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These lecture notes cover advanced topics in linear regression, with an in-depth exploration of the existence, uniqueness, relations, computation, and non-asymptotic properties of the most prominent estimators in this setting. The covered estimators include least squares, ridgeless, ridge, and lasso. The content follows a proposition-proof structure, making it suitable for students seeking a formal and rigorous understanding of the statistical theory underlying machine learning methods.

The mathematical modeling of crowds is complicated by the fact that crowds possess the behavioral ability to develop and adapt moving strategies in response to the context. For example, in emergency situations, people tend to alter their walking strategy in response to fear. To be able to simulate these situations, we consider a kinetic model of crowd dynamics that features the level of stress as a parameter and propose to estimate this key parameter by solving an inverse crowd dynamics problem. This paper states the mathematical problem and presents a method for its numerical solution. We show some preliminary results based on a synthetic data set, i.e., test cases where the exact stress level is known and the crowd density data are generated numerically by solving a forward crowd dynamics problem., Comment: 31 pages, 18 figures

We present a novel reduced-order Model (ROM) that leverages optimal transport (OT) theory and displacement interpolation to enhance the representation of nonlinear dynamics in complex systems. While traditional ROM techniques face challenges in this scenario, especially when data (i.e., observational snapshots) is limited, our method addresses these issues by introducing a data augmentation strategy based on OT principles. The proposed framework generates interpolated solutions tracing geodesic paths in the space of probability distributions, enriching the training dataset for the ROM. A key feature of our approach is its ability to provide a continuous representation of the solution's dynamics by exploiting a virtual-to-real time mapping. This enables the reconstruction of solutions at finer temporal scales than those provided by the original data. To further improve prediction accuracy, we employ Gaussian Process Regression to learn the residual and correct the representation between the interpolated snapshots and the physical solution. We demonstrate the effectiveness of our methodology with atmospheric mesoscale benchmarks characterized by highly nonlinear, advection-dominated dynamics. Our results show improved accuracy and efficiency in predicting complex system behaviors, indicating the potential of this approach for a wide range of applications in computational physics and engineering.

We develop and analyze a splitting method for fluid-poroelastic structure interaction. The fluid is described using the Stokes equations and the poroelastic structure is described using the Biot equations. The transmission conditions on the interface are mass conservation, balance of stresses, and the Beavers-Joseph-Saffman condition. The splitting method involves single and decoupled Stokes and Biot solves at each time step. The subdomain problems use Robin boundary conditions on the interface, which are obtained from the transmission conditions. The Robin data is represented by an auxiliary interface variable. We prove that the method is unconditionally stable and establish that the time discretization error is $\mathcal{O}(\sqrt{T}\Delta t)$, where $T$ is the final time and $\Delta t$ is the time step. We further study the iterative version of the algorithm, which involves an iteration between the Stokes and Biot sub-problems at each time step. We prove that the iteration converges to a monolithic scheme with a Robin Lagrange multiplier used to impose the continuity of the velocity. Numerical experiments are presented to illustrate the theoretical results.

The two-layer quasi-geostrophic equations (2QGE) is a simplified model that describes the dynamics of a stratified, wind-driven ocean in terms of potential vorticity and stream function. Its numerical simulation is plagued by a high computational cost due to the size of the typical computational domain and the need for high resolution to capture the full spectrum of turbulent scales. In this paper, we present a data-driven reduced order model (ROM) for the 2QGE that drastically reduces the computational time to predict ocean dynamics, especially when there are variable physical parameters. The main building blocks of our ROM are: i) proper orthogonal decomposition (POD) and ii) long short-term memory (LSTM) recurrent neural networks. Snapshots data are collected from a high-resolution simulation for part of the time interval of interest and for given parameter values in the case of variable parameters. POD is applied to each field variable to extract the dominant modes and a LSTM model is trained on the modal coefficients associated with the snapshots for each variable. Then, the trained LSTM models predict the modal coefficients for the remaining part of the time interval of interest and for a new parameter value. To illustrate the predictive performance of our POD-LSTM ROM and the corresponding time savings, we consider an extension of the so-called double-gyre wind forcing test. We show that the POD-LSTM ROM is accurate in predicting both time-averaged fields and time-dependent quantities (modal coefficients, enstrophy, and kinetic energy), even when retaining only 10-20\% of the singular value energy of the system. The computational speed up for the prediction is about up to 1E+07 compared to a finite volume based full order method., Comment: 41 pages, 23 figures

