A Regularization Homotopy Strategy for the Constrained Parameter Inversion of Partial Differential Equations.

Area of Science:

• Applied Mathematics
• Computational Science
• Geophysics

Background:

• Parameter inversion of partial differential equations (PDEs) is challenging due to local minima in cost functions.
• Standard iterative methods often fail to find the global minimum without a close initial guess.
• Constraints can aid convergence but do not fully resolve local minima entrapment.

Purpose of the Study:

• To develop a robust method for parameter inversion of PDEs that overcomes local minima.
• To enhance the convergence of inversion techniques using constraints and regularization.
• To address the ill-posed nature of inverse problems in scientific modeling.

Main Methods:

• A novel homotopy strategy is designed, leveraging constraints to guide the inversion process.
• Standard Tikhonov regularization is incorporated to handle the ill-posedness inherent in inverse problems.
• The method's efficacy is demonstrated through coefficient inversion in a two-phase porous media saturation equation.

Main Results:

• The proposed homotopy strategy effectively avoids local minima, leading to reliable convergence.
• The integration of Tikhonov regularization ensures stability and accuracy in ill-posed inverse problems.
• Successful coefficient inversion in the porous media saturation equation validates the method's efficiency.

Conclusions:

• The developed homotopy strategy offers a significant advancement in solving parameter inversion problems for PDEs.
• This approach provides a more reliable and accurate solution for inverse problems, particularly in complex systems like porous media.
• The study highlights the importance of combining advanced strategies like homotopy with regularization for effective inverse problem solving.