# Gaussian Process Regression and Conditional Polynomial Chaos for Parameter Estimation

## TL;DR

This paper introduces a novel Bayesian parameter estimation method combining Gaussian Process regression and conditional Polynomial Chaos to efficiently estimate unknown parameters in PDE models, reducing computational costs and improving accuracy.

## Contribution

The paper presents a new data-driven surrogate modeling approach that integrates Gaussian Process regression with conditional Polynomial Chaos for PDE parameter estimation.

## Key findings

- Reduces dimensionality of parameter space.
- Replaces expensive PDE solutions with surrogate models.
- Improves accuracy in estimating hydraulic conductivity.

## Abstract

We present a new approach for constructing a data-driven surrogate model and using it for Bayesian parameter estimation in partial differential equation (PDE) models. We first use parameter observations and Gaussian Process regression to condition the Karhunen-Lo\'{e}ve (KL) expansion of the unknown space-dependent parameters and then build the conditional generalized Polynomial Chaos (gPC) surrogate model of the PDE states. Next, we estimate the unknown parameters by computing coefficients in the KL expansion minimizing the square difference between the gPC predictions and measurements of the states using the Markov Chain Monte Carlo method. Our approach addresses two major challenges in the Bayesian parameter estimation. First, it reduces dimensionality of the parameter space and replaces expensive direct solutions of PDEs with the conditional gPC surrogates. Second, the estimated parameter field exactly matches the parameter measurements. In addition, we show that the conditional gPC surrogate can be used to estimate the states variance, which, in turn, can be used to guide data acquisition. We demonstrate that our approach improves its accuracy with application to one- and two-dimensional Darcy equation with (unknown) space-dependent hydraulic conductivity. We also discuss the effect of hydraulic conductivity and head locations on the accuracy of the hydraulic conductivity estimations.