The Discrete Adjoint Method for Exponential Integration

TL;DR

This paper develops a discrete adjoint method for exponential integrators used in stiff PDEs, enabling efficient sensitivity analysis and parameter estimation, with applications demonstrated on pattern formation models.

Contribution

It introduces a general adjoint exponential integration method for sensitivity analysis of semi-linear PDEs solved by ETD schemes, including derivatives of $\

Findings

Derived derivatives of $\

Implemented adjoint exponential integrator for Krogstad scheme

Applied methods to Swift-Hohenberg pattern formation model

Abstract

The implementation of the discrete adjoint method for exponential time differencing (ETD) schemes is considered. This is important for parameter estimation problems that are constrained by stiff time-dependent PDEs when the discretized PDE system is solved using an exponential integrator. We also discuss the closely related topic of computing the action of the sensitivity matrix on a vector, which is required when performing a sensitivity analysis. The PDE system is assumed to be semi-linear and can be the result of a linearization of a nonlinear PDE, leading to exponential Rosenbrock-type methods. We discuss the computation of the derivatives of the $\varphi$-functions that are used by ETD schemes and find that the derivatives strongly depend on the way the $\varphi$-functions are evaluated numerically. A general adjoint exponential integration method, required when computing the gradients, is developed and its implementation is illustrated by applying it to the Krogstad scheme. The applicability of the methods developed here to pattern formation problems is demonstrated using the Swift-Hohenberg model.