Publication 25-CNA-023

On the Variational Dual Formulation of the Nash System and an Adaptive Convex Gradient-Flow Approach to Nonlinear PDEs

Dmitry Vorotnikov
CMUC, Department of Mathematics
University of Coimbra
Coimbra, Portugal
mitvorot@mat.uc.pt

Amit Acharya
Department of Civil & Environmental Engineering
Center for Nonlinear Analysis
Carnegie Mellon University
Pittsburgh, PA 15213
acharyaamit@cmu.edu

Abstract: We investigate the influence of base states on the consistency of the dual variational formulation for quadratic systems of PDEs, which are not necessarily conservative (typical examples include the noise-free Nash system with a quadratic Hamiltonian and multiple players). We identify a sufficient condition under which consistency holds over large time intervals. In particular, in the single-player case, there exists a sequence of base states (each exhibiting full consistency) that converges in mean to zero. We also prove existence of variational dual solutions to the noise-free Nash system for arbitrary base states. Furthermore, we propose a scheme based on Hilbertian gradient flows that, starting from an arbitrary base state, generates a sequence of new base states that is expected to converge to a solution of the original PDE.

Get the paper in its entirety as  25-CNA-023.pdf

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