# Variational method for multiple parameter identification in elliptic PDEs

## TL;DR

This paper introduces a variational approach with Tikhonov regularization for simultaneously identifying multiple parameters in elliptic PDEs from limited measurement data, ensuring convergence and error analysis.

## Contribution

It presents a novel variational method combined with finite element discretization for multi-parameter inverse problems in elliptic PDEs, with proven convergence and error bounds.

## Key findings

- Convergence of the proposed method is established.
- Error bounds for the parameter estimates are derived.
- The approach effectively handles weaker measurement data.

## Abstract

In the present paper we investigate the inverse problem of identifying simultaneously the diffusion matrix, source term and boundary condition as well as the state in the Neumann boundary value problem for an elliptic partial differential equation (PDE) from a measurement data, which is weaker than required of the exact state. A variational method based on energy functions with Tikhonov regularization is here proposed to treat the identification problem. We discretize the PDE with the finite element method and prove the convergence as well as analyse error bounds of this approach.