# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1907.07806