# Preconditioned pseudo-time continuation for parameterized inverse problems

## TL;DR

This paper introduces an efficient pseudo-time continuation method with a novel adaptive quasi-Newton preconditioner to solve parameterized inverse problems constrained by PDEs, significantly reducing computational costs for uncertainty quantification.

## Contribution

We develop a new adaptive quasi-Newton preconditioner for pseudo-time continuation, improving the efficiency of solving parameterized inverse PDE problems.

## Key findings

- Preconditioner accelerates convergence of inverse problem solutions.
- Method effectively handles nonlinear inverse problems.
- Framework enables efficient uncertainty quantification.

## Abstract

We consider parameterized variational inverse problems that are constrained by partial differential equations (PDEs). We seek to efficiently compute the solution of the inverse problem when auxiliary model parameters, which appear in the governing PDE, are varied. Computing the solution of the inverse problem for different auxiliary parameter values is crucial for uncertainty quantification. This, however, is computationally challenging since it requires solving many optimization problems for different realizations of the auxiliary parameters. We leverage pseudo-time continuation and solve an initial value problem to evolve the optimal solution along an auxiliary parameter path. This article introduces the use of an adaptive quasi-Newton Hessian preconditioner to accelerate the computation. Our proposed preconditioner exploits properties of the pseudo-time continuation process to achieve reliable and efficient computation. We elaborate our proposed framework and elucidate its properties for two nonlinear inverse problems.