shift-invariant (DSI) kernel of adaptive smoothness, which combines multiple higher-order DSI kernels through a weighted sum of smoothness parameters, for use with fast digital net GPs. A series of numerical experiments illustrate the performance of our fast Bayesian MLQMC method and error estimates for both single-level problems and multilevel problems with a fixed number of levels. The Bayesian error estimates obtained using digital nets are found to be reliable, although, in some cases, mildly conservative.

stat.CO↗

arXiv2025

Extrapolating Phase-Field Simulations in Space and Time with Purely Convolutional Architectures

Phase-field models of liquid metal dealloying (LMD) can resolve rich microstructural dynamics but become intractable for large domains or long time horizons. We present a conditionally parameterized, fully convolutional U-Net surrogate that generalizes far beyond its training window in both space and time. The design integrates convolutional self-attention and physics-aware padding, while parameter conditioning enables variable time-step skipping and adaptation to diverse alloy systems. Although trained only on short, small-scale simulations, the surrogate exploits the translational invariance of convolutions to extend predictions to much longer horizons than traditional solvers. It accurately reproduces key LMD physics, with relative errors typically under 5% within the training regime and below 10% when extrapolating to larger domains and later times. The method accelerates computations by up to 16,000 times, cutting weeks of simulation down to seconds, and marks an early step toward scalable, high-fidelity extrapolation of LMD phase-field models.

cs.CE↗

arXiv2025

A Comparison of Surrogate Constitutive Models for Viscoplastic Creep Simulation of HT-9 Steel

Mechanistic microstructure-informed constitutive models for the mechanical response of polycrystals are a cornerstone of computational materials science. However, as these models become increasingly more complex - often involving coupled differential equations describing the effect of specific deformation modes - their associated computational costs can become prohibitive, particularly in optimization or uncertainty quantification tasks that require numerous model evaluations. To address this challenge, surrogate constitutive models that balance accuracy and computational efficiency are highly desirable. Data-driven surrogate models, that learn the constitutive relation directly from data, have emerged as a promising solution. In this work, we develop two local surrogate models for the viscoplastic response of a steel: a piecewise response surface method and a mixture of experts model. These surrogates are designed to adapt to complex material behavior, which may vary with material parameters or operating conditions. The surrogate constitutive models are applied to creep simulations of HT-9 steel, an alloy of considerable interest to the nuclear energy sector due to its high tolerance to radiation damage, using training data generated from viscoplastic self-consistent (VPSC) simulations. We define a set of test metrics to numerically assess the accuracy of our surrogate models for predicting viscoplastic material behavior, and show that the mixture of experts model outperforms the piecewise response surface method in terms of accuracy.

physics.comp-ph↗

arXiv2023

Multi-fidelity uncertainty quantification for homogenization problems in structure-property relationships from crystal plasticity finite elements

Crystal plasticity finite element method (CPFEM) has been an integrated computational materials engineering (ICME) workhorse to study materials behaviors and structure-property relationships for the last few decades. These relations are mappings from the microstructure space to the materials properties space. Due to the stochastic and random nature of microstructures, there is always some uncertainty associated with materials properties, for example, in homogenized stress-strain curves. For critical applications with strong reliability needs, it is often desirable to quantify the microstructure-induced uncertainty in the context of structure-property relationships. However, this uncertainty quantification (UQ) problem often incurs a large computational cost because many statistically equivalent representative volume elements (SERVEs) are needed. In this paper, we apply a multi-level Monte Carlo (MLMC) method to CPFEM to study the uncertainty in stress-strain curves, given an ensemble of SERVEs at multiple mesh resolutions. By using the information at coarse meshes, we show that it is possible to approximate the response at fine meshes with a much reduced computational cost. We focus on problems where the model output is multi-dimensional, which requires us to track multiple quantities of interest (QoIs) at the same time. Our numerical results show that MLMC can accelerate UQ tasks around 2.23x, compared to the classical Monte Carlo (MC) method, which is widely known as the ensemble average in the CPFEM literature.

math.NA↗

arXiv2023

Bayesian calibration with summary statistics for the prediction of xenon diffusion in UO2 nuclear fuel

