arXiv subjects

Ruanui Nicholson

Publications and source records attributed to Ruanui Nicholson.

16 recordsLinked to original sources

arXiv2026

Online Gate-Driven Flow Control in Resin Transfer Moulding Using a Neural-Network Surrogate

In resin transfer moulding, complete saturation of the fibre preform is necessary before the resin front reaches the outlet vent(s), to prevent dry-spot formation. In practice, the flow front rarely advances uniformly due to race-tracking effects. We propose a combined estimation and control strategy to address this issue. We use pressure-sensor data collected during filling to estimate the unknown race-tracking strengths via an iterated extended Kalman filter, and to simultaneously optimise auxiliary gate pressures to prevent the resin from arriving at the vent before complete saturation occurs. To make this approach feasible in real time, we replace the expensive finite element-control volume model with neural network surrogate models. The Bayesian approximation error framework is used to account for the discrepancy between the surrogates and the original model. To make the training of the surrogates feasible with time-dependent control actions, we prove that the flow-front geometry depends only on the time-averaged gate pressures. We demonstrate the methodology on a representative forked geometry, showing that the combined estimation-and-control framework can substantially reduce the number of unfilled nodes at the end of filling compared with an uncontrolled baseline. However, the level of improvement depends strongly on the location and number of auxiliary gates.

math.OC↗

arXiv2026

Multi-type Sensor Placement for PDE-based Bayesian Inverse Problems

We address optimal placement of multi-type sensors for Bayesian inverse problems governed by partial differential equations (PDEs). The proposed framework allows for sensors with different accuracies and observation types. We formulate the optimal experimental design (OED) problem as a knapsack-constrained binary optimization problem for maximizing expected information gain (EIG). To approximately solve the resulting optimization problems, we propose a stochastic cost-benefit greedy algorithm, which admits theoretical guarantees for monotone submodular set functions. Specifically, these guarantees apply in the case of linear Gaussian inverse problems with uncorrelated measurement errors, where the EIG admits a convenient closed-form expression. For nonlinear inverse problems, we develop a non-intrusive approach that uses the Bayesian approximation error framework to define an observation model with an error-corrected global linear model. We show that the corresponding approximate EIG is a lower bound for the exact EIG and thus provides a principled surrogate objective for the OED problem. The effectiveness of the proposed methods is demonstrated in two model inverse problems governed by PDEs.

math.NA↗

arXiv2026

Beyond Independence: on Jointly Normal Priors in Bayesian Inversion

We consider joint inversion for two or more unknown parameters from observational data in the Bayesian framework. Standard approaches often either treat the parameters as independent or impose structural similarity through regularisation terms that can be difficult to interpret statistically. We instead construct jointly Gaussian prior models with prescribed Gaussian marginals, so that correlation between the parameters can be incorporated without altering the marginal prior distributions. We propose a joint covariance construction that preserves the marginals, allows spatially varying cross-correlation, and supports uncertainty and inference in the correlation itself. The construction is valid for any strict contraction encoding the desired cross-correlation and is optimal in a canonical correlation sense under the principal square root factorisation. We demonstrate the method using prior sampling and several inference examples: a low-dimensional illustrative example and two higher-dimensional examples, including a PDE-constrained problem. The examples highlight both the potential pitfalls of ignoring or neglecting uncertainty in the correlation as well as reinforcing a key principle of the Bayesian paradigm: unknown quantities included in a model should be treated as random variables.

stat.ME↗

arXiv2026

Shape Derivative-Informed Neural Operators with Application to Risk-Averse Shape Optimization

