We study numerical methods for one-dimensional, advection-dominated transport problems, focusing on minimizing numerical diffusion. We compare explicit finite difference (FD) schemes, a global random walk (GRW) algorithm, and a method of lines (MOL) approach. Our results show that FD and GRW methods perform similarly at certain Courant numbers, while the GRW—free from numerical diffusion—effectively preserves solution shape across various Courant values. These findings highlight GRW and MOL as efficient methods for controlling numerical diffusion in advection-dominated cases.

The numerical reconstruction of the solution to the system of linear (an)isotropic elasticity in a doubly-connected three-dimensional domain from overdetermined data, also referred to as Cauchy data, available on a part of the boundary is investigated. A regularising iterative method is employed to generate a stable numerical approximation to this severely ill-posed inverse problem. The method proposed herein is an extension of the Landweber iteration for an operator equation equivalent to the Cauchy problem. Numerical examples are presented in order to confirm the accuracy, convergence, stability and robustness of the algorithm

Coagulation equations describe the evolution in time of a system of particles that are characterized by their volume. In this talk, we introduce the standard coagulation model. Depending on the interaction speed of the particles, solutions may lose mass instantaneously. We then analyze alternative frameworks and see how this property changes.

Random Feature Methods (RFMs) and their variants such as extreme learning machine finite-basis physics-informed neural networks (ELM-FBPINNs) offer a scalable approach for solving partial differential equations (PDEs) by using localized, overlapping and randomly initialized neural network basis functions to approximate the PDE solution and training them to minimize PDE residuals through solving structured least-squares problems. This combination leverages the approximation power of randomized neural networks, the parallelism of domain decomposition, and the accuracy and efficiency of least-squares solvers. However, the resulting structured least-squares systems are often {severely ill-conditioned}, due to local redundancy among random basis functions and correlation introduced by subdomain overlaps, which significantly affects the convergence of standard solvers. In this work, we introduce a block rank-revealing QR (RRQR) filtering and preconditioning}strategy that operates directly on the structured least-squares problem. First, local RRQR factorizations identify and remove redundant basis functions while preserving numerically informative ones, reducing problem size, and improving conditioning. Second, we use these factorizations to construct a right preconditioner for the global problem which preserves block-sparsity and numerical stability. Third, we derive deterministic bounds of the condition number of the preconditioned system, with probabilistic refinements for small overlaps.

Joint work with Jan Willem van Beek (TU Eindhoven) and Ben Moseley (Imperial College London).

Parametric data-driven reduced-order models (ROMs) that embed dependencies in a large number of input parameters are crucial for enabling many-query tasks in large-scale problems. These tasks, including design optimization, control, and uncertainty quantification, are essential for developing digital twins in real-world applications. However, standard training data generation methods are computationally prohibitive due to the curse of dimensionality, as their cost scales exponentially with the number of inputs.

We consider the computational efficiency of Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods applied to elliptic PDEs with random coefficients. These arise, for example, in groundwater flow modelling, where a commonly used model for the unknown parameter is a random field. We make use of the circulant embedding procedure for sampling from the aforementioned coefficient. To reduce the computational complexity of the MLMC estimator in the case of highly oscillatory random fields, we devise and implement a smoothing technique integrated into the circulant embedding method. This allows choosing the coarsest mesh on the first level of MLMC independently of the correlation length of the covariance function of the random field, leading to significant savings in computational cost. We illustrate this with numerical experiments, where we see a saving of up to a factor of 5-10 in computational cost for accuracies of practical interest.

We study the H1-stability for all positive time of a family of A-stable time-stepping methods for the 2D Navier–Stokes equations. More precisely, we discretize in time using one-leg second-order accurate A-stable methods, and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the numerical scheme admits this stability.

This work is related to pharmacological studies. It aims to develop mathematical models that reproduce experimental data of arterial compliance. The ROTSAC experimental setup [1] investigates how arterial stiffness is influenced by vasoconstrictors and vasodilators. In this experiment, aortic segments are
mounted on two parallel metal hooks and stretched with an imposed dynamic load.

I will present a first mathematical model based oh 3d Shells [2] with active fibers describing the behavior of the tissue and validate it against experimental
results. The model parameters involved in the constitutive laws are identified using real data by means of an optimization method. The resulting model is
able to reproduce the experimental data and predict the system’s behavior in different settings beyond those used for parameter estimation. This enables the
assessment of different scenarios concerning the impact of the molecules on the active or passive contributions of the arterial wall.

I will then present a more complete mathematical model for simulating the aforementioned ex vivo setup. It includes a contact mechanics model to account
for the interactions between the tissue and rigid components. This is a joint work with Sara Costa Faya, Damiano Lombardi and Miguel
Fernandez. The experiments were performed at the University of Antwerp by Pieter-Jan Guns and Callan Wesley as part of the the INSPIRE European Training Network.

References
[1] Leloup, A. J. A., Van Hove, C. E., De Moudt, S., De Meyer, G. R. Y., De Keulenaer, G. W., and Fransen, P. (2019). Vascular smooth muscle
cell contraction and relaxation in the isolated aorta: a critical regulator of large artery compliance. Physiol Rep, 7(4):e13934.
[2] Chapelle, D., Ferent, A., and Bathe, K. J. (2004). 3D shell elements and their underlying mathematical model. Math. Models Methods Appl. Sci.,
14(1):105–142.