Continuous analogue to iterative optimization for PDE-constrained inverse problems.

Area of Science:

• Computational Mathematics
• Optimization Theory
• Scientific Computing

Background:

• Parameter estimation for physical processes often relies on experimental data.
• Iterative methods like steepest descent are commonly used but can be computationally intensive.
• Continuous analogues of iterative methods exist for specific problem classes.

Purpose of the Study:

• To extend the application of continuous analogues to function spaces.
• To develop and analyze continuous analogues for partial differential equation-constrained optimization problems.
• To investigate the convergence and stability of these novel methods.

Main Methods:

• Derivation of coupled ordinary differential equation-partial differential equation models as continuous analogues.
• Mathematical proof of convergence to the optimum under mild assumptions.
• Analysis of local stability and convergence bounds for the retraction parameter.

Main Results:

• Successful derivation of a class of continuous analogues for PDE-constrained optimization.
• Demonstrated convergence properties of the derived models.
• Established bounds for local stability and convergence related to the retraction parameter.

Conclusions:

• Continuous analogues, specifically coupled ODE-PDE models, are effective for parameter estimation in function spaces.
• These methods offer a viable alternative to traditional iterative optimization techniques.
• The study validates the approach through application to a biological tissue gradient formation model.