A Sobolev gradient of a real-valued functional on a Hilbert space is a gradient of that
functional taken relative to an underlying Sobolev norm. This book shows how descent …

The numerical solution of linear elliptic partial differential equations most often involves a
finite element or finite difference discretization. To preserve sparsity, the arising system is …

One of the most important problems in numerical simulations is the preservation of
qualitative properties of solutions of the mathematical models by computed approximations …

Tight frames can be characterized as those frames which possess optimal numerical stability
properties. In this paper, we consider the question of modifying a general frame to generate …

Recent experiments have shown that deep networks can approximate solutions to high-
dimensional PDEs, seemingly escaping the curse of dimensionality. However, questions …

In this paper we combine concepts from Riemannian optimization P.-A. Absil, R. Mahony,
and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press …

Data-driven machine learning approaches are being increasingly used to solve partial
differential equations (PDEs). They have shown particularly striking successes when training …

We propose the first hybridizable discontinuous Galerkin method for the p-Laplacian
equation. When using polynomials of degree k≧0 for the approximation spaces of u, ∇u …

In order to have reliable numerical simulations it is very important to preserve basic
qualitative properties of solutions of mathematical models by computed approximations. For …

Discrete maximum principles (DMPs) are established for finite element approximations of
systems of nonlinear parabolic partial differential equations with mixed boundary and …