Reduced order models (ROMs) have achieved a lot of success in reducing the computational cost of traditional numerical methods across many disciplines. For convection-dominated (e.g., turbulent) flows, however, standard ROMs generally yield inaccurate results, usually affected by spurious oscillations. Thus, ROMs are usually equipped with numerical stabilization or closure models to account for the effect of the discarded modes. The literature on ROM closures and stabilizations is large and growing fast. In this paper, we focus on one particular type of ROM closures and stabilizations that are inspired by Large Eddy Simulation (LES). These ROMs, which we call LES-ROMs, are extremely easy to implement, very efficient, and accurate. Carefully tuned LES-ROMs can accurately capture the average physical quantities of interest in challenging convection-dominated flows in many applications. LES-ROM are constructed by leveraging spatial filtering, i.e., the same principle used to build classical LES models. This ensures a modeling consistency between LES-ROMs and the approaches that generated the data used to train them. It also ``bridges'' two distinct research fields (LES and ROMs), disconnected until now. This paper is a review of LES-ROMs. It starts with a description of a versatile LES strategy called evolve-filter-relax (EFR) that has been successfully used as a full order method. We then show how the EFR strategy, and spatial filtering in general, can be leveraged to construct LES-ROMs. Several applications of LES-ROMs are presented. Finally, we draw conclusions and outline several research directions and open questions in the LES-ROM development. While we do not claim this review to be comprehensive, we certainly hope it serves as a brief and friendly introduction to this exciting research area, which has a lot of potential in practical numerical simulation of convection-dominated flows.

We present a comparative computational study of two stabilized Reduced Order Models (ROMs) for the simulation of convection-dominated incompressible flow (Reynolds number of the order of a few thousands). Representative solutions in the parameter space, which includes either time only or time and Reynolds number, are computed with a Finite Volume method and used to generate a reduced basis via Proper Orthogonal Decomposition (POD). Galerkin projection of the Navier-Stokes equations onto the reduced space is used to compute the ROM solution. To ensure computational efficiency, the number of POD modes is truncated and ROM solution accuracy is recovered through two stabilization methods: i) adding a global constant artificial viscosity to the reduced dimensional model, and ii) adding a different value of artificial viscosity for the different POD modes. We test the stabilized ROMs for fluid flow in an idealized medical device consisting of a conical convergent, a narrow throat, and a sudden expansion. Both stabilization methods significantly improve the ROM solution accuracy over a standard (non-stabilized) POD-Galerkin model.

We develop, and implement in a Finite Volume environment, a density-based approach for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. The well-balancing of the approach is ensured by a local hydrostatic reconstruction updated in runtime during the simulation to keep the numerical error under control. To approximate the solution of the Riemann problem, we consider four methods: Roe-Pike, HLLC, AUSM+-up and HLLC-AUSM. We assess our density-based approach and compare the accuracy of these four approximated Riemann solvers using two two classical benchmarks, namely the smooth rising thermal bubble and the density current., Comment: 22 pages

Although the two-layer quasi-geostrophic equations (2QGE) are a simplified model for the dynamics of a stratified, wind-driven ocean, their numerical simulation is still plagued by the need for high resolution to capture the full spectrum of turbulent scales. Since such high resolution would lead to unreasonable computational times, it is typical to resort to coarse low-resolution meshes combined with the so-called eddy viscosity parameterization to account for the diffusion mechanisms that are not captured due to mesh under-resolution. We propose to enable the use of further coarsened meshes by adding a (linear or nonlinear) differential low-pass to the 2QGE, without changing the eddy viscosity coefficient. While the linear filter introduces constant (additional) artificial viscosity everywhere in the domain, the nonlinear filter relies on an indicator function to determine where and how much artificial viscosity is needed. Through several numerical results for a double-gyre wind forcing benchmark, we show that with the nonlinear filter we obtain accurate results with very coarse meshes, thereby drastically reducing the computational time (speed up ranging from 30 to 300)., Comment: 34 pages, 13 figures

