The analysis of singular perturbed differential equations began early in the twentieth
century, when approximate solutions were constructed from asymptotic expansions …

The focus of this book is the large-scale statistical behavior of solutions of divergence-form
elliptic equations with random coefficients, which is closely related to the long-time …

We lay out a program for constructing discontinuous Petrov–Galerkin (DPG) schemes
having test function spaces that are automatically computable to guarantee stability. Given a …

A model-constrained adaptive sampling methodology is proposed for the reduction of large-
scale systems with high-dimensional parametric input spaces. Our model reduction method …

These notes serve as an introduction to a subject of study in computational mathematics
referred to as domain decomposition methods. It concerns divide and conquer methods for …

This monograph is the first to provide a comprehensive, self-contained and rigorous
presentation of some of the most powerful preconditioning methods for solving finite element …

This paper studies an unsupervised deep learning-based numerical approach for solving
partial differential equations (PDEs). The approach makes use of the deep neural network to …

We consider the application of least-squares variational principles to the numerical solution
of partial differential equations. Our main focus is on the development of least-squares finite …

We consider the solution of the system of linear algebraic equations which arises from the
finite element discretization of boundary value problems associated to the differential …

We introduce a Robust version of the Variational Physics-Informed Neural Networks method
(RVPINNs). As in VPINNs, we define the quadratic loss functional in terms of a Petrov …