Shape optimization under uncertainty (OUU) is computationally intensive for classical PDE-based methods due to the high cost of repeated sampling-based risk evaluation across many uncertainty realizations and varying geometries, while standard neural surrogates often fail to provide accurate and efficient sensitivities for optimization. We introduce Shape-DINO, a derivative-informed neural operator framework for learning PDE solution operators on families of varying geometries, with a particular focus on accelerating PDE-constrained shape OUU. Shape-DINOs encode geometric variability through diffeomorphic mappings to a fixed reference domain and employ a derivative-informed operator learning objective that jointly learns the PDE solution and its Fr\'echet derivatives with respect to design variables and uncertain parameters, enabling accurate state predictions and reliable gradients for large-scale OUU. We establish a priori error bounds linking surrogate accuracy to optimization error and prove universal approximation results for multi-input reduced basis neural operators in suitable $C^1$ norms. We demonstrate efficiency and scalability on three representative shape OUU problems, including boundary design for a Poisson equation and shape design governed by steady-state Navier-Stokes exterior flows in two and three dimensions. Across these examples, Shape-DINOs produce more reliable optimization results than operator surrogates trained without derivative information. In our examples, Shape-DINOs achieve 3-8 orders-of-magnitude speedups in state and gradient evaluations. Counting training data generation, Shape-DINOs reduce necessary PDE solves by 1-2 orders-of-magnitude compared to a strictly PDE-based approach for a single OUU problem. Moreover, Shape-DINO construction costs can be amortized across many objectives and risk measures, enabling large-scale shape OUU for complex systems.

math.OC↗

arXiv2025

Taylor Approximation Variance Reduction for Approximation Errors in PDE-constrained Bayesian Inverse Problems

In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors (between the accurate parameter-to-observable mapping and the surrogate). The statistics of these errors are in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.

math.OC↗

arXiv2025

Invariant Image Reparameterisation: Bridging Symbolic and Numerical Methods for Identifiability Analysis, Model Reduction, and Prediction

Structural and practical parameter non-identifiability issues are common when mathematical models are used to interpret data. Such issues motivate model reparameterisation and reduction methods. Here, we consider Invariant Image Reparameterisation (IIR), which asks when symbolic reparameterisation conditions can be replaced by numerical derivative calculations at a single reference point. The central object is the invariant image: a reduced, basis-independent representation of the parameter combinations controlling observable model behaviour. We show that when a one-to-one componentwise transformation makes observable behaviour depend only on fixed linear combinations of the transformed parameters, a single numerical Jacobian determines the associated lower-dimensional reparameterisation space. This includes models depending on monomial combinations of the original parameters. We also give a first-order invariance condition that distinguishes minimal from non-minimal but exact reductions via the invariant part of the local null space. In structurally identifiable but practically weakly informed settings, the same calculations separate strongly and weakly informed parameter combinations. The invariant image admits multiple coordinate representations: the SVD gives a default orthonormal basis ordered by local identifiability, while sparse monomial bases are often more interpretable. Treating these coordinates as interest parameters in Profile-Wise Analysis gives likelihood-based uncertainty quantification and prediction. We demonstrate the method on parameterised normal models with Poisson-limit, extended Poisson-limit, and non-limit cases, and on the repressilator, a nonlinear differential equation model of gene regulation. A Julia implementation of IIR, with these and further examples, is available at https://github.com/omaclaren/reparam.

stat.AP↗

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↗

arXiv2024

Non-intrusive optimal experimental design for large-scale nonlinear Bayesian inverse problems using a Bayesian approximation error approach

We consider optimal experimental design (OED) for nonlinear inverse problems within the Bayesian framework. Optimizing the data acquisition process for large-scale nonlinear Bayesian inverse problems is a computationally challenging task since the posterior is typically intractable and commonly-encountered optimality criteria depend on the observed data. Since these challenges are not present in OED for linear Bayesian inverse problems, we propose an approach based on first linearizing the associated forward problem and then optimizing the experimental design. Replacing an accurate but costly model with some linear surrogate, while justified for certain problems, can lead to incorrect posteriors and sub-optimal designs if model discrepancy is ignored. To avoid this, we use the Bayesian approximation error (BAE) approach to formulate an A-optimal design objective for sensor selection that is aware of the model error. In line with recent developments, we prove that this uncertainty-aware objective is independent of the exact choice of linearization. This key observation facilitates the formulation of an uncertainty-aware OED objective function using a completely trivial linear map, the zero map, as a surrogate to the forward dynamics. The base methodology is also extended to marginalized OED problems, accommodating uncertainties arising from both linear approximations and unknown auxiliary parameters. Our approach only requires parameter and data sample pairs, hence it is particularly well-suited for black box forward models. We demonstrate the effectiveness of our method for finding optimal designs in an idealized subsurface flow inverse problem and for tsunami detection.

