SIAM/ASA Journal on Uncertainty Quantification

Engineering and applied science rely on computational experiments to rigorously study physical systems. The mathematical models used to probe these systems are highly complex, and sampling-intensive studies often require prohibitively many simulations for acceptable accuracy. Surrogate models provide a means of circumventing the high computational expense of sampling such complex models. In particular, polynomial chaos expansions (PCEs) have been successfully used for uncertainty quantification studies of deterministic models where the dominant source of uncertainty is parametric. We discuss an extension to conventional PCE surrogate modeling to enable surrogate construction for stochastic computational models that have intrinsic noise in addition to parametric uncertainty. We develop a PCE surrogate on a joint space of intrinsic and parametric uncertainty, enabled by Rosenblatt transformations, which are evaluated via kernel density estimation of the associated conditional cumulative distributions. Furthermore, we extend the construction to random field data via the Karhunen–Loève expansion. We then take advantage of closed-form solutions for computing PCE Sobol indices to perform a global sensitivity analysis of the model which quantifies the intrinsic noise contribution to the overall model output variance. Additionally, the resulting joint PCE is generative in the sense that it allows generating random realizations at any input parameter setting that are statistically approximately equivalent to realizations from the underlying stochastic model. The method is demonstrated on a chemical catalysis example model and a synthetic example controlled by a parameter that enables a switch from unimodal to bimodal response distributions.

In a multifidelity setting, data are available under the same conditions from two (or more) sources, e.g., computer codes, one being lower-fidelity but computationally cheaper, and the other higher-fidelity and more expensive. This paper studies for which low-fidelity outputs one should obtain high-fidelity outputs if the goal is to estimate the probability density function of the latter, especially when it comes to the distribution tails and extremes. It is suggested to approach this problem from the perspective of the importance sampling of low-fidelity outputs according to some proposal distribution, combined with special considerations for the distribution tails based on extreme value theory. The notion of an optimal proposal distribution is introduced and investigated, in both theory and simulations. The approach is motivated and illustrated with an application to estimate the probability density function of record extremes of ship motions, obtained through two computer codes of different fidelities.

We introduce a Fourier-based fast algorithm for Gaussian process regression in low dimensions. It approximates a translationally invariant covariance kernel by complex exponentials on an equispaced Cartesian frequency grid of \(M\) nodes. This results in a weight-space \(M\times M\) system matrix with Toeplitz structure, which can thus be applied to a vector in \({\mathcal O}(M \log{M})\) operations via the fast Fourier transform (FFT), independent of the number of data points \(N\). The linear system can be set up in \({\mathcal O}(N+M \log{M})\) operations using nonuniform FFTs. This enables efficient massive-scale regression via an iterative solver, even for kernels with fat-tailed spectral densities (large \(M\)). We provide bounds on both kernel approximation and posterior mean errors. Numerical experiments for squared-exponential and Matérn kernels in one, two, and three dimensions often show 1–2 orders of magnitude acceleration over state-of-the-art rank-structured solvers at comparable accuracy. Our method allows two-dimensional Matérn-\(\frac{3}{2}\) regression from \(N=10^9\) data points to be performed in two minutes on a standard desktop, with posterior mean accuracy \(10^{-3}\). This opens up spatial statistics applications 100 times larger than previously possible.

Recent studies have confirmed cardiovascular diseases remain responsible for the highest mortality rate among noncommunicable diseases. The accurate left ventricular (LV) volume estimation is critical for valid diagnosis and management of various cardiovascular conditions, but poses a significant challenge due to inherent uncertainties associated with the segmentation algorithms in magnetic resonance imaging. Recent machine learning advancements, particularly U-Net-like convolutional networks, have facilitated automated segmentation for medical images, but struggles under certain pathologies and/or different scanner vendors and imaging protocols. This study proposes a novel methodology for post-hoc uncertainty estimation in the LV volume prediction using Itô stochastic differential equations to model pathwise behavior for the prediction error. The model describes the area of the left ventricle along the heart’s long axis. The method is agnostic to the underlying segmentation algorithm, facilitating its use with various existing and future segmentation technologies. The proposed approach provides a mechanism for quantifying uncertainty, enabling medical professionals to intervene for unreliable predictions. This is of utmost importance in critical applications such as medical diagnosis, where prediction accuracy and reliability can directly impact patient outcomes. The method is also robust to dataset changes, enabling application for medical centers with limited access to labeled data. Our findings highlight the proposed uncertainty estimation methodology’s potential to enhance automated segmentation robustness and generalizability, paving the way for more reliable and accurate LV volume estimation in clinical settings as well as opening new avenues for uncertainty quantification in biomedical image segmentation, providing promising directions for future research.

We address the computational efficiency of finding the A-optimal Bayesian experimental design, where the observation map is based on partial differential equations and thus computationally expensive to evaluate. A-optimality is a widely used and easily interpreted criterion, that seeks the optimal experimental design by minimizing the expected conditional variance. Our study presents a novel likelihood-free approach to the A-optimal experimental design that does not require sampling or integration over the Bayesian posterior distribution. In our proposed approach, we estimate the expected conditional variance via the variance of the conditional expectation and approximate the conditional expectation using its orthogonal projection property. We derive an asymptotic error estimate for the proposed estimator of the expected conditional variance and verify it with numerical experiments. Furthermore, we extend our approach to the case where the domain of the experimental design parameters is continuous. Specifically, we propose a nonlocal approximation of the conditional expectation using an artificial neural network and apply transfer learning and data augmentation to reduce the number of evaluations of the measurement model. Through numerical experiments, we demonstrate that our method greatly reduces the number of measurement model evaluations compared with widely used importance sampling-based approaches. Code is available at https://github.com/vinh-tr-hoang/DOEviaPACE.