We discuss a new approach for the numerical computation of self-similar blow-up solutions of certain nonlinear partial differential equations. These solutions become unbounded in finite time at a single point at which there is a growing and increasingly narrow peak. Our main focus is on a quasilinear parabolic problem in one space dimension, but our approach can also be applied to other problem...

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient multi-layer neural networks, while Hessian is approximated automatic differentiation of at previous step. This methodology extends to case approach recently proposed in Huré et al. (Math ...

Recently developed kinetic theory for neuronal network dynamics has been demonstrated to be a powerful theoretical framework for investigating coarse-grained dynamical properties of neuronal networks. The kinetic equations are a system of (1+1)dimensional nonlinear partial differential equations (PDE) on a bounded domain with the following features: (i) the boundary conditions are nonlinear and...

This paper is concerned with developing accurate and efficient nonstandard discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary goal of the paper to develop a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity...

Abstract We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The combines functional train (FTT) series expansions, operator splitting time integration, and algorithm based on thresholding criterion that limits component PDE velocity vector normal FTT manifold. This yields scheme can add or remove modes adaptively from as integratio...