A Comprehensive Study of Adjoint-Based Optimization of Non-Linear Systems with Application to Burgers' Equation

TL;DR

This paper investigates the convergence and stability of adjoint-based optimization methods applied to nonlinear conservation laws, specifically Burgers' equation, highlighting the effects of numerical scheme choices on solution accuracy.

Contribution

It provides a detailed analysis of how different numerical schemes and differentiation approaches impact the convergence of adjoint solutions in nonlinear systems.

Findings

Convergence of adjoint equations depends on the numerical scheme used.

Incomplete differentiation can lead to inaccuracies in sensitivity analysis.

Time discretization inconsistencies affect the stability of adjoint solutions.

Abstract

In the context of adjoint-based optimization, nonlinear conservation laws pose significant problems regarding the existence and uniqueness of both direct and adjoint solutions, as well as the well-posedness of the problem for sensitivity analysis and gradient-based optimization algorithms. In this paper we will analyze the convergence of the adjoint equations to known exact solutions of the inviscid Burgers' equation for a variety of numerical schemes. The effect of the non-differentiability of the underlying approximate Riemann solver, complete vs. incomplete differentiation of the discrete schemes and inconsistencies in time advancement will be discussed.