The evolution and release of fission gas impacts the performance of UO2 nuclear fuel. We have created a Bayesian framework to calibrate a novel model for fission gas transport that predicts diffusion rates of uranium and xenon in UO2 under both thermal equilibrium and irradiation conditions. Data sets are taken from historical diffusion, gas release, and thermodynamic experiments. These data sets consist invariably of summary statistics, including a measurement value with an associated uncertainty. Our calibration strategy uses synthetic data sets in order to estimate the parameters in the model, such that the resulting model predictions agree with the reported summary statistics. In doing so, the reported uncertainties are effectively reflected in the inferred uncertain parameters. Furthermore, to keep our approach computationally tractable, we replace the fission gas evolution model by a polynomial surrogate model with a reduced number of parameters, which are identified using global sensitivity analysis. We discuss the efficacy of our calibration strategy, and investigate how the contribution of the different data sets, taken from multiple sources in the literature, can be weighted in the likelihood function constructed as part of our Bayesian calibration setup, in order to account for the different number of data points in each set of data summaries. Our results indicate a good match between the calibrated diffusivity and non-stoichiometry predictions and the given data summaries. We demonstrate a good agreement between the calibrated xenon diffusivity and the established fit from Turnbull et al. (1982), indicating that the dominant uranium vacancy diffusion mechanism in the model is able to capture the trends in the data.

stat.AP↗

arXiv2023

Global Sensitivity Analysis of a coupled multiphysics model to predict surface evolution in fusion plasma-surface interactions

We construct a global sensitivity analysis framework for a coupled multiphysics model used to predict the changes in material properties and surface morphology of helium plasma-facing components in future fusion reactors. The model combines the particle dynamics simulator F-TRIDYN, that predicts the helium implantation profile, with the cluster dynamics simulator Xolotl, that predicts the growth and evolution of subsurface helium gas bubbles. In order to keep the sensitivity analysis tractable, we first construct a sparse, high-dimensional polynomial chaos expansion surrogate model for each output quantity of interest, which allows the efficient extraction of sensitivity information. The sensitivity analysis is performed for two problem settings: one for ITER-like conditions, and one that resembles conditions inside the PISCES-A linear plasma device. We present a systematic comparison of important parameters, for both F-TRIDYN and Xolotl in isolation as well as for the coupled model, and discuss the physical interpretation of these results.

physics.plasm-ph↗

arXiv2023

Multi-fidelity microstructure-induced uncertainty quantification by advanced Monte Carlo methods

Quantifying uncertainty associated with the microstructure variation of a material can be a computationally daunting task, especially when dealing with advanced constitutive models and fine mesh resolutions in the crystal plasticity finite element method (CPFEM). Numerous studies have been conducted regarding the sensitivity of material properties and performance to the mesh resolution and choice of constitutive model. However, a unified approach that accounts for various fidelity parameters, such as mesh resolutions, integration time-steps, and constitutive models simultaneously is currently lacking. This paper proposes a novel uncertainty quantification (UQ) approach for computing the properties and performance of homogenized materials using CPFEM, that exploits a hierarchy of approximations with different levels of fidelity. In particular, we illustrate how multi-level sampling methods, such as multi-level Monte Carlo (MLMC) and multi-index Monte Carlo (MIMC), can be applied to assess the impact of variations in the microstructure of polycrystalline materials on the predictions of homogenized materials properties. We show that by adaptively exploiting the fidelity hierarchy, we can significantly reduce the number of microstructures required to reach a certain prescribed accuracy. Finally, we show how our approach can be extended to a multi-fidelity framework, where we allow the underlying constitutive model to be chosen from either a phenomenological plasticity model or a dislocation-density-based model.

cond-mat.mtrl-sci↗

arXiv2023

The multifaceted nature of uncertainty in structure-property linkage with crystal plasticity finite element model

Uncertainty quantification (UQ) plays a critical role in verifying and validating forward integrated computational materials engineering (ICME) models. Among numerous ICME models, the crystal plasticity finite element method (CPFEM) is a powerful tool that enables one to assess microstructure-sensitive behaviors and thus, bridge material structure to performance. Nevertheless, given its nature of constitutive model form and the randomness of microstructures, CPFEM is exposed to both aleatory uncertainty (microstructural variability), as well as epistemic uncertainty (parametric and model-form error). Therefore, the observations are often corrupted by the microstructure-induced uncertainty, as well as the ICME approximation and numerical errors. In this work, we highlight several ongoing research topics in UQ, optimization, and machine learning applications for CPFEM to efficiently solve forward and inverse problems.

