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In this paper, we consider an online basis enrichment mixed generalized multiscale method with oversampling, for solving flow problems in highly heterogeneous porous media. This is an exten- sion of the online mixed generalized multiscale…

In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally…

In this paper, we consider the offline and online Constraint Energy Minimizing Generalized Mul- tiscale Finite Element Method (CEM-GMsFEM) for high-contrast linear elasticity problem. Offline basis construction starts with an auxiliary…

In this paper, we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media. The mathematical model is described by the system of Richards equations for each continuum…

The oversampling multiscale finite element method (MsFEM) is one of the most popular methods for simulating composite materials and flows in porous media which may have many scales. But the method may be inapplicable or inefficient in some…

In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous…

This paper proposes a non-intrusive, data-driven reduced-order modeling framework for stochastic optimal control problems governed by partial differential equations. The control problem is formulated with a quadratic cost functional and…

Generative models have achieved remarkable progress with the emergence of flow matching (FM). It has demonstrated strong generative capabilities and attracted significant attention as a simulation-free flow-based framework capable of…

This paper presents a novel generative model to synthesize fluid simulations from a set of reduced parameters. A convolutional neural network is trained on a collection of discrete, parameterizable fluid simulation velocity fields. Due to…

The mathematical formulation of sign-changing problems involves a linear second-order partial differential equation in the divergence form, where the coefficient can assume positive and negative values in different subdomains. These…

Fine-scale simulation of complex systems governed by multiscale partial differential equations (PDEs) is computationally expensive and various multiscale methods have been developed for addressing such problems. In addition, it is…

From a mathematical perspective, the extraordinary properties of metamaterials are often reflected in the coefficients of the governing partial differential equations (PDEs). These coefficients may fall outside the assumptions of classical…

In this paper, we consider the incompressible Stokes flow problem in a perforated domain and employ the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to solve this problem. The proposed method…

This paper explores the application of the multiscale finite element method (MsFEM) to address steady-state Stokes-Darcy problems with BJS interface conditions in highly heterogeneous porous media. We assume the existence of multiscale…

In this paper, we consider flow and transport problems in thin domains. The mathematical model considered in the paper is described by a system of equations for velocity, pressure, and concentration, where the flow is described by the…

In this work, we employ the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) to solve the problem of linear heterogeneous poroelasticity with coefficients of high contrast. The proposed method makes use…

In this paper, we develop the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions (Dirichlet and Neumann) for the elasticity equations in high contrast media. By a special…

Estimation of spatially-varying parameters for computationally expensive forward models governed by partial differential equations is addressed. A novel multiscale Bayesian inference approach is introduced based on deep probabilistic…

We introduce a generative learning framework to model high-dimensional parametric systems using gradient guidance and virtual observations. We consider systems described by Partial Differential Equations (PDEs) discretized with structured…

In this paper, we consider the quasi-gas-dynamic (QGD) model in a multiscale environment. The model equations can be regarded as a hyperbolic regularization and are derived from kinetic equations. So far, the research on QGD models has been…