We consider three conservative forms of the mildly compressible Euler equations, called CE1, CE2 and CE3, with the goal of understanding which leads to the most accurate and robust pressure-based solver in a finite volume environment. Forms CE1 and CE2 are both written in density, momentum, and specific enthalpy, but employ two different treatments of the buoyancy and pressure gradient terms: for CE1 it is the standard pressure splitting implemented in open-source finite volume solvers (e.g., OpenFOAM), while for CE2 it is the typical pressure splitting found in computational atmospheric studies. Form CE3 is written in density, momentum, and potential temperature, with the buoyancy and pressure terms addressed as in CE2. For each formulation, we adopt a computationally efficient splitting approach. The three formulations are thoroughly assessed and compared through six benchmark tests involving dry air flow over a flat terrain or orography. We found that all three models are able to provide accurate results for the tests with a flat terrain, although the solvers based on the CE2 and CE3 forms are more robust. As for the mountain tests, CE1 solutions become unstable, while the CE2 and CE3 models provide results in very good agreement with data in the literature, the CE3 model being the most accurate. Hence, the CE3 model is the most accurate, reliable, and robust for the simulation of mesoscale atmospheric flows when using a pressure-based approach and space discretization by a finite volume method., Comment: 23 pages, 15 figures, 3 tables

Dynamical encounters of stellar-mass black holes (BHs) in a disc of compact objects around a supermassive BH (SMBH) can accelerate the formation and coalescence of BH binaries. It has been proposed that binary-single encounters among BHs in such discs can lead to an excess of highly-eccentric BH mergers. However, previous studies have neglected how the disc velocity dispersion and the SMBH's tidal field affect the 3-body dynamics. We investigate the outcomes of binary-single encounters considering different values of the disc velocity dispersion, and examine the role of the SMBH's tidal field. We then demonstrate how their inclusion affects the properties of merging BH binaries. We perform simulations of 4-body encounters (i.e. with the SMBH as fourth particle) using the highly-accurate, regularized code TSUNAMI, which includes post-Newtonian corrections up to order 3.5PN. The disc velocity dispersion controls how orbits in the disc are aligned and circular, and determines the relative velocity of the binary-single pair before the encounter. As the velocity dispersion decreases, the eccentricity of post-encounter binaries transitions from thermal to superthermal, and binaries experience enhanced hardening. The transition between these two regimes happens at disc eccentricities and inclinations of order e ~ i ~ 10^-4. These distinct regimes correspond to a disc dominated by random motions, and one dominated by the Keplerian shear. The inclusion of the SMBH's tidal field and the disc velocity dispersion can significantly affect the number of GW mergers, and especially the number of highly-eccentric inspirals. These can be up to ~2 times higher at low velocity dispersion, and ~12 times lower at high velocity dispersions. The spin-orbit alignment is influenced by the tidal field exclusively at high velocity dispersions, effectively inhibiting the formation of anti-aligned binary BHs., Comment: 12 pages, 7 figures, accepted for publication in Astronomy & Astrophysics

This paper studies the fusogenicity of cationic liposomes in relation to their surface distribution of cationic lipids and utilizes membrane phase separation to control this surface distribution. It is found that concentrating the cationic lipids into small surface patches on liposomes, through phase-separation, can enhance liposome's fusogenicity. Further concentrating these lipids into smaller patches on the surface of liposomes led to an increased level of fusogenicity. These experimental findings are supported by numerical simulations using a mathematical model for phase-separated charged liposomes. Findings of this study may be used for design and development of highly fusogenic liposomes with minimal level of toxicity.

