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In this paper, we discuss a general framework for multicontinuum homogenization. Multicontinuum models are widely used in many applications and some derivations for these models are established. In these models, several macroscopic variables at each macroscale point are defined and the resulting multicontinuum equations are formulated. In this paper, we propose a general formulation and associated ingredients that allow performing multicontinuum homogenization. Our derivation consists of several main parts. In the first part, we propose a general expansion, where the solution is expressed via the product of multiple macro variables and associated cell problems. The second part consists of formulating the cell problems. The cell problems are formulated as saddle point problems with constraints for each continua. Defining the continua via test functions, we set the constraints as an integral representation. Finally, substituting the expansion to the original system, we obtain multicontinuum systems. We present an application to the mixed formulation of elliptic equations. This is a challenging system as the system does not have symmetry. We discuss the local problems and various macroscale representations for the solution and its gradient. Using various order approximations, one can obtain different systems of equations. We discuss the applicability of multicontinuum homogenization and relate this to high contrast in the cell problem. Numerical results are presented. • We derive multicontinuum methods for general systems. • We discuss the application of the framework. • We discuss cell solutions. • We consider applications to several equations, including systems and non-symmetric problems. • We present numerical implementation for a mixed formulation. [ABSTRACT FROM AUTHOR]

In this paper, we present a general derivation of multicontinuum equations and discuss cell problems. We present constraint cell problem formulations in a representative volume element and oversampling techniques that allow reducing boundary effects. We discuss different choices of constraints for cell problems. We present numerical results that show how oversampling reduces boundary effects. Finally, we discuss the relation of the proposed methods to our previously developed methods, Nonlocal Multicontinuum Approaches. • We derive multicontinuum methods using a homogenization-like expansion and present constraint cell problem formulations. • Constraint cell problems allow using averages for different continua and give a flexibility to the framework. • We discuss appropriate local boundary conditions in representative volume elements and introduce oversampling. • The resulting multicontinuum equations show that local averages of the solution will differ among each other. • The average constraints, discussed in this paper, are easy to set and guarantee exponential decay. [ABSTRACT FROM AUTHOR]

Splitting method is a powerful method to handle application problems by splitting physics, scales, domain, and so on. Many splitting algorithms have been designed for efficient temporal discretization. In this paper, our goal is to use temporal splitting concepts in designing machine learning algorithms and, at the same time, help splitting algorithms by incorporating data and speeding them up. We propose a machine learning assisted splitting scheme which improves the efficiency of the scheme meanwhile preserves the accuracy. We consider a recently introduced multiscale splitting algorithms, where the multiscale problem is solved on a coarse grid. To approximate the dynamics, only a few degrees of freedom are solved implicitly, while others explicitly. This splitting concept allows identifying degrees of freedom that need implicit treatment. In this paper, we use this splitting concept in machine learning and propose several strategies. First, the implicit part of the solution can be learned as it is more difficult to solve, while the explicit part can be computed. This provides a speed-up and data incorporation for splitting approaches. Secondly, one can design a hybrid neural network architecture because handling explicit parts requires much fewer communications among neurons and can be done efficiently. Thirdly, one can solve the coarse grid component via PDEs or other approximation methods and construct simpler neural networks for the explicit part of the solutions. We discuss these options and implement one of them by interpreting it as a machine translation task. This interpretation of the splitting scheme successfully enables us using the Transformer since it can perform model reduction for multiple time series and learn the connection between them. We also find that the splitting scheme is a great platform to predict the coarse solution with insufficient information of the target model: the target problem is partially given and we need to solve it through a known problem which approximates the target. Our machine learning model can incorporate and encode the given information from two different problems and then solve the target problems. We conduct four numerical examples and the results show that our method is stable and accurate. • We have designed a machine learning strategy that is based on stable splitting for multiscale problems. • Our approach uses implicit and explicit discretization combined with the Transformer for multiscale problems to solution prediction. • Numerical results are presented that show the proposed methods provide good prediction and accuracy. [ABSTRACT FROM AUTHOR]