math.NA↗

arXiv2022

Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty

We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.

math.NA↗

arXiv2021

Joint Estimation of Robin Coefficient and Domain Boundary for the Poisson Problem

We consider the problem of simultaneously inferring the heterogeneous coefficient field for a Robin boundary condition on an inaccessible part of the boundary along with the shape of the boundary for the Poisson problem. Such a problem arises in, for example, corrosion detection, and thermal parameter estimation. We carry out both linearised uncertainty quantification, based on a local Gaussian approximation, and full exploration of the joint posterior using Markov chain Monte Carlo (MCMC) sampling. By exploiting a known invariance property of the Poisson problem, we are able to circumvent the need to re-mesh as the shape of the boundary changes. The linearised uncertainty analysis presented here relies on a local linearisation of the parameter-to-observable map, with respect to both the Robin coefficient and the boundary shape, evaluated at the maximum a posteriori (MAP) estimates. Computation of the MAP estimate is carried out using the Gauss-Newton method. On the other hand, to explore the full joint posterior we use the Metropolis-adjusted Langevin algorithm (MALA), which requires the gradient of the log-posterior. We thus derive both the Fr\'{e}chet derivative of the solution to the Poisson problem with respect to the Robin coefficient and the boundary shape, and the gradient of the log-posterior, which is efficiently computed using the so-called adjoint approach. The performance of the approach is demonstrated via several numerical experiments with simulated data.

math.OC↗

arXiv2019

Coupled Elastic-Acoustic Modelling for Quantitative Photoacoustic Tomography

Quantitative photoacoustic tomography (qPAT) is an imaging technique aimed at estimating chromophore concentrations inside tissues from photoacoustic images, which are formed by combining optical information and ultrasonic propagation. The application of qPAT as a transcranial imaging modality is complicated by shear waves that can be produced when ultrasound waves travel from soft tissue to bone. Because of this, the estimation of chromophores distributions near the skull can be problematic. In this paper, we take steps towards compensating for aberrations of the recorded photoacoustic signals caused by elastic wave propagation. With photoacoustic data simulated in a coupled elastic-acoustic domain, we conduct inversions in a purely acoustic domain. Estimation of the posterior density of the initial pressure is achieved by inversion under the Bayesian framework. We utilize the Bayesian approximation error approach to compensate for the modelling errors arising from approximating a coupled elastic-acoustic domain with a purely fluid domain. The resulting reconstructions and corresponding uncertainty estimates are then used to evaluate the posterior density of the optical absorption parameter. In the sense of the posterior uncertainty, the results show that the Bayesian approximation error approach yields a more feasible estimate for the posterior model of the initial pressure which, in turn, yields a more feasible estimate for the posterior model of the absorption coefficient.

physics.comp-ph↗

arXiv2019

What can be estimated? Identifiability, estimability, causal inference and ill-posed inverse problems

We consider basic conceptual questions concerning the relationship between statistical estimation and causal inference. Firstly, we show how to translate causal inference problems into an abstract statistical formalism without requiring any structure beyond an arbitrarily-indexed family of probability models. The formalism is simple but can incorporate a variety of causal modelling frameworks, including 'structural causal models', but also models expressed in terms of, e.g., differential equations. We focus primarily on the structural/graphical causal modelling literature, however. Secondly, we consider the extent to which causal and statistical concerns can be cleanly separated, examining the fundamental question: 'What can be estimated from data?'. We call this the problem of estimability. We approach this by analysing a standard formal definition of 'can be estimated' commonly adopted in the causal inference literature -- identifiability -- in our abstract statistical formalism. We use elementary category theory to show that identifiability implies the existence of a Fisher-consistent estimator, but also show that this estimator may be discontinuous, and thus unstable, in general. This difficulty arises because the causal inference problem is, in general, an ill-posed inverse problem. Inverse problems have three conditions which must be satisfied to be considered well-posed: existence, uniqueness, and stability of solutions. Here identifiability corresponds to the question of uniqueness; in contrast, we take estimability to mean satisfaction of all three conditions, i.e. well-posedness. Lack of stability implies that naive translation of a causally identifiable quantity into an achievable statistical estimation target may prove impossible. Our article is primarily expository and aimed at unifying ideas from multiple fields, though we provide new constructions and proofs.

