Carlos D. Acosta, Raimund Bürger, Carlos E. Mejia:
Numerical identification of parameters for strongly degenerate parabolic equations by discrete mollification
Numerical methods for the reliable and efficient identification of parameters defining the flux function and the diffusion coefficient of a strongly degenerate parabolic equation are of great importance in applicative areas including a sedimentation-consolidation model and a diffusively corrected kinematic traffic model. This problem can be treated by repeatedly solving the corresponding direct problem under systematic variation of an initially guessed set of model parameters, with the aim of successively minimizing a cost functional that measures the distance between a space- or time-dependent observation and the corresponding numerical solution. The direct problem is solved herein by a version of a well-known explicit, monotone three-point finite difference scheme. This version is obtained by replacing the standard conservative three-point finite difference discretization of the diffusive term by a formula that involves a discrete mollification operator. The mollified scheme occupies a larger stencil but converges under a less restrictive CFL condition, which allows to employ a larger time step than for the basic scheme. By numerical experiments it is demonstrated that despite additional computational effort, the mollified scheme leads to gains in CPU time for the parameter identification procedure. Moreover, results are also favorable compared with the basic scheme in terms of the error level and the sensitivity with respect to the initial guess and noise in the context of parameter recognition problems
This preprint gave rise to the following definitive publication(s):
Carlos D. ACOSTA, Raimund BüRGER, Carlos E. MEJIA: A stability and sensitivity analysis of parametric functions in a sedimentation model. Dyna, vol. 81, 183, pp. 22-30, (2014).
Carlos D. ACOSTA, Raimund BüRGER, Carlos E. MEJIA: Efficient parameter estimation in a macroscopic traffic flow model by discrete mollification. Transportmetrica A: Transport Science, vol. 11, 8, pp. 702-715, (2015).