Despite great progress in simulating multiphysics problems using the numerical
discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate …

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

We introduce a simple, rigorous, and unified framework for solving nonlinear partial
differential equations (PDEs), and for solving inverse problems (IPs) involving the …

We present a general kernel-based framework for learning operators between Banach
spaces along with a priori error analysis and comprehensive numerical comparisons with …

The incorporation of physical information in machine learning frameworks is opening and
transforming many application domains. Here the learning process is augmented through …

Over forty years ago average-case error was proposed in the applied mathematics literature
as an alternative criterion with which to assess numerical methods. In contrast to worst-case …

In this paper, we propose Constraint Energy Minimizing Generalized Multiscale Finite
Element Method (CEM-GMsFEM). The main goal of this paper is to design multiscale basis …

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

The objective of this book is to introduce the reader to the Localized Orthogonal
Decomposition (LOD) method for solving partial differential equations with multiscale data …

Learning can be seen as approximating an unknown function by interpolating the training
data. Although Kriging offers a solution to this problem, it requires the prior specification of a …