A MEASURE-THEORETIC COMPUTATIONAL METHOD FOR INVERSE SENSITIVITY PROBLEMS I: METHOD AND ANALYSIS.

Area of Science:

• Computational Mathematics
• Data Science
• Applied Probability

Background:

• Inverse problems are crucial in science and engineering for parameter estimation and model calibration.
• Quantifying input uncertainty in deterministic models is essential for reliable predictions.
• Traditional methods often struggle with ill-posed inverse problems and complex input distributions.

Purpose of the Study:

• To develop a novel approach for inverse sensitivity analysis in deterministic maps.
• To quantify input uncertainty given specified output uncertainty in a linear functional.
• To provide an efficient computational method for approximating probability measures on input spaces.

Main Methods:

• Utilized the law of total probability to reformulate the inverse problem in terms of probability measures.
• Applied the implicit function theorem to derive an approximation for the set-valued inverse.
• Developed an efficient computational algorithm for measure-theoretic approximation of input probability distributions.

Main Results:

• Successfully derived a method for approximating the set-valued inverse, creating a quotient space representation of the input space.
• Developed an efficient computational approach to approximate the probability measure on the input space.
• The method effectively addresses the generally ill-posed nature of inverse sensitivity analysis problems.

Conclusions:

• The proposed method offers a robust framework for inverse sensitivity analysis and model calibration.
• This computational approach enhances the ability to quantify input uncertainty in complex deterministic systems.
• The findings contribute to more reliable parameter estimation and uncertainty quantification in scientific modeling.