Identification of the forcing term in hyperbolic equations

Abstract

We investigate the problem of recovering the possibly both space and time-dependent forcing term along with the temperature in hyperbolic systems from many integral observations. In practice, these average weighted integral observations can be considered as generalized interior point measurements. This linear but ill-posed problem is solved using the Tikhonov regularization method in order to obtain the closest stable solution to a given a priori known initial estimate. We prove the Fréchet differentiability of the Tikhonov regularization functional and derive a formula for its gradient. This minimization problem is solved iteratively using the conjugate gradient method. The numerical discretization of the well-posed problems, that are: the direct, adjoint and sensitivity problems that need to be solved at each iteration is performed using finite-difference methods. Numerical results are presented and discussed for one and two-dimensional problems.

Keywords:

• Inverse force problem
• hyperbolic equations
• integral observations
• Tikhonov's regularization
• conjugate gradient method

2010 Mathematics Subject Classification:

• 65M32

Acknowledgments

M. Alosaimi would like to thank Taif University in Saudi Arabia and the United Kingdom Saudi Arabian Cultural Bureau (UKSACB) in London for supporting his PhD studies at the University of Leeds. D. Lesnic would like to acknowledge some small financial support received from the EPSRC funded research network on Inverse Problems EP/P005985/1 for a couple of days research visit of Professor Dinh Nho Hào to Leeds in July 2019.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

D. Lesnic would like to acknowledge some small financial support received from the EPSRC funded research network on Inverse Problems EP/P005985/1 for a couple of days research visit of Professor Dinh Nho Hào to Leeds in July 2019.