Nonlinear emergent macroscale PDEs, with error bound, for nonlinear microscale systems

Numerical Error Estimation for Nonlinear Hyperbolic PDEs via Nonlinear Error Transport

The estimation of discretization error in numerical simulations is a key component in the development of uncertainty quantification. In particular, there exists a need for reliable, robust estimators for finite volume and finite difference discretizations of hyperbolic partial differential equations. The approach espoused here, often called the error transport approach in the literature, is to ...

Singular Control-Invariance PDEs for Nonlinear Systems

In the present study a new method is proposed that allows the derivation of control laws capable of enforcing the desirable dynamics on an invariant manifold in state space. The problem of interest naturally surfaces in broad classes of physical and chemical systems whose dynamic behavior needs to be controlled and favorably shaped by external driving forces. In particular, the formulation of t...

Simulating macroscale brain circuits with microscale resolution

Nonlinear Diffusion Pdes

The main theory behind nonlinear diffusion models is to use nonlinear PDEs to create a scale space representation that consists of gradually simplified images where some image features such as edges are maintained or even enhanced. The earliest nonlinear diffusion model proposed in image processing is the so-called anisotropic diffusion1 by Perona and Malik [2]. In their formulation, they repla...

A Generalization Error Estimate for Nonlinear Systems