Solving parameter estimation problems with discrete adjoint exponential integrators

Abstract

The solution of inverse problems in a variational setting finds best estimates of the model parameters by minimizing a cost function that penalizes the mismatch between model outputs and observations. The gradients required by the numerical optimization process are computed using adjoint models. Exponential integrators are a promising family of time discretization schemes for evolutionary partial differential equations. In order to allow the use of these discretization schemes in the context of inverse problems, adjoints of exponential integrators are required. This work derives the discrete adjoint formulae for W-type exponential propagation iterative methods of Runge–Kutta type (EPIRK-W). These methods allow arbitrary approximations of the Jacobian while maintaining the overall accuracy of the forward integration. The use of Jacobian approximation matrices that do not depend on the model state avoids the complex calculation of Hessians in the discrete adjoint formulae. The adjoint code itself is generated efficiently via algorithmic differentiation and used to solve inverse problems with the Lorenz-96 model and a model from computational magnetics. Numerical results are encouraging and indicate the suitability of exponential integrators for this class of problems.

Keywords:

• exponential integrators
• discrete adjoints
• algorithmic differentiation
• 4D-Var data assimilation

AMS Subject Classification:

• 34H05
• 34K29
• 34K35

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ulrich Römer http://orcid.org/0000-0002-1277-7509

Mahesh Narayanamurthi http://orcid.org/0000-0002-4410-4499

Additional information

Funding

The work of M. Narayanamurthi and A. Sandu was supported in part by awards NSF Division of Mathematical Sciences DMS-1419003, NSF Division of Computing and Communication Foundations CCF-1613905, Air Force Office of Scientific Research(AFOSR) DDDAS 15RT1037 and by the Computational Science Laboratory at Virginia Tech. Part of the work was performed during U. Römer's visit at Virginia Tech.