Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0708.1758

Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations

Xiaobing Feng·Michael Neilan
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper concerns with numerical approximations of solutions of second order fully nonlinear partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for second order fully nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called vanishing moment method, hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods with "guaranteed" convergence. The main idea of the proposed vanishing moment method is to approximate a second order fully nonlinear PDE by a higher order, in particular, a fourth order quasilinear PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.

Explore related subjects

math.NAcs.NAmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xiaobing Feng, Michael Neilan. 2007-08-13. Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations. https://arxiv.org/abs/0708.1758

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A general framework for the OSRC-preconditioned EFIE in computational electromagnetics

We investigate on-surface radiation condition (OSRC) preconditioners for the electric field integral equation (EFIE), considering Electric-to-Magnetic (EtM) and Magnetic-to-Electric (MtE) approximations applied from both the left and the right. A pseudodifferential analysis establishes the well-posedness of the four resulting formulations and shows that, locally, the OSRC EtM and MtE approximations present a similar structure to the EFIE itself and its inverse, respectively. At the discrete level, both MtE formulations are compatible with the standard RWG/SNC discretization and can be implemented without barycentric refinement, whereas the EtM formulations require an appropriate discrete duality map and BC/RBC spaces. Numerical experiments show that the OSRC preconditioners substantially improve GMRES convergence compared with the unpreconditioned EFIE, with the refinement-free MtE formulations also providing the lowest computational cost among the formulations considered.

math.NA↗
arXiv2026

MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data

Collective behaviors that emerge from interactions are fundamental to numerous biological systems. To learn such interacting forces from observations, we introduce a measure-valued neural network that infers measure-dependent interaction (drift) terms directly from particle-trajectory observations. The proposed architecture generalizes standard neural networks to operate on probability measures by learning cylindrical features, using an embedding network that produces scalable distribution-to-vector representations. On the theory side, we establish well-posedness of the resulting dynamics and prove propagation-of-chaos for the associated interacting-particle system. We further show universal approximation and quantitative approximation rates under a low-dimensional measure-dependence assumption. Numerical experiments on first and second order systems, including deterministic and stochastic Motsch-Tadmor dynamics, two-dimensional attraction-repulsion aggregation, Cucker-Smale dynamics, and a hierarchical multi-group system, demonstrate accurate prediction and strong out-of-distribution generalization.

math.NA↗