In this work, we address the problem of Bayesian calibration of an expensive computer code, assumed to be without model discrepancy. In a calibration process involving costly physical measurement acquisition and high computational time of the computer code, the estimation accuracy and cost mitigation must guide the selection of the designs of physical and numerical experiments. To this end, we propose a unified strategy to select both physical and numerical experimental designs with the aim of approximating the posterior density of calibration parameters. We first build an initial Gaussian process emulator, which is used to compute an optimality criterion for the design of physical experiments. Then we combine physical observations with available computer code evaluations to sequentially add new points to the design of numerical experiments in order to improve the Gaussian process emulator for calibration purpose. We introduce new optimality criteria for selecting the design of physical experiments based on either the posterior density or the behavior of the computer code, as well as a new acquisition function for selecting the design of numerical experiments inspired by the sequential uncertainty reduction (SUR) paradigm. An illustration and a performance analysis, together with a comparison to a space-filling approach, are conducted across four settings: two analytical test cases, an example involving the calibration of a harmonic oscillator, and a more realistic application concerning the calibration of the Level E model.
We describe a stochastic method to characterize uncertainties in numerical models that have a single real-valued output (observable). The rationale estimates the probability density function (PDF) of the observable by inverting the characteristic function (CF), which is computed by quadrature. This approach is seldom used due to the highly oscillatory integrals that define the CF, which cause traditional quadrature algorithms to fail. Instead, we overcome these high oscillations by using an oscillatory Levin quadrature algorithm. We specifically choose the implementation proposed by Webster and Evans for the one-dimensional case and generalize it to the multivariate case. Using the Webster-Evans/Levin rule (WEL), we build a probabilistic adjoint operator (PrAdO).
This work aims to enhance the small amount of data obtained from simulations using statistical methods, in order to improve the analysis of the underlying physical properties of plasma behavior, particularly electron density and electron temperature in the boundary region of the tokamak. Electron density and electron temperature are important observables for characterizing plasma behavior in the boundary region of the tokamak, as they influence transport properties and confinement regimes. While the simulations used in this work are fully deterministic, we aim to extract statistical information by interpreting numerical simulations as samples from underlying random fields. This approach allows for the construction of marginal and joint probability density functions (PDFs) that provide physical insight beyond standard deterministic interpretation and capture key structural features of the plasma behavior in the boundary region of the tokamak, including the role of triangularity, the impact of the diffusion coefficient, and the implications for plasma behavior. Numerical simulations are performed with the Gkeyll gyrokinetic code. Despite their resolution, the computational cost limits the number of available simulation data, suggesting a role for advanced analysis techniques capable of extracting meaningful physical insights from a small dataset. To this end, we explore a statistical framework based on probabilistic learning on manifolds (PLoM), a nonparametric method designed for small-sample inference.
Inverse uncertainty quantification is essential in applied mechanics, particularly for characterizing the uncertainty in model parameters based on uncertain observations of a model's output. Approaches via Bayesian inference often struggle with the intractability of the likelihood due to the complexities of real-world simulators and experiments. This paper addresses these challenges by proposing a novel approach to tractable inverse uncertainty quantification using probabilistic circuits. The method circumvents the need for an explicit likelihood by directly learning the joint distribution of system inputs and observations through density estimation. Additionally, this approach allows for various probabilistic queries, including marginals, posterior distributions, and sensitivity analysis on a single model, while ensuring computational tractability. Through numerical experiments on representative problems in applied mechanics, we demonstrate the effectiveness and flexibility of our method to real-world problems.