cond-mat.mtrl-sci↗

arXiv2021

On the Selection of Random Field Evaluation Points in the p-MLQMC Method

Engineering problems are often characterized by significant uncertainty in their material parameters. A typical example coming from geotechnical engineering is the slope stability problem where the soil's cohesion is modeled as a random field. An efficient manner to account for this uncertainty is the novel sampling method called p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). The p-MLQMC method uses a hierarchy of p-refined Finite Element meshes combined with a deterministic Quasi-Monte Carlo sampling rule. This combination yields a significant computational cost reduction with respect to classic Multilevel Monte Carlo. However, in previous work, not enough consideration was given how to incorporate the uncertainty, modeled as a random field, in the Finite Element model with the p-MLQMC method. In the present work we investigate how this can be adequately achieved by means of the integration point method. We therefore investigate how the evaluation points of the random field are to be selected in order to obtain a variance reduction over the levels. We consider three different approaches. These approaches will be benchmarked on a slope stability problem in terms of computational runtime. We find that for a given tolerance the Local Nested Approach yields a speedup up to a factor five with respect to the Non-Nested approach.

math.NA↗

arXiv2020

Multilevel Asymptotic-Preserving Monte Carlo for Particle Simulations

We develop a novel multilevel asymptotic-preserving Monte Carlo method, called Multilevel Kinetic-Diffusion Monte Carlo (ML-KDMC), for simulating the kinetic Boltzmann transport equation with a Bhatnagar-Gross-Krook (BGK) collision operator. This equation occurs, for instance, in mathematical models of the neutral particles in the plasma edge of nuclear fusion reactors. In this context, the Kinetic-Diffusion Monte Carlo method is known to maintain accuracy both in the low-collisional and the high-collisional limit, without an exploding simulation cost in the latter. We show that, by situating this method within a Multilevel Monte Carlo (MLMC) framework, using a hierarchy of larger time step sizes, the simulation cost is reduced even further. The different levels in our ML-KDMC method are connected via a new and improved recipe for correlating particle trajectories with different time step sizes. Furthermore, a new and more general level selection strategy is presented. We illustrate the efficiency of our ML-KDMC method by applying it to a one-dimensional test case with nonhomogeneous and anisotropic plasma background. Our method yields significant speedups compared to the single-level KDMC scheme, both in the low and high collisional regime. In the high-collisional case, our ML-KDMC outperforms the single-level KDMC method by several orders of magnitude.

math.NA↗

arXiv2019

Enhanced Multi-Index Monte Carlo by means of Multiple Semi-Coarsened Multigrid for Anisotropic Diffusion Problems

In many models used in engineering and science, material properties are uncertain or spatially varying. For example, in geophysics, and porous media flow in particular, the uncertain permeability of the material is modelled as a random field. These random fields can be highly anisotropic. Efficient solvers, such as the Multiple Semi-Coarsened Multigrid (MSG) method, see [15, 16, 17], are required to compute solutions for various realisations of the uncertain material. The MSG method is an extension of the classic Multigrid method, that uses additional coarse grids that are coarsened in only a single coordinate direction. In this sense, it closely resembles the extension of Multilevel Monte Carlo (MLMC) [6] to Multi-Index Monte Carlo (MIMC) [9]. We present an unbiased MIMC method that reuses the MSG coarse solutions, similar to the work in [11]. Our formulation of the estimator can be interpreted as the problem of learning the unknown distribution of the number of samples across all indices, and unifies the previous work on adaptive MIMC [19] and unbiased estimation [18]. We analyse the cost of this new estimator theoretically and present numerical experiments with various anisotropic random fields, where the unknown coefficients in the covariance model are considered as hyperparameters. We illustrate its robustness and superiority over unbiased MIMC without sample reuse.

math.NA↗

arXiv2019

h- and p-refined Multilevel Monte Carlo Methods for Uncertainty Quantification in Structural Engineering

