We study a reaction-diffusion model posed on two distinct spatial scales that accounts for diffusion, aggregation, fragmentation, and deposition of populations of colloidal particles within a porous material. In this model, the macroscopic transport of the particles is described by an effective equation whose transport coefficients are determined by cell problems posed on the underlying pore scale. The internal pore geometry can change over time due to deposition or detachment of colloidal particles. We represent the evolving microstructure as solid cores whose phase boundaries can grow or shrink over time. As deposition progresses, neighbouring growing cores may come into contact, leading to local clogging of the pore space. We investigate how such evolving microstructures influence the effective transport and storage properties of porous layers. We establish basic analytical results concerning the weak solvability of the resulting multiscale evolution problem, which takes the form of a strongly non-linear parabolic system, in the non-clogging regime. For the numerical approximation of weak solutions we propose a two-scale finite element discretization. Numerical experiments illustrate how local clogging affects the effective dispersion tensor and quantify the resulting trade-off between transport efficiency and storage capacity.

We present an optimal control problem to guide the selection of morphology classes arising in organic solar cells. The study focuses on phase separation processes in polymer solvent mixtures, with particular attention to solvent evaporation as a mechanism to arrest morphology formation. We establish the existence of optimal controls and analyze the Frechet derivative of the control to state mapping. Finally, we derive the first order necessary optimality condition via the corresponding adjoint system.

We consider the mathematical analysis and homogenization of a moving boundary problem posed for a highly heterogeneous, periodically perforated domain. More specifically, we are looking at a one-phase thermo-elasticity system with phase transformations where small inclusions, initially periodically distributed, are growing or shrinking based on a kinetic under-cooling-type law and where surface stresses are created based on the curvature of the phase interface. This growth is assumed to be uniform in each individual cell of the the perforated domain. After transforming to the initial reference configuration (utilizing the Hanzawa transformation), we use the contraction mapping principle to show the existence of a unique solution for a possibly small but $\varespilon$-independent time interval ($\varespilon$ is here the scale of heterogeneity). In the homogenization limit, we discover a macroscopic thermo-elasticity problem which is strongly non-linearly coupled (via an internal parameter called height function) to local changes in geometry. As a direct byproduct of the mathematical analysis work, we present an alternative equivalent formulation which lends itself to an effective precomputing strategy that is very much needed as the limit problem is computationally expensive.

We present a hybrid stochastic-continuum model to study the sulphation of calcium carbonate and the consequent formation of gypsum, a key phenomenon driving marble deterioration. While calcium carbonate and gypsum are continuous random fields evolving according to random ordinary differential equations, the dynamics of sulfuric acid particles follow Itô-type stochastic differential equations. The particle evolution incorporates both strong repulsion between particles via the Lennard‒Jones potential and non-local interactions with the continuum environment. The particle–continuum coupling is also achieved through a chemical reaction, which is modelled as a Poisson counting process. We simulate the spatiotemporal evolution of this corrosion process using the Euler‒Maruyama algorithm with varying initial data combined with finite elements to address spatial discretization. Despite symmetric initial data, our simulations highlight an uneven progression of corrosion due to stochastic influences in the model.

Universal scaling in phase separation is typically assumed to be isotropic in systems with conserved dynamics. Here we show that boundary forcing alone can break this dynamical scaling symmetry, leading to anisotropic coarsening, with different effective global growth laws observed parallel and perpendicular to the boundary. We consider a ternary mixture with two conserved components and a passive species undergoing surface evaporation, which provides a simple setting to investigate this effect. In this case, evaporation leads to a progressive mass loss and to the formation of macroscopic concentration gradients, which, in turn, drive anisotropic coarsening, with different effective growth laws observed parallel and perpendicular to the boundary. At the same time, bulk regions appear to retain the standard Model B scaling, suggesting that the observed anisotropy is mainly induced by boundary fluxes rather than by changes in the intrinsic dynamics. Our results indicate that boundary conditions can play an important role in breaking scaling symmetry and may offer a way to influence coarsening behavior in nonequilibrium phase separation.

We consider a two-scale parabolic problem describing the water-induced swelling for a class of porous materials with elongated internal structures. The system of evolution equations we are considering here consists of a parabolic equation describing the evolution of the moisture content into a macroscopic domain coupled in a two-scale fashion to a free boundary problem capturing a microscopic swelling process. The macroscopic domain is a three-dimensional object (the target porous material), while the microscopic domains are a stack of elongated pores modeled as one-dimensional halflines connected at an edge to the macroscopic domain. By imposing a flux boundary condition at the edge of each pore, we allow the moisture content to intrude into the respective microscopic domain. In this work, we prove the existence and uniqueness of a solution to our two-scale problem. One key ingredient in our proof is the guarantee that the microscopic solution is measurable with respect to variable pointing out to the macroscopic domain. By using the Banach’s fixed-point theorem, we establish the local-in-time well-posedness of our two-scale problem.