The simulation of atmospheric flows by means of traditional discretization methods remains computationally intensive, hindering the achievement of high forecasting accuracy in short time frames. In this paper, we apply three reduced order models that have successfully reduced the computational time for different applications in computational fluid dynamics while preserving accuracy: Dynamic Mode Decomposition (DMD), Hankel Dynamic Mode Decomposition (HDMD), and Proper Orthogonal Decomposition with Interpolation (PODI). The three methods are compared in terms of computational time and accuracy in the simulation of two well-known benchmarks for mesoscale flow. The accuracy of the DMD and HDMD solutions deteriorates rather quickly as the forecast time window expands, although these methods are designed to predict the dynamics of a system. The reason is likely the strong nonlinearity in the benchmark flows. The PODI solution is accurate for the entire duration of the time interval of interest thanks to the use of interpolation with radial basis functions. This holds true also when the model features a physical parameter expected to vary in a given range, as is typically the case in weather prediction.

The paper studies a scalar auxiliary variable (SAV) method to solve the Cahn-Hilliard equation with degenerate mobility posed on a smooth closed surface {\Gamma}. The SAV formulation is combined with adaptive time stepping and a geometrically unfitted trace finite element method (TraceFEM), which embeds {\Gamma} in R3. The stability is proven to hold in an appropriate sense for both first- and second-order in time variants of the method. The performance of our SAV method is illustrated through a series of numerical experiments, which include systematic comparison with a stabilized semi-explicit method., Comment: 23 pages, 12 figures

We present a filter stabilization technique for the mildly compressible Euler equations that relies on a linear or nonlinear indicator function to identify the regions of the domain where artificial viscosity is needed and determine its amount. For the realization of this technique, we adopt a three step algorithm called Evolve-Filter-Relax (EFR), which at every time step evolves the solution (i.e., solves the Euler equations on a coarse mesh), then filters the computed solution, and finally performs a relaxation step to combine the filtered and non-filtered solutions. We show that the EFR algorithm is equivalent to an eddy-viscosity model in Large Eddy Simulation. Three indicator functions are considered: a constant function (leading to a linear filter), a function proportional to the norm of the velocity gradient (recovering a Smagorinsky-like model), and a function based on approximate deconvolution operators. Through well-known benchmarks for atmospheric flow, we show that the deconvolution-based filter yields stable solutions that are much less dissipative than the linear filter and the Samgorinsky-like model and we highlight the efficiency of the EFR algorithm.

We introduce GEA (Geophysical and Environmental Applications), a new open-source atmosphere and ocean modeling framework within the finite volume C++ library OpenFOAM. Here, we present the development of a non-hydrostatic atmospheric model consisting of a pressure-based solver for the Euler equations written in conservative form using density, momentum, and total energy as variables. We validate the solver for two idealized test cases involving buoyancy driven flows: smooth and non-smooth rising thermal bubble. Through qualitative and quantitative comparisons against numerical data available in the literature, we show that our approach is accurate., Comment: 8 pages, 2 figures, 2 tables. arXiv admin note: substantial text overlap with arXiv:2302.04836

Within OpenFOAM, we develop a pressure-based solver for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. For the stabilization of the Euler equations and to capture sub-grid processes, we consider two Large Eddy Simulation models: the classical Smagorinsky model and the one equation eddy-viscosity model. To achieve high computational efficiency, our solver uses a splitting scheme that decouples the computation of each variable. The numerical results obtained with our solver are validated against numerical data available in the literature for two classical benchmarks: the rising thermal bubble and the density current. Through qualitative and quantitative comparisons, we show that our approach is accurate. This work is meant to lay the foundation for a new open source package specifically created for quick assessment of new computational approaches for the simulation of atmospheric flows at the mesoscale level

We propose a regularization for Reduced Order Models (ROMs) of the quasi-geostrophic equations (QGE) to increase accuracy when the Proper Orthogonal Decomposition (POD) modes retained to construct the reduced basis are insufficient to describe the system dynamics. Our regularization is based on the so-called BV-alpha model, which modifies the nonlinear term in the QGE and adds a linear differential filter for the vorticity. To show the effectiveness of the BV-alpha model for ROM closure, we compare the results computed by a POD-Galerkin ROM with and without regularization for the classical double-gyre wind forcing benchmark. Our numerical results show that the solution computed by the regularized ROM is more accurate, even when the retained POD modes account for a small percentage of the eigenvalue energy. Additionally, we show that, although computationally more expensive that the ROM with no regularization, the regularized ROM is still a competitive alternative to full order simulations of the QGE., Comment: 19 pages, 4 tables, 9 figures

