Optimization of a partial differential equation on a complex network

TL;DR

This paper develops a robust finite element method with error bounds and preconditioning for optimizing solutions to differential equations on complex networks, applicable to physical and social phenomena.

Contribution

It introduces a novel discretization approach with rigorous error analysis and an efficient preconditioning strategy for large-scale network optimization problems.

Findings

Method performs robustly across various examples.

Provides rigorous error bounds for discretization.

Efficient preconditioning improves large-scale computations.

Abstract

Differential equations on metric graphs can describe many phenomena in the physical world but also the spread of information on social media. To efficiently compute the solution is a hard task in numerical analysis. Solving a design problem, where the optimal setup for a desired state is given, is even more challenging. In this work, we focus on the task of solving an optimization problem subject to a differential equation on a metric graph with the control defined on a small set of Dirichlet nodes. We discuss the discretization by finite elements and provide rigorous error bounds as well as an efficient preconditioning strategy to deal with the large-scale case. We show in various examples that the method performs very robustly.