math.ST↗

arXiv2018

Incorporating Posterior-Informed Approximation Errors into a Hierarchical Framework to Facilitate Out-of-the-Box MCMC Sampling for Geothermal Inverse Problems and Uncertainty Quantification

We consider geothermal inverse problems and uncertainty quantification from a Bayesian perspective. Our main goal is to make standard, `out-of-the-box' Markov chain Monte Carlo (MCMC) sampling more feasible for complex simulation models by using suitable approximations. To do this, we first show how to pose both the inverse and prediction problems in a hierarchical Bayesian framework. We then show how to incorporate so-called posterior-informed model approximation error into this hierarchical framework, using a modified form of the Bayesian approximation error (BAE) approach. This enables the use of a `coarse', approximate model in place of a finer, more expensive model, while accounting for the additional uncertainty and potential bias that this can introduce. Our method requires only simple probability modelling, a relatively small number of fine model simulations, and only modifies the target posterior -- any standard MCMC sampling algorithm can be used to sample the new posterior. These corrections can also be used in methods that are not based on MCMC sampling. We show that our approach can achieve significant computational speed-ups on two geothermal test problems. We also demonstrate the dangers of naively using coarse, approximate models in place of finer models, without accounting for the induced approximation errors. The naive approach tends to give overly confident and biased posteriors while incorporating BAE into our hierarchical framework corrects for this while maintaining computational efficiency and ease-of-use.

stat.CO↗

arXiv2018

An Additive Approximation to Multiplicative Noise

Multiplicative noise models are often used instead of additive noise models in cases in which the noise variance depends on the state. Furthermore, when Poisson distributions with relatively small counts are approximated with normal distributions, multiplicative noise approximations are straightforward to implement. There are a number of limitations in existing approaches to marginalize over multiplicative errors, such as positivity of the multiplicative noise term. The focus in this paper is in large dimensional (inverse) problems for which sampling type approaches have too high computational complexity. In this paper, we propose an alternative approach to carry out approximative marginalization over the multiplicative error by embedding the statistics in an additive error term. The approach is essentially a Bayesian one in that the statistics of the additive error is induced by the statistics of the other unknowns. As an example, we consider a deconvolution problem on random fields with different statistics of the multiplicative noise. Furthermore, the approach allows for correlated multiplicative noise. We show that the proposed approach provides feasible error estimates in the sense that the posterior models support the actual image.

math.OC↗

arXiv2018

Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field

We consider the reconstruction of a heterogeneous coefficient field in a Robin boundary condition on an inaccessible part of the boundary in a Poisson problem with an uncertain (or unknown) inhomogeneous conductivity field in the interior of the domain. To account for model errors that stem from the uncertainty in the conductivity coefficient, we treat the unknown conductivity as a nuisance parameter and carry out approximative premarginalization over it, and invert for the Robin coefficient field only. We approximate the related modelling errors via the Bayesian approximation error (BAE) approach. The uncertainty analysis presented here relies on a local linearization of the parameter-to-observable map at the maximum a posteriori (MAP) estimates, which leads to a normal (Gaussian) approximation of the parameter posterior density. To compute the MAP point we apply an inexact Newton conjugate gradient approach based on the adjoint methodology. The construction of the covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. Two numerical experiments are considered: one where the prior covariance on the conductivity is isotropic, and one where the prior covariance on the conductivity is anisotropic. Results are compared to those based on standard error models, with particular emphasis on the feasibility of the posterior uncertainty estimates. We show that the BAE approach is a feasible one in the sense that the predicted posterior uncertainty is consistent with the actual estimation errors, while neglecting the related modelling error yields infeasible estimates for the Robin coefficient. In addition, we demonstrate that the BAE approach is approximately as computationally expensive (measured in the number of PDE solves) as the conventional error approach.

math.OC↗