Numerical weather prediction is pushing the envelope of grid resolution at local and global scales alike. Aiming to model topography with higher precision, a handful of articles introduced unstructured vertical grids and tested them for dry atmospheres. The next step towards effective high-resolution unstructured grids for atmospheric modeling requires that also microphysics is independent of any vertical columns, in contrast to what is ubiquitous across operational and research models. In this paper, we present a non-column based continuous and discontinuous spectral element implementation of Kessler's microphysics with warm rain as a first step towards fully unstructured atmospheric models. We test the proposed algorithm against standard three-dimensional benchmarks for precipitating clouds and show that the results are comparable with those presented in the literature across all of the tested effective resolutions. While presented for both continuous and discontinuous spectral elements in this paper, the method that we propose can very easily be adapted to any numerical method utilized in other research and legacy codes.

We consider an optimal flow control problem in a patient-specific coronary artery bypass graft with the aim of matching the blood flow velocity with given measurements as the Reynolds number varies in a physiological range. Blood flow is modelled with the steady incompressible Navier-Stokes equations. The geometry consists in a stenosed left anterior descending artery where a single bypass is performed with the right internal thoracic artery. The control variable is the unknown value of the normal stress at the outlet boundary, which is need for a correct set-up of the outlet boundary condition. For the numerical solution of the parametric optimal flow control problem, we develop a data-driven reduced order method that combines proper orthogonal decomposition (POD) with neural networks. We present numerical results showing that our data-driven approach leads to a substantial speed-up with respect to a more classical POD-Galerkin strategy proposed in [59], while having comparable accuracy.

We build a unifying convex analysis framework characterizing the statistical properties of a large class of penalized estimators, both under a regular and an irregular design. Our framework interprets penalized estimators as proximal estimators, defined by a proximal operator applied to a corresponding initial estimator. We characterize the asymptotic properties of proximal estimators, showing that their asymptotic distribution follows a closed-form formula depending only on (i) the asymptotic distribution of the initial estimator, (ii) the estimator's limit penalty subgradient and (iii) the inner product defining the associated proximal operator. In parallel, we characterize the Oracle features of proximal estimators from the properties of their penalty's subgradients. We exploit our approach to systematically cover linear regression settings with a regular or irregular design. For these settings, we build new $\sqrt{n}-$consistent, asymptotically normal Ridgeless-type proximal estimators, which feature the Oracle property and are shown to perform satisfactorily in practically relevant Monte Carlo settings.

We investigate various data-driven methods to enhance projection-based model reduction techniques with the aim of capturing bifurcating solutions. To show the effectiveness of the data-driven enhancements, we focus on the incompressible Navier-Stokes equations and different types of bifurcations. To recover solutions past a Hopf bifurcation, we propose an approach that combines proper orthogonal decomposition with Hankel dynamic mode decomposition. To approximate solutions close to a pitchfork bifurcation, we combine localized reduced models with artificial neural networks. Several numerical examples are shown to demonstrate the feasibility of the presented approaches.

We present a Large Eddy Simulation (LES) approach based on a nonlinear differential low-pass filter for the simulation of two-dimensional barotropic flows with under-refined meshes. For the implementation of such model, we choose a segregated three-step algorithm combined with a computationally efficient Finite Volume method. We assess the performance of our approach on the classical double-gyre wind forcing benchmark. The numerical experiments we present demonstrate that our nonlinear filter is an improvement over a linear filter since it is able to recover the four-gyre pattern of the time-averaged stream function even with extremely coarse meshes. In addition, our LES approach provides an average kinetic energy that compares well with the one computed with a Direct Numerical Simulation., Comment: 16 pages, 11 figures

This work introduces a novel approach for data-driven model reduction of time-dependent parametric partial differential equations. Using a multi-step procedure consisting of proper orthogonal decomposition, dynamic mode decomposition and manifold interpolation, the proposed approach allows to accurately recover field solutions from a few large-scale simulations. Numerical experiments for the Rayleigh-B\'{e}nard cavity problem show the effectiveness of such multi-step procedure in two parametric regimes, i.e.~medium and high Grashof number. The latter regime is particularly challenging as it nears the onset of turbulent and chaotic behaviour. A major advantage of the proposed method in the context of time-periodic solutions is the ability to recover frequencies that are not present in the sampled data.

