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We present an introduction to periodic and stochastic homogenization of ellip- tic partial differential equations. The first part is concerned with the qualitative theory, which we present for equations with periodic and random coefficients…
Numerical multiscale methods usually rely on some coupling between a macroscopic and a microscopic model. The macroscopic model is incomplete as effective quantities, such as the homogenized material coefficients or fluxes, are missing in…
This work introduces a new approach for accelerating the numerical analysis of time-domain partial differential equations (PDEs) governing complex physical systems. The methodology is based on a combination of a classical reduced-order…
In this paper, we propose an equation-based parametric Reduced Order Model (ROM), whose accuracy is improved with data-driven terms added into the reduced equations. These additions have the aim of reintroducing contributions that in…
In this paper, we combine concepts of the generalized multiscale finite element method and mode decomposition methods to construct a robust local-global approach for model reduction of flows in high-contrast porous media. This is achieved…
We analyze neural network solutions to partial differential equations obtained with Physics Informed Neural Networks. In particular, we apply tools of classical finite element error analysis to obtain conclusions about the error of the Deep…
In this study, an efficient reanalysis strategy for dynamic topology optimization is proposed. Compared with other related studies, an online successive dynamic reanalysis method and POD-based approximate dynamic displacement strategy are…
We propose a locally one dimensional (LOD) finite difference method for multidimensional Riesz fractional diffusion equation with variable coefficients on a finite domain. The numerical method is second-order convergent in both space and…
Finite element based simulation of phenomena governed by partial differential equations is a standard tool in many engineering workflows today. However, the simulation of complex geometries is computationally expensive. Many engineering…
We consider proper orthogonal decomposition (POD) methods to approximate the incompressible Navier-Stokes equations. We study the case in which one discretization for the nonlinear term is used in the snapshots (that are computed with a…
We use the local orthogonal decomposition technique to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale diffusion coefficient. We consider nonsmooth initial data and a backward…
Models with dominant advection always posed a difficult challenge for projection-based reduced order modelling. Many methodologies that have recently been proposed are based on the pre-processing of the full-order solutions to accelerate…
We present a method that employs physics-informed deep learning techniques for parametrically solving partial differential equations. The focus is on the steady-state heat equations within heterogeneous solids exhibiting significant phase…
We present a new hybrid numerical method for multiscale partial differential equations, which simultaneously captures the global macroscopic information and resolves the local microscopic events over regions of relatively small size. The…
In recent years, large-scale numerical simulations played an essential role in estimating the effects of explosion events in urban environments, for the purpose of ensuring the security and safety of cities. Such simulations are…
We describe and analyze a hybrid finite element/neural network method for predicting solutions of partial differential equations. The methodology is designed for obtaining fine scale fluctuations from neural networks in a local manner. The…
We propose a novel numerical homogenization method based on the edge multiscale approach for solving indefinite time-harmonic Maxwell equations in heterogeneous media with large wavenumber. Numerical methods for these equations in…
Solving PDEs with machine learning techniques has become a popular alternative to conventional methods. In this context, Neural networks (NNs) are among the most commonly used machine learning tools, and in those models, the choice of an…
This paper is the companion article of [Gloria, M3AS, 21 (2011), No. 3, pp 1601-1630]. One common drawback among numerical homogenization methods is the presence of the so-called resonance error, which roughly speaking is a function of the…
In recent years, there has been a growing interest in understanding complex microstructures and their effect on macroscopic properties. In general, it is difficult to derive an effective constitutive law for such microstructures with…