• We have designed a generalized multiscale model for heterogeneous Cosserat media without scale separation. • We have designed multiscale spaces with and without coupling of processes and investigated them. • Numerical results are presented that show the proposed methods provide a good accuracy with a few degrees of freedom. In this paper, we develop a multiscale computational approach for heterogeneous Cosserat media without scale separation and high contrast. Cosserat medium has been used in many applications to describe materials whose particles possess rotational degrees of freedom. Many previous findings investigate Cosserat medium in the presence of heterogeneities. Approaches that include homogenization and numerical homogenization have been developed for problems with scale separation, such as periodicity. In these approaches, macroscopic models are derived that show the resulting equations can be Cauchy medium or Cosserat medium depending on the Cosserat intrinsic length scale. In our paper, we consider computational macroscopic models for Cosserat medium without scale separation. We use Generalized Multiscale Finite Element Method (GMsFEM) to derive a macroscopic model on a coarse computational grid, that does not resolve the small scales and the contrast. In this approach, multiple macroscopic basis functions are used to describe the small scales. The resulting macroscopic model is similar to multicontinua model as it contains multiple macroscopic parameters at each macroscopic point (or grid). We present numerical results for different heterogeneous media types. Our numerical results show that using a few multiscale basis functions (i.e., macroscopic parameters), we can achieve a good accuracy. [ABSTRACT FROM AUTHOR]

• Designed coupled implicit-explicit temporal and spatial splitting method. • Formulated conditions for stability and designed appropriate spaces. • Multiscale spaces with CEM-GMsFEM are used in constructing the splitting method. • Numerical results confirm our theoretical findings. Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extent are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is d t = H 2 / κ m a x , where κ m a x is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The contrast-independent stability of these algorithms requires a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented. [ABSTRACT FROM AUTHOR]

This paper presents a novel Generalized Finite Element Method with global-local enrichment (GFEMgl) to solve time-dependent parabolic and hyperbolic problems with subscale features in both space and time. Compared with currently available GFEMgl, the proposed method requires a solution of local problems to solve transient PDEs, possibly with a different time integrator from the global problem or other local problems. The fine-scale solution can be easily retrieved. The proposed method is tested for the solution of the heat, advection, and advection-diffusion equations whose accuracy, stability, scalability, and convergence are thoroughly analyzed. Compared to direct analysis, numerical results show that the proposed method has the following properties 1) the accuracy closely matches direct analysis result with a fine mesh; 2) critical time step size is loosened; 3) optimal convergence rate is achieved. Moreover, the proposed method allows local problems to be turned off if their solutions are not helpful for the current global time step, which can further improve its efficiency. • Subscale features of parabolic and hyperbolic problems are captured on coarse meshes. • Method is optimally convergent on uniform meshes. • No a-priori knowledge about the solution is required. • Critical time step is larger than for methods that provide similar accuracy. • Adaptive enrichment reduces problem size. [ABSTRACT FROM AUTHOR]

This paper presents a lattice Boltzmann framework for accurate and efficient simulation of free-surface granular flows. The granular assembly is treated as a viscoplastic fluid, whose apparent viscosity varies locally with the shear rate and pressure according to a regularized μ (I) -rheology. A single-phase free-surface model is employed to track the evolution of the particle-air interface. The lattice Boltzmann implementation is first validated by simulating a steady-state granular flow on a rough inclined plane and a good agreement with the analytical solution is achieved. The validated model is then applied to simulate a transient granular column collapse problem. Compared to a companion discrete element simulation, the lattice Boltzmann model with the μ (I) -rheology is able to capture the overall dynamic behaviors of granular column collapse. However, a different behavior is observed when a similar Bingham viscoplastic model with a fixed yield stress is applied, highlighting the pressure dependent nature of granular flows. The proposed lattice Boltzmann formulation is highly efficient compared to the conventional computational fluid dynamics, and has the potential to conduct three-dimensional continuum simulation of large-scale geophysical flows with microscopic granular physics. • We extend the lattice Boltzmann framework that enables accurate and efficient simulation of free-surface granular flows. • We treat granular flows as viscoplastic fluids following the μ (I) rheology. • We verify the numerical model against analytical solutions and results from discrete element simulation. • We demonstrate the calculation efficiency by comparing to an open-source Navier-Stokes solver using the finite volume method. [ABSTRACT FROM AUTHOR]