We develop a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for the efficient numerical simulation of the parametric Navier-Stokes equations in the stream function-vorticity formulation. Unlike previous works, we choose different reduced coefficients for the vorticity and stream function fields. In addition, for parametric studies we use a global POD basis space obtained from a database of time dependent full order snapshots related to sample points in the parameter space. We test the performance of our ROM strategy with the vortex merger benchmark. Accuracy and efficiency are assessed for both time reconstruction and physical parametrization., Comment: 19 pages, 18 figures

Liposomes that achieve a heterogeneous and spatially organized surface through phase separation have been recognized to be a promising platform for delivery purposes. However, their design and optimization through experimentation can be expensive and time-consuming. To assist with the design and reduce the associated cost, we propose a computational platform for modeling membrane coarsening dynamics based on the principles of continuum mechanics and thermodynamics. This model couples phase separation to lateral flow and accounts for different membrane fluidity within the different phases, which is known to affect the coarsening dynamics on lipid membranes. The simulation results are in agreement with the experimental data in terms of liquid ordered domains area fraction, total domains perimeter over time and total number of domains over time for two different membrane compositions (DOPC:DPPC with a 1:1 molar ratio with 15% Chol and DOPC:DPPC with a 1:2 molar ratio with 25% Chol) that yield opposite and nearly inverse phase behavior. This quantitative validation shows that the developed platform can be a valuable tool in complementing experimental practice. Keywords: Multicomponent Membranes; Membrane fluidity; Membrane Phase Separation; Computational Modeling; Fluorescence Microscopy; Liposomes, Comment: 21 pages, 13 figures

We consider phase-field models with and without lateral flow for the numerical simulation of lateral phase separation and coarsening in lipid membranes. For the numerical solution of these models, we apply an unfitted finite element method that is flexible in handling complex and possibly evolving shapes in the absence of an explicit surface parametrization. Through several numerical tests, we investigate the effect of the presence of lateral flow on the evolution of phases. In particular, we focus on understanding how variable line tension, viscosity, membrane composition, and surface shape affect the pattern formation. Keywords: Lateral phase separation, surface Cahn-Hilliard equation, lateral flow, surface Navier-Stokes-Cahn-Hilliard system, TraceFEM, Comment: 21 pages, 12 figures

We develop a Reduced Order Model (ROM) for a Large Eddy Simulation (LES) approach that combines a three-step algorithm called Evolve-Filter-Relax (EFR) with a computationally efficient finite volume method. The main novelty of our ROM lies in the use within the EFR algorithm of a nonlinear, deconvolution-based indicator function that identifies the regions of the domain where the flow needs regularization. The ROM we propose is a hybrid projection/data-driven strategy: a classical Proper Orthogonal Decomposition Galerkin projection approach for the reconstruction of the velocity and the pressure fields and a data-driven reduction method to approximate the indicator function used by the nonlinear differential filter. This data-driven technique is based on interpolation with Radial Basis Functions. We test the performance of our ROM approach on two benchmark problems: 2D and 3D unsteady flow past a cylinder at Reynolds number 0 <= Re <= 100. The accuracy of the ROM is assessed against results obtained with the full order model for velocity, pressure, indicator function and time evolution of the aerodynamics coefficients., Comment: 27 pages, 19 figures, 4 tables. arXiv admin note: text overlap with arXiv:2009.13593

We focus on the modeling and simulation of an infectious disease spreading in a medium size population occupying a confined environment, such as an airport terminal, for short periods of time. Because of the size of the crowd and venue, we opt for a kinetic type model. The paper is divided into two parts. In the first part, we adopt the simplifying assumption that people's walking speed and direction are given. The resulting kinetic model features a variable that denotes the level of exposure to people spreading the disease, a parameter describing the contagion interaction strength, and a kernel function that is a decreasing function of the distance between a person and a spreading individual. Such model is tested on problems involving a small crowd in a square walkable domain. In the second part, ideas from the simplified model are used to incorporate disease spreading in a kinetic theory approach for crowd dynamics, i.e. the walking speed and direction result from interaction with other people and the venue., Comment: 20 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:2103.15151

