Numerical homogenization is a methodology for the computational solution of multiscale
partial differential equations. It aims at reducing complex large-scale problems to simplified …

In this paper, we propose a general approach called Generalized Multiscale Finite Element
Method (GMsFEM) for performing multiscale simulations for problems without scale …

The aim of this monograph is to describe the main concepts and recent-vances in
multiscale? nite element methods. This monograph is intended for …

This introductory graduate text is based on a graduate course the author has taught
repeatedly over the last ten years to students in applied mathematics, engineering sciences …

Mathematics is playing an ever more important role in the physical and biological sciences,
provoking a blurring of boundaries between scienti? c disciplines and a resurgence of …

Although additive manufacturing has offered substantially new opportunities and flexibility
for fabricating 3D complex lattice structures, effective design of such sophisticated structures …

Numerical homogenization, ie, the finite-dimensional approximation of solution spaces of
PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements …

We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/
multiresolution method for PDEs with rough (L^∞) coefficients with rigorous a priori …

We consider divergence form elliptic operators in dimension n≥ 2 with L∞ coefficients.
Although solutions of these operators are only Hölder-continuous, we show that they are …

The objective of this paper is to design novel multi-layer neural networks for multiscale
simulations of flows taking into account the observed fine data and physical modeling …