Motivated by questions related to morphology formation in 3D involving interacting ternary mixtures, we propose a finite volume scheme to approximate numerically the unique weak solution to a coupled system of parab olic equations with nonlinear and nonlocal drift. The special feature of our system is that the coupling takes place precisely via the structure of the drift terms. We prove the convergence of the scheme towards the unique solution of the target evolution system and explore as well the stability of the solution with respect to selected parameters. We illustrate numerically in 3D the appearance of the wanted morphologies and compute as well the empirical order of convergence of the numerical approximations towards their limit.

When designing schools and highways, engineers need to know how people might move through these structures. If they can accurately describe how crowds will react to their designs before they build the various structures, they can prepare for emergencies, avoid traffic jams, and make the flow of crowds more efficient. In this article, we provide an introduction to the mathematical modeling of traffic flow by describing how individual people might act while moving in large crowds. We provide links to an experiment with real people, a model of the experiment, and interactive simulations for readers to see how well the mathematical model mimics a real crowd and for the readers to explore the features of similar models.

We consider a classical model of non-equilibrium statistical mechanics accounting for non-Markovian effects, which is referred to as the Generalized Langevin Equation in the literature. We derive reduced Markovian descriptions obtained through the neglection of inertial terms and/or heat bath variables. The adopted reduction scheme relies on the framework of the Invariant Manifold method, which allows to retain the slow degrees of freedom from a multiscale dynamical system. Our approach is also rooted on the Fluctuation-Dissipation Theorem, which helps preserve the proper dissipative structure of the reduced dynamics. We highlight the appropriate time scalings introduced within our procedure, and also prove the commutativity of selected reduction paths.

Synthetic data generation is increasingly used in applications involving privacy preservation, data sharing, and data scarcity. In many situations, preserving the dependence structure of the original data is of central interest. In this work, we propose a lightweight postprocessing methodology for synthetic tabular data based on the Orthogonal Procrustes problem. Starting from an already generated synthetic dataset, our approach constructs the closest dataset that restores the Pearson correlation structure of the original data. On the theoretical side, we show that preserving Pearson correlation is equivalent to the action of linear orthogonal maps in the centered-data subspace, and then deploy the Orthogonal Procrustes problem. However, in order for this to hold, we first establish a result ensuring that applying the Orthogonal Procrustes step remains in the aforementioned subspace under suitable assumptions. Applications to several datasets and synthetic data generators illustrate the effectiveness of the proposed approach. In particular, the numerical experiments indicate that the correlation structure can be restored while largely preserving the individual feature distributions, the geometry of the data, and the performance of downstream classification tasks.

We investigate a two-scale system featuring an upscaled parabolic dispersion-reaction equation intimately linked to a family of elliptic cell problems. The system is strongly coupled through a dispersion tensor, which depends on the solutions to the cell problems, and via the cell problems themselves, where the solution of the parabolic problem interacts nonlinearly with the drift term. This particular mathematical structure is motivated by a rigorously derived upscaled reaction-diffusion-convection model that describes the evolution of a population of interacting particles pushed by a large drift through an array of periodically placed obstacles (i.e., through a regular porous medium). We prove the existence and uniqueness of weak solutions to our system by means of an iterative scheme, where particular care is needed to ensure the uniform positivity of the dispersion tensor. Additionally, we use finite element-based approximations for the same iteration scheme to perform multiple simulation studies. Finally, we highlight how the choice of micro-geometry (building the regular porous medium) and of the nonlinear drift coupling affects the macroscopic dispersion of particles.

Film formation from solvent evaporation in polymer ternary solutions is relevant for several technological applications, such as the fabrication of organic solar cells. The performance of the final device will strongly depend on the internal morphology of the obtained film, which, in turn, is affected by the processing conditions. We are interested in modeling morphology formation in 3D for ternary mixtures using both a lattice model and its continuous counterpart in the absence of evaporation. In our previous works, we found that, in 2D, both models predict the existence of two distinct regimes: (i) a low-solvent regime, characterized by two interpenetrated domains of the two polymers, and (ii) a high-solvent regime, where isolated polymer domains are dispersed in the solvent background. In the significantly more intriguing 3D case, we observe a comparable scenario both for the discrete and the continuous model. The lattice model reveals its ability to describe morphology formation even in the high solvent content 3D case, in which the three-dimensional nature of space could have prevented cluster formation.

Inspired by experimental evidence collected when processing thin films from ternary solutions made of two solutes, typically polymers, and one solvent, we computationally study the morphology formation of domains obtained in three-state systems using both a lattice model and a continuum counterpart. The lattice-based approach relies on the Blume-Capel nearest neighbor model with bulk conservative Kawasaki dynamics, whereas as continuum system we consider a coupled system of evolution equations that is derived as hydrodynamic limit when replacing the nearest neighbor interaction in the lattice case by a suitable Kac potential. We explore how the obtained morphology depends on the solvent content in the mixture. In particular, we study how these scenarios change when the solvent is allowed to evaporate.