We present a stabilized POD-Galerkin reduced order method (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. In both steps of the EF algorithm, velocity and pressure fields are approximated using different POD basis and coefficients. To achieve pressure stabilization, we consider and compare two strategies: the pressure Poisson equation and the supremizer enrichment of the velocity space. We show that the evolve and filtered velocity spaces have to be enriched with the supremizer solutions related to both evolve and filter pressure fields in order to obtain stable and accurate solutions with the supremizer enrichment method. We test our ROM approach on 2D unsteady flow past a cylinder at Reynolds number 0 <= Re <= 100. We find that both stabilization strategies produce comparable errors in the reconstruction of the lift and drag coefficients, with the pressure Poisson equation method being more computationally efficient., Comment: 18 pages, 4 figures, 3 tables. arXiv admin note: substantial text overlap with arXiv:2009.13593

This paper studies a model of two-phase flow with an immersed material viscous interface and a finite element method for numerical solution of the resulting system of PDEs. The interaction between the bulk and surface media is characterized by no-penetration and slip with friction interface conditions. The system is shown to be dissipative and a model stationary problem is proved to be well-posed. The finite element method applied in this paper belongs to a family of unfitted discretizations. The performance of the method when model and discretization parameters vary is assessed. Moreover, an iterative procedure based on the splitting of the system into bulk and surface problems is introduced and studied numerically.

The paper considers a thermodynamically consistent phase-field model of a two-phase flow of incompressible viscous fluids. The model allows for a non-linear dependence of fluid density on the phase-field order parameter. Driven by applications in biomembrane studies, the model is written for tangential flows of fluids constrained to a surface and consists of (surface) Navier-Stokes-Cahn-Hilliard type equations. We apply an unfitted finite element method to discretize the system and introduce a fully discrete time-stepping scheme with the following properties: (i) the scheme decouples the fluid and phase-field equation solvers at each time step, (ii) the resulting two algebraic systems are linear, and (iii) the numerical solution satisfies the same stability bound as the solution of the original system under some restrictions on the discretization parameters. Numerical examples are provided to demonstrate the stability, accuracy, and overall efficiency of the approach. Our computational study of several two-phase surface flows reveals some interesting dependencies of flow statistics on the geometry., Comment: 22 pages, 5 figures, 1 table

We present some ideas on how to extend a kinetic type model for crowd dynamics to account for an infectious disease spreading. We focus on a medium size crowd occupying a confined environment where the disease is easily spread. The kinetic theory approach we choose uses tools of game theory to model the interactions of a person with the surrounding people and the environment and it features a parameter to represent the level of stress. It is known that people choose different walking strategies when subjected to fear or stressful situations. To demonstrate that our model for crowd dynamics could be used to reproduce realistic scenarios, we simulate passengers in one terminal of Hobby Airport in Houston. In order to model disease spreading in a walking crowd, we introduce a variable that denotes the level of exposure to people spreading the disease. In addition, we introduce a parameter that describes the contagion interaction strength and a kernel function that is a decreasing function of the distance between a person and a spreading individual. We test our contagion model on a problem involving a small crowd walking through a corridor., Comment: 27 pages, 12 figures. arXiv admin note: text overlap with arXiv:1901.07620

The goal of this paper is to test solids4Foam, the fluid-structure interaction (FSI) toolbox developed for foam-extend (a branch of OpenFOAM), and assess its flexibility in handling more complex flows. For this purpose, we consider the interaction of an incompressible fluid described by a Leray model with a hyperelastic structure modeled as a Saint Venant-Kirchhoff material. We focus on a strongly coupled, partitioned fluid-structure interaction (FSI) solver in a finite volume environment, combined with an arbitrary Lagrangian-Eulerian approach to deal with the motion of the fluid domain. For the implementation of the Leray model, which features a nonlinear differential low-pass filter, we adopt a three-step algorithm called Evolve-Filter-Relax. We validate our approach against numerical data available in the literature for the 3D cross flow past a cantilever beam at Reynolds number 100 and 400., Comment: 18 pages, 4 figures, 3 tables