• Transport equations with memory terms are rewritten as equations without memory. • We investigate the relation to homogenization and show that the de-memorized equations are still homogenized equations. • We propose various algorithms for solving multiscale de-memorized equations. • Stability is investigated and numerical results are presented. In this paper, we consider a class of convection-diffusion equations with memory effects. These equations arise as a result of homogenization or upscaling of linear transport equations in heterogeneous media and play an important role in many applications. First, we present a dememorization technique for these equations. We show that the convection-diffusion equations with memory effects can be written as a system of standard convection diffusion reaction equations. This allows removing the memory term and simplifying the computations. We consider a relation between dememorized equations and micro-scale equations, which do not contain memory terms. We note that dememorized equations differ from micro-scale equations and constitute a macroscopic model. Next, we consider both implicit and partially explicit methods. The latter is introduced for problems in multiscale media with high-contrast properties. Because of high-contrast, explicit methods are restrictive and require time steps that are very small (scales as the inverse of the contrast). We show that, by appropriately decomposing the space, we can treat only a few degrees of freedom implicitly and the remaining degrees of freedom explicitly. We present a stability analysis. Numerical results are presented that confirm our theoretical findings about partially explicit schemes applied to dememorized systems of equations. [ABSTRACT FROM AUTHOR]

In this work, we design and investigate contrast-independent partially explicit time discretizations for wave equations in heterogeneous high-contrast media. We consider multiscale problems, where the spatial heterogeneities are at subgrid level and are not resolved. In our previous work Chung et al. (2021) [25] , we have introduced contrast-independent partially explicit time discretizations and applied to parabolic equations. The main idea of contrast-independent partially explicit time discretization is to split the spatial space into two components: contrast dependent ("fast") and contrast independent ("slow") spaces defined via multiscale space decomposition. Using this decomposition, our goal is to appropriately introduce time splitting such that the resulting scheme is stable and can guarantee contrast-independent discretization under some suitable conditions. In this paper, we propose contrast-independent partially explicitly scheme for wave equations, where the splitting requires a careful design. We prove that the proposed splitting is unconditionally stable under some suitable conditions formulated for the second space ("slow"). This condition requires the second space to be contrast independent. Our approaches identify local features that need implicit treatment and the latter is done on a coarse grid using appropriate splitting algorithms. We note that the proposed work extends our previous work Chung et al. (2021) [25] to wave equations. This extension consists of (1) a different discretization (both spatial and temporal), (2) energy conservation property (instead of energy decay), and (3) a different proof strategy. We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast. • Coupled temporal and spatial splitting method is designed for multiscale wave equations. We identify stability conditions for proposed method. • We design multiscale spaces using CEM-GMsFEM to achieve a stability on the coarse grid. • Numerical results are presented that show the proposed methods converge with the time step that is independent of the contrast. [ABSTRACT FROM AUTHOR]

