arXiv subjects

Alex de Beer

Publications and source records attributed to Alex de Beer.

3 recordsLinked to original sources

arXiv2026

Multiscale Structural Reliability Analysis in high dimensions with Tensor Trains and Physics-Augmented Neural Networks

Structural reliability evaluation for composites constitutes a fundamentally high-dimensional multiscale problem, as microscale material uncertainties must propagate to the macroscale and can be quantified as high-dimensional random fields. Conventional approaches are computationally intractable, as they rely on repeatedly solving coupled partial differential equation systems across scales while contending with the exponential complexity inherent in high-dimensional uncertainty quantification. This work introduces a scalable and physically consistent framework that addresses both bottlenecks simultaneously in the case of separation of scales and (anisotropic) linear elasticity. In particular, we couple a physics-augmented Voigt--Reuss Neural Network (VRNN) with the Deep Inverse Rosenblatt Transport (DIRT) method to estimate the posterior probability of structural failure. The VRNN is used to resolve the computationally expensive FE$^2$ scheme by providing a near-instantaneous evaluation of the homogenized stiffness tensor that is guaranteed to be symmetric, positive-definite, and strictly bounded within the Voigt--Reuss limits, enabling fast evaluation of the homogenized responses. The DIRT method constructs a sequence of functional tensor train approximations to efficiently store an approximation of the high-dimensional optimal importance sampling distribution for estimating the probability of failure. This mitigates the curse of dimensionality arising from the Karhunen--Lo\`eve expansion of the random fields. The framework is demonstrated on a three-dimensional heterogeneous benchmark problem, where the uncertainty in the microscale material properties is characterized by a Bayesian posterior distribution obtained from limited strain observations. Our results show that the proposed framework can provide low-variance estimates of failure probabilities in dimensions up to 150.

cs.CE↗

arXiv2024

Ensemble Kalman Inversion for Geothermal Reservoir Modelling

Numerical models of geothermal reservoirs typically depend on hundreds or thousands of unknown parameters, which must be estimated using sparse, noisy data. However, these models capture complex physical processes, which frequently results in long run-times and simulation failures, making the process of estimating the unknown parameters a challenging task. Conventional techniques for parameter estimation and uncertainty quantification, such as Markov chain Monte Carlo (MCMC), can require tens of thousands of simulations to provide accurate results and are therefore challenging to apply in this context. In this paper, we study the ensemble Kalman inversion (EKI) algorithm as an alternative technique for approximate parameter estimation and uncertainty quantification for geothermal reservoir models. EKI possesses several characteristics that make it well-suited to a geothermal setting; it is derivative-free, parallelisable, robust to simulation failures, and in many cases requires far fewer simulations to provide an accurate characterisation of the posterior than conventional uncertainty quantification techniques such as MCMC. We illustrate the use of EKI in a reservoir modelling context using a combination of synthetic and real-world case studies. Through these case studies, we also demonstrate how EKI can be paired with flexible parametrisation techniques capable of accurately representing prior knowledge of the characteristics of a reservoir and adhering to geological constraints, and how the algorithm can be made robust to simulation failures. Our results demonstrate that EKI provides a reliable and efficient means of obtaining accurate parameter estimates for large-scale, two-phase geothermal reservoir models, with appropriate characterisation of uncertainty.

stat.AP↗

arXiv2024

Data Space Inversion for Efficient Predictions and Uncertainty Quantification for Geothermal Models

The ability to make accurate predictions with quantified uncertainty provides a crucial foundation for the successful management of a geothermal reservoir. Conventional approaches for making predictions using geothermal reservoir models involve estimating unknown model parameters using field data, then propagating the uncertainty in these estimates through to the predictive quantities of interest. However, the unknown parameters are not always of direct interest; instead, the predictions are of primary importance. Data space inversion (DSI) is an alternative methodology that allows for the efficient estimation of predictive quantities of interest, with quantified uncertainty, that avoids the need to estimate model parameters entirely. In this paper, we illustrate the applicability of DSI to geothermal reservoir modelling. We first review the processes of model calibration, prediction and uncertainty quantification from a Bayesian perspective, and introduce data space inversion as a simple, efficient technique for approximating the posterior predictive distribution. We then introduce a modification of the typical DSI algorithm that allows us to sample directly and efficiently from the DSI approximation to the posterior predictive distribution, and apply the algorithm to two model problems in geothermal reservoir modelling. We evaluate the accuracy and efficiency of our DSI algorithm relative to other common methods for uncertainty quantification and study how the number of reservoir model simulations affects the resulting approximation to the posterior predictive distribution. Our results demonstrate that data space inversion is a robust and efficient technique for making predictions with quantified uncertainty using geothermal reservoir models, providing a useful alternative to more conventional approaches.

stat.AP↗