We present an isoparametric unfitted finite element approach of the CutFEM or Nitsche-XFEM family for the simulation of two-phase Stokes problems with slip between phases. For the unfitted generalized Taylor--Hood finite element pair $\mathbf{P}_{k+1}-P_k$, $k\ge1$, we show an inf-sup stability property with a stability constant that is independent of the viscosity ratio, slip coefficient, position of the interface with respect to the background mesh and, of course, mesh size. In addition, we prove stability and optimal error estimates that follow from this inf-sup property. We provide numerical results in two and three dimensions to corroborate the theoretical findings and demonstrate the robustness of our approach with respect to the contrast in viscosity, slip coefficient value, and position of the interface relative to the fixed computational mesh.

In this paper, we present a computational modeling approach for the dynamics of human crowds, where the spreading of an emotion (specifically fear) has an influence on the pedestrians' behavior. Our approach is based on the methods of the kinetic theory of active particles. The model allows us to weight between two competing behaviors depending on fear level: the search for less congested areas and the tendency to follow the stream unconsciously (herding). The fear level of each pedestrian influences her walking speed and is influenced by the fear levels of her neighbors. Numerically, we solve our pedestrian model with emotional contagion using an operator splitting scheme. We simulate evacuation scenarios involving two groups of interacting pedestrians to assess how domain geometry and the details of fear propagation impact evacuation dynamics. Further, we reproduce the evacuation dynamics of an experimental study involving distressed ants., Comment: 26 pages, 10 figures. arXiv admin note: text overlap with arXiv:1901.07620

This paper focuses on reduced-order models (ROMs) built for the efficient treatment of PDEs having solutions that bifurcate as the values of multiple input parameters change. First, we consider a method called local ROM that uses k-means algorithm to cluster snapshots and construct local POD bases, one for each cluster. We investigate one key ingredient of this approach: the local basis selection criterion. Several criteria are compared and it is found that a criterion based on a regression artificial neural network (ANN) provides the most accurate results for a channel flow problem exhibiting a supercritical pitchfork bifurcation. The same benchmark test is then used to compare the local ROM approach with the regression ANN selection criterion to an established global projection-based ROM and a recently proposed ANN based method called POD-NN. We show that our local ROM approach gains more than an order of magnitude in accuracy over the global projection-based ROM. However, the POD-NN provides consistently more accurate approximations than the local projection-based ROM.

We propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. The main novelty of the proposed approach relies in applying spatial filtering both for the collection of the snapshots and in the reduced order model, as well as in considering the pressure field at reduced level. In both steps of the EF algorithm, velocity and pressure fields are approximated by using different POD basis and coefficients. For the reconstruction of the pressures fields, we use a pressure Poisson equation approach. We test our ROM on two benchmark problems: 2D and 3D unsteady flow past a cylinder at Reynolds number 0 <= Re <= 100. The accuracy of the reduced order model is assessed against results obtained with the full order model. For the 2D case, a parametric study with respect to the filtering radius is also presented., Comment: 29 pages, 16 figures, 9 tables

Membrane phase-separation is a mechanism that biological membranes often use to locally concentrate specific lipid species in order to organize diverse membrane processes. Phase separation has also been explored as a tool for the design of liposomes with heterogeneous and spatially organized surfaces. These "patchy" liposomes are promising platforms for delivery purposes, however their design and optimization through experimentation can be expensive and time-consuming. We developed a computationally efficient method based on the surface Cahn-Hilliard phase-field model to complement experimental investigations in the design of patchy liposomes. The method relies on thermodynamic considerations to set the initial state for numerical simulations. We show that our computational approach delivers not only qualitative pictures, but also accurate quantitative information about the dynamics of the membrane organization. In particular, the computational and experimental results are in excellent agreement in terms of raft area fraction, total raft perimeter over time and total number of rafts over time for two different membrane compositions (DOPC:DPPC with a 2:1 molar ratio with 20% Chol and DOPC:DPPC with a 3:1 molar ratio with 20% Chol). Thus, the computational phase-field model informed by experiments has a considerable potential to assist in the design of liposomes with spatially organized surfaces, thereby containing the cost and time required by the design process.