Adjoint sensitivity analysis of nonstandard objective functions

Area of Science:

• Computational Science
• Mathematical Modeling
• Dynamic Systems Analysis

Background:

• Adjoint sensitivity analysis (ASA) is widely used for parameter estimation and optimal control in dynamic models.
• Standard ASA typically requires objective functions to be of the Bolza form (sum of running and terminal costs).
• Time-dependent sensitivity analysis (TDSA) analyzes sensitivities to time-limited perturbations.

Purpose of the Study:

• To extend the applicability of Adjoint sensitivity analysis (ASA) to non-Bolza objective functions.
• To demonstrate that non-Bolza objectives can be reformulated as equivalent Bolza objectives for ASA.
• To showcase the utility of the extended ASA in practical dynamic modeling scenarios.

Main Methods:

• Developed a method to relax the Bolza form assumption in Adjoint sensitivity analysis.
• Showcased the equivalence of sensitivities between non-Bolza and reformulated Bolza objectives.
• Applied the extended ASA to Time-dependent sensitivity analysis (TDSA) problems.

Main Results:

• Successfully demonstrated that Adjoint sensitivity analysis can be applied to non-Bolza objective functions.
• Identified practical non-Bolza objectives, such as extremum objectives (e.g., peak infections) and first passage times (e.g., population thresholds).
• Showed that these non-standard objectives can be replaced by equivalent Bolza objectives for sensitivity calculation.

Conclusions:

• The Adjoint sensitivity analysis framework is more versatile than previously assumed, accommodating a broader range of objective functions.
• This extension enables more accurate and efficient sensitivity analysis for complex dynamic models with non-standard objectives.
• The approach is validated through applications in epidemic modeling, fishery dynamics, and invasive species spread.