A Nonlinear Multigrid Method for the Parameter Identification Problem of Partial Differential Equations with Constraints

In this paper, we consider the parameter identification problem of partial differential equations with constraints. A nonlinear multigrid method is introduced to the process of parameter inversion. By keeping the objective functions on coarse grids consistent with those on fine grids, the proposed method reduces the dimensions of objective functions enormously and mitigates the risk of trapping in local minima effectively. Furthermore, constraints significantly improve the convergence ability of the method. We performed the numerical simulation based on the porosity identification of elastic wave equations in the fluid-saturated porous media, which suggests that the nonlinear multigrid method with constraints decreases the computational expenditure, suppresses the noise, and improves the inversion results.

Automated summary

This paper investigates the parameter identification problem of partial differential equations with constraints and introduces a nonlinear multigrid method for parameter inversion. The results show that this method reduces the dimensions of objective functions, effectively mitigates the risk of trapping in local minima, and significantly improves the convergence ability of the method through constraints.