# Adding Constraints to Bayesian Inverse Problems

## TL;DR

This paper introduces a Bayesian approach to inverse problems that incorporates constraints, improving parameter estimation accuracy by enforcing known conditions and extending to ensemble Kalman filter methods.

## Contribution

It proposes a novel framework for integrating equality and inequality constraints into Bayesian inverse problems, enhancing parameter identification accuracy.

## Key findings

- Constraints improve parameter estimation accuracy.
- The method effectively handles both equality and inequality constraints.
- Numerical simulations demonstrate improved identification of true parameters.

## Abstract

Using observation data to estimate unknown parameters in computational models is broadly important. This task is often challenging because solutions are non-unique due to the complexity of the model and limited observation data. However, the parameters or states of the model are often known to satisfy additional constraints beyond the model. Thus, we propose an approach to improve parameter estimation in such inverse problems by incorporating constraints in a Bayesian inference framework. Constraints are imposed by constructing a likelihood function based on fitness of the solution to the constraints. The posterior distribution of the parameters conditioned on (1) the observed data and (2) satisfaction of the constraints is obtained, and the estimate of the parameters is given by the maximum a posteriori estimation or posterior mean. Both equality and inequality constraints can be considered by this framework, and the strictness of the constraints can be controlled by constraint uncertainty denoting a confidence on its correctness. Furthermore, we extend this framework to an approximate Bayesian inference framework in terms of the ensemble Kalman filter method, where the constraint is imposed by re-weighing the ensemble members based on the likelihood function. A synthetic model is presented to demonstrate the effectiveness of the proposed method and in both the exact Bayesian inference and ensemble Kalman filter scenarios, numerical simulations show that imposing constraints using the method presented improves identification of the true parameter solution among multiple local minima.