The left panel displays the critical manifold for our pattern forming model, which describes the establishment of a morphogenetic gradient and its influence on intracellular cell state dynamics over a growing tissue. The system trajectories, shown in grey, display two types of behaviour corresponding to the acquisition of one of two cell fates. One type of trajectory remains always on the upper (attracting) surface of the critical manifold, whilst the other type "fall off" onto the lower surface. The stable manifold of a folded saddle (green line) acts as a separatix between regions of tissue adopting the cell fates corresponding to the upper and lower surfaces, which is confirmed in the numerical simulations shown on the right. [Display omitted] • We present a two-stage model of cell differentiation in a growing domain • We find sufficient conditions for robust patterning of the domain in terms of parameters that determine timescales within the model • We show that the success of patterning in the model can be understood in terms of a folded singularity for system with two slow and one fast variable. One of the fundamental mechanisms in embryogenesis is the process by which cells differentiate and create tissues and structures important for functioning as a multicellular organism. Morphogenesis involves diffusive process of chemical signalling involving morphogens that pre-pattern the tissue. These morphogens influence cell fate through a highly nonlinear process of transcriptional signalling. In this paper, we consider this multiscale process in an idealised model for a growing domain. We focus on intracellular processes that lead to robust differentiation into two cell lineages through interaction of a single morphogen species with a cell fate variable that undergoes a bifurcation from monostability to bistability. In particular, we investigate conditions that result in successful and robust pattern formation into two well-separated domains, as well as conditions where this fails and produces a pinned boundary wave where only one part of the domain grows. We show that successful and unsuccessful patterning scenarios can be characterised in terms of presence or absence of a folded saddle singularity for a system with two slow variables and one fast variable; this models the interaction of slow morphogen diffusion, slow parameter drift through bifurcation and fast transcription dynamics. We illustrate how this approach can successfully model acquisition of three cell fates to produce three-domain "French flag" patterning, as well as for a more realistic model of the cell fate dynamics in terms of two mutually inhibiting transcription factors. [ABSTRACT FROM AUTHOR]

• We propose a data-driven method to discover multiscale chemical reactions governed by the law of mass action. • We propose a partial-parameters-freezing (PPF) technique to learn the multiscale reactions. • We use a single matrix to represent the stoichiometric coefficients for a system without catalysis reactions. In this paper, we propose a data-driven method to discover multiscale chemical reactions governed by the law of mass action. First, we use a single matrix to represent the stoichiometric coefficients for both the reactants and products in a system without catalysis reactions. The negative entries in the matrix denote the stoichiometric coefficients for the reactants and the positive ones for the products. Second, we find that the conventional optimization methods usually get stuck in the local minima and could not find the true solution in learning the multiscale chemical reactions. To overcome this difficulty, we propose a partial-parameters-freezing (PPF) technique to progressively determine the network parameters by using the fact that the stoichiometric coefficients are integers. With such a technique, the dimension of the searching space is gradually reduced in the training process and the global minima can be eventually obtained. Several numerical experiments including the classical Michaelis–Menten kinetics, the hydrogen oxidation reactions and the simplified GRI-3.0 mechanism verify the good performance of our algorithm in learning the multiscale chemical reactions. The code is available at https://github.com/JuntaoHuang/multiscale-chemical-reaction. [ABSTRACT FROM AUTHOR]

• A new algorithm incorporates model reduction in the spatial and temporal domains. • We study parabolic problems with heterogeneous coefficients and rough initial data. • We derive convergence analysis that weakly depends on the heterogeneous coefficients. • The convergence is rigorously studied, which greatly improves the current result. • Extensive numerical tests are performed to show the fast convergence of our algorithm. We propose in this paper the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm to solve parabolic equations with heterogeneous coefficients efficiently. This algorithm combines the advantages of multiscale methods that can deal with heterogeneity in the spatial domain effectively, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step and the fine-scale time step. Extensive numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly. [ABSTRACT FROM AUTHOR]

In this paper, a Floquet Model Order Reduction (MOR) method is proposed for the modelling of finite periodic structures subjected to harmonic or transient loads. This Floquet MOR generates frequency-independent projection matrices of the full-scale structure using a Floquet expansion of the local state-vectors computed on a single unit-cell with the Forced Wave Finite Element framework. This two-step MOR method can handle large multi-scale waveguides regardless of the scale and the number of unit-cells. A comparison with a standard sub-structuring approach based on unit-cell wave-mode reduction demonstrates that Floquet MOR is able to create reduced models of finite periodic structures with an unequalled computational efficiency. The method's ability to produce small and accurate models suitable for time-domain simulations presents a high potential for virtual sensing, real-time Structural Health Monitoring (SHM) and digital twins for large periodic structures and metamaterials. • We introduce an original MOR technique for transient simulations in periodic media. • The reduced model of the overall structure is built from a single unit-cell analysis. • The numerical framework is based on WFEM and exploits a POD-like Floquet expansion. • The MOR handles complex waveguiding effects and is validated on a metamaterial model. [ABSTRACT FROM AUTHOR]