We provide conditions under which we prove for measure-valued transport equations with non-linear reaction term in the space of finite signed Radon measures, that positivity is preserved, as well as absolute continuity with respect to Lebesgue measure, if the initial condition has that property. Moreover, if the initial condition has L^p regular density, then the solution has the same property.

This work presents global random walk approximations of solutions to one-dimensional Stefan-type moving-boundary problems. We are particularly interested in the case when the moving boundary is driven by an explicit representation of its speed. This situation is usually referred to in the literature as moving-boundary problem with kinetic condition. As a direct application, we propose a numerical scheme to forecast the penetration of small diffusants into a rubber-based material. To check the quality of our results, we compare the numerical results obtained by global random walks either using the analytical solution to selected benchmark cases or relying on finite element approximations with a priori known convergence rates. It turns out that the global random walk concept can be used to produce good quality approximations of the weak solutions to the target class of problems.

We study a nonlinear coupled parabolic system with non-local drift terms modeling at the continuum level the inter-species interaction within a ternary mixture that allows the evaporation of one of the species. In the absence of evaporation, the proposed system coincides with the hydrodynamic limit of a stochastic interacting particle system of Blume–Capel-type driven by the Kawasaki dynamics. Similar governing dynamics are found in models used to study morphology formation in the design of organic solar cells, thin adhesive bands, and other applications. We investigate the well-posedness of the target system and present preliminary numerical simulations which incorporate ‘from the top’ evaporation into the model. We employ a finite volumes scheme to construct approximations of the weak solution and illustrate how the evaporation process can affect the shape and connectivity of the evolving-in-time morphologies.

We propose a two-scale finite element method designed for heterogeneous microstructures. Our approach exploits domain diffeomorphisms between the microscopic structures to gain computational efficiency. By using a conveniently constructed pullback operator, we are able to model the different microscopic domains as macroscopically dependent deformations of a reference domain. This allows for a relatively simple finite element framework to approximate the underlying PDE system with a parallel computational structure. We apply this technique to a model problem where we focus on transport in plant tissues. We illustrate the accuracy of the implementation with convergence benchmarks and show satisfactory parallelization speed-ups. We further highlight the effect of the heterogeneous microscopic structure on the output of the two-scale systems. Our implementation (publicly available on GitHub) builds on the deal.II FEM library. Application of this technique allows for an increased capacity of microscopic detail in multiscale modeling, while keeping running costs manageable.

The effective, fast transport of matter through porous media is often characterized by complex dispersion effects. To describe in mathematical terms such situations, instead of a simple macroscopic equation (as in the classical Darcy's law), one may need to consider two-scale boundary-value problems with full coupling between the scales where the macroscopic transport depends non-linearly on local (i.e. microscopic) drift interactions, which are again influenced by local concentrations. Such two-scale problems are computationally very expensive as numerous elliptic partial differential equations (cell problems) have to constantly be recomputed. In this work, we investigate such an effective two-scale model involving a suitable nonlinear dispersion term and explore numerically the behavior of its weak solutions. We introduce two distinct numerical schemes dealing with the same non-linear scale-coupling: (i) a Picard-type iteration and (ii) a time discretization decoupling. In addition, we propose a precomputing strategy where the calculations of cell problems are pushed into an offline phase. Our approach works for both schemes and significantly reduces computation times. We prove that the proposed precomputing strategy converges to the exact solution. Finally, we test our schemes via several numerical experiments that illustrate dispersion effects introduced by specific choices of microstructure and model ingredients.

We propose a method to generate statistically representative synthetic data from a given dataset. The main goal of our method is for the created data set to mimic the inter–feature correlations present in the original data, while also offering a tunable parameter to influence the privacy level. In particular, our method constructs a statistical map by using the empirical conditional distributions between the features of the original dataset. Part of the tunability is achieved by limiting the depths of conditional distributions that are being used. We describe in detail our algorithms used both in the construction of a statistical map and how to use this map to generate synthetic observations. This approach is tested in three different ways: with a hand calculated example; a manufactured dataset; and a real world energy-related dataset of consumption/production of households in Madeira Island. We evaluate the method by comparing the datasets using the Pearson correlation matrix with different levels of resolution and depths of correlation. These two considerations are being viewed as tunable parameters influencing the resulting datasets fidelity and privacy. The proposed methodology is general in the sense that it does not rely on the used test dataset. We expect it to be applicable in a much broader context than indicated here.

We present a finite-volume based numerical scheme for a nonlocal Cahn-Hilliard equation which combines ideas from recent numerical schemes for gradient flow equations and nonlocal Cahn-Hilliard equations. The equation of interest is a special case of a previously derived and studied system of equations which describes phase separation in ternary mixtures. We prove the scheme is both energy stable and respects the analytical bounds of the solution. Furthermore, we present numerical demonstrations of the theoretical results using both the Flory-Huggins (FH) and Ginzburg-Landau (GL) free-energy potentials.