Clemens V. Verhoosel

Aricia Rinkens, Rodrigo L. S. Silva, Erik Quaeghebeur et al.

Employing Bayesian inference to calibrate constitutive model parameters has grown substantially in recent years. Among the available techniques, Markov Chain Monte Carlo (MCMC) sampling remains one of the most widely used approaches for estimating the posterior distribution. Nevertheless, the selection of a specific MCMC algorithm is often driven by practical considerations, such as software availability or prior user experience. To support sampler selection, we present a comparison of three prominent samplers in the context of two distinct physical systems: a thermal conduction system and a viscous flow system. Calibration data are obtained through tailor-made experimental setups. We use the Kullback-Leibler (KL) divergence, which quantifies the statistical distance between the sampled posterior and the reference ('true') posterior, as a measure of convergence to compare the performance of the following MCMC sampling methods: the Metropolis-Hastings (MH) sampler, the Affine Invariant Stretch Move (AISM) sampler, and the No-U-Turn Sampler (NUTS). We study how this metric correlates to heuristic indicators such as the Gelman-Rubin diagnostic and the effective sample size. In addition, we assess the samplers' computational effort in terms of required number of model evaluations. Based on the results, we find that the heuristic convergence and performance indicators provide a good qualitative measure for KL-divergence for both systems. Regarding computational effort, the NUTS is net beneficial for the viscous flow system, as the high effective sample size outweighs the additional effort required for gradient-based proposal generation. For the thermal conduction system, which involves more expensive model evaluations, the NUTS is not advantageous. Thus, the computational efficiency of gradient evaluations is an important argument in sampler selection.

Zhijian Wang, Stein K. F. Stoter, Clemens V. Verhoosel et al.

In this work, we proposes a CO2-temperature network model that links multi-zone mass transport and thermal dynamics through shared latent drivers, airflow and occupancy. The thermal component is formulated as a resistance-capacitance (RC) network augmented with airflow-driven convective exchange, while the CO2 component is governed by inter-zonal convective transport. To calibrate the model and track time-varying operating conditions based on sparse sensing, we introduce a moving-window Bayesian inference procedure that jointly estimates thermal parameters, airflow and occupancy trajectories. The estimation also provides room-specific sensor noise levels, yielding posterior predictive forecasts with credible intervals. The framework is assessed using a controlled synthetic benchmark, and a scaled physical validation experiment using CO2 and temperature sensing. In both settings, the posterior accurately reconstructs trajectories within windows and delivers low forecast errors. When inference windows overlap abrupt regime transitions, the widened uncertainty bands and increased inferred noise levels provide an interpretable diagnostic of model-data mismatch, followed by rapid recovery once the new regime is observed. Overall, coupling CO2-informed airflow with thermal dynamics yields a robust approach for conductive and advective temperature prediction, supporting practical monitoring and energy-performance assessment under limited sensing.

Rodrigo L. S. Silva, Clemens Verhoosel, Erik Quarghebeur

We present a Bayesian model calibration framework for inferring nonlinear constitutive relationships in heat conduction problems, with a focus on temperature-dependent thermal conductivity. The proposed framework integrates gradient-based optimization and uncertainty quantification (UQ) to address the inverse problem of estimating the conductivity function from transient temperature measurements. A key contribution is an adaptive algorithm that sequentially refines both the numerical discretization for model simulation, and the model complexity used to represent the conductivity curve. The discretization is optimized through the minimization of a loss function, and Morozov's discrepancy principle is used as an uncertainty-motivated stopping criterion. The model complexity is selected using an approach that balances maximizing the likelihood of the data with penalizing excessive model complexity. As a result, the numerical and modeling biases remain of the same order as the uncertainty imposed by the measurement noise, leading to robust and computationally efficient inference. The methodology is demonstrated on both synthetic and experimental data, showing that it enables accurate calibration of nonlinear constitutive models with minimal overfitting and limited computational cost.

Robin Willems, Peter Förster, Sebastian Schöps et al.

Cardio-mechanical models can be used to support clinical decision-making. Unfortunately, the substantial computational effort involved in many cardiac models hinders their application in the clinic, despite the fact that they may provide valuable information. In this work, we present a probabilistic reduced-order modeling (ROM) framework to dramatically reduce the computational effort of such models while providing a credibility interval. In the online stage, a fast-to-evaluate generalized one-fiber model is considered. This generalized one-fiber model incorporates correction factors to emulate patient-specific attributes, such as local geometry variations. In the offline stage, Bayesian inference is used to calibrate these correction factors on training data generated using a full-order isogeometric cardiac model (FOM). A Gaussian process is used in the online stage to predict the correction factors for geometries that are not in the training data. The proposed framework is demonstrated using two examples. The first example considers idealized left-ventricle geometries, for which the behavior of the ROM framework can be studied in detail. In the second example, the ROM framework is applied to scan-based geometries, based on which the application of the ROM framework in the clinical setting is discussed. The results for the two examples convey that the ROM framework can provide accurate online predictions, provided that adequate FOM training data is available. The uncertainty bands provided by the ROM framework give insight into the trustworthiness of its results. Large uncertainty bands can be considered as an indicator for the further population of the training data set.

Maarten M. Hodzelmans, René R. Hiemstra, Joris J. C. Remmers et al.

For higher-order discretizations of explicit dynamics problems, Isogeometric Analysis (IGA) has several favorable properties as compared to classical Finite Element Analysis (FEA). While FEA produces spurious modes at orders beyond linear, this is not the case for IGA. Consequently, fewer degrees of freedom are required for comparable accuracy, larger timesteps can be taken, and the method is more robust for nonlinear problems. If outlier modes are removed, the timestep even becomes virtually independent of the order. In this paper, we investigate how these advantages apply to the poroelastic continuum model. We consider both a primal formulation, wherein our variables are the displacement of the matrix material, the fluid displacement, and the pressure, as well as a reduced form wherein the pressure is eliminated. For our discretizations, we employ divergence-conforming spline spaces. Conforming spline spaces for the fluid displacement ensure inf-sup stability for the primal form, as well as a correct null space in the reduced form. Furthermore, we prove and demonstrate that the two formulations coincide when both displacements are discretized with conforming spline spaces. Through spectral analysis, we find that the aforementioned benefits of IGA do carry over directly to the context of poroelasticity. In 1D, we split the discrete spectrum into fast and slow waves. When normalized against an analytical solution, each of these sub-spectra closely resembles results known in elasticity. Consequently, when poroelasticity is discretized with outlier-free IGA, the timestep is essentially independent of the order. We show this timestep scaling in 2D as well.