# A non-intrusive reduced basis EKI for time-fractional diffusion inverse problems

## TL;DR

This paper introduces a non-intrusive reduced basis method combined with ensemble Kalman inversion to efficiently solve time-fractional diffusion inverse problems, significantly reducing computational costs while maintaining accuracy.

## Contribution

The paper presents a novel POD-DSRBF surrogate modeling approach that accelerates EKI for TFDIPs by decoupling offline and online computations.

## Key findings

- Achieves significant computational speed-up.
- Maintains high accuracy in inverse problem solutions.
- Demonstrated effectiveness on nonlinear TFDIPs.

## Abstract

In this study, we consider an ensemble Kalman inversion (EKI) for the numerical solution of time-fractional diffusion inverse problems (TFDIPs). Computational challenges in the EKI arise from the need for repeated evaluations of the forward model. We address this challenge by introducing a non-intrusive reduced basis (RB) method for constructing surrogate models to reduce computational cost. In this method, a reduced basis is extracted from a set of full-order snapshots by the proper orthogonal decomposition (POD), and a doubly stochastic radial basis function (DSRBF) is used to learn the projection coefficients. The DSRBF is carried out in the offline stage with a stochastic leave-one-out cross-validation algorithm to select the shape parameter, and the outputs for new parameter values can be obtained rapidly during the online stage. Due to the complete decoupling of the offline and online stages, the proposed non-intrusive RB method -- referred to as POD-DSRBF -- provides a powerful tool to accelerate the EKI approach for TFDIPs. We demonstrate the practical performance of the proposed strategies through two nonlinear time-fractional diffusion inverse problems. The numerical results indicate that the new algorithm can achieve significant computational gains without sacrificing accuracy.