Practical structural engineering problems are often characterized by significant uncertainties. Historically, one of the prevalent methods to account for this uncertainty has been the standard Monte Carlo (MC) method. Recently, improved sampling methods have been proposed, based on the idea of variance reduction by employing a hierarchy of mesh refinements. We combine an h- and p-refinement hierarchy with the Multilevel Monte Carlo (MLMC) and Multilevel Quasi-Monte Carlo (MLQMC) method. We investigate the applicability of these novel combination methods on three structural engineering problems, for which the uncertainty resides in the Young's modulus: the static response of a cantilever beam with elastic material behavior, its static response with elastoplastic behavior, and its dynamic response with elastic behavior. The uncertainty is either modeled by means of one random variable sampled from a univariate Gamma distribution or with multiple random variables sampled from a gamma random field. This random field results from a truncated Karhunen-Loève (KL) expansion. In this paper, we compare the computational costs of these Monte Carlo methods. We demonstrate that MQLMC and MLMC have a significant speedup with respect to MC, regardless of the mesh refinement hierarchy used. We empirically demonstrate that the MLQMC cost is optimally proportional to 1/epsilon under certain conditions, where epsilon is the tolerance on the root-mean-square error (RMSE). In addition, we show that, when the uncertainty is modeled as a random field, the multilevel methods combined with p-refinement have a significant lower computation cost than their counterparts based on h-refinement. We also illustrate the effect the uncertainty models have on the uncertainty bounds in the solutions.

math.NA↗

arXiv2018

Multilevel Monte Carlo for uncertainty quantification in structural engineering

Practical structural engineering problems often exhibit a significant degree of uncertainty in the material properties being used, the dimensions of the modeled structures, etc. In this paper, we consider a cantilever beam and a beam clamped at both ends, both subjected to a static and a dynamic load. The material uncertainty resides in the Young's modulus, which is modeled by means of one random variable, sampled from a univariate Gamma distribution, or with multiple random variables, sampled from a Gamma random field. Three different responses are considered: the static elastic, the dynamic elastic and the static elastoplastic response. In the first two cases, we simulate the spatial displacement of a concrete beam and its frequency response in the elastic domain. The third case simulates the spatial displacement of a steel beam in the elastoplastic domain. In order to compute the statistical quantities of the static deflection and frequency response function, Multilevel Monte Carlo (MLMC) is combined with a Finite Element solver. In this paper, the computational costs and run times of the MLMC method are compared with those of the classical Monte Carlo method, demonstrating a significant speedup of up to several orders of magnitude for the studied cases.

cs.CE↗

arXiv2018

Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method

The Multilevel Monte Carlo method is an efficient variance reduction technique. It uses a sequence of coarse approximations to reduce the computational cost in uncertainty quantification applications. The method is nowadays often considered to be the method of choice for solving PDEs with random coefficients when many uncertainties are involved. When using Full Multigrid to solve the deterministic problem, coarse solutions obtained by the solver can be recycled as samples in the Multilevel Monte Carlo method, as was pointed out by Kumar, Oosterlee and Dwight [Int. J. Uncertain. Quantif., 7 (2017), pp. 57--81]. In this article, an alternative approach is considered, using Quasi-Monte Carlo points, to speed up convergence. Additionally, our method comes with an improved variance estimate which is also valid in case of the Monte Carlo based approach. The new method is illustrated on the example of an elliptic PDE with lognormal diffusion coefficient. Numerical results for a variety of random fields with different smoothness parameters in the Mat\'ern covariance function show that sample recycling is more efficient when the input random field is nonsmooth.

math.NA↗

arXiv2017

A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger

We present an adaptive version of the Multi-Index Monte Carlo method, introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with coefficients that are random fields. A classical technique for sampling from these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm is based on the adaptive algorithm used in sparse grid cubature as introduced by Gerstner and Griebel (2003), and automatically chooses the number of terms needed in this expansion, as well as the required spatial discretizations of the PDE model. We apply the method to a simplified model of a heat exchanger with random insulator material, where the stochastic characteristics are modeled as a lognormal random field, and we show consistent computational savings.

math.NA↗