# Goal-oriented adaptive surrogate construction for stochastic inversion

## TL;DR

This paper introduces an adaptive surrogate modeling method for stochastic inverse problems that reduces errors efficiently by guiding refinement with local error indicators, demonstrated on complex vibroacoustics problems.

## Contribution

It presents a novel adaptive surrogate construction approach that balances stochastic and deterministic errors using adjoint techniques and multi-level surrogates.

## Key findings

- Significantly reduces computational cost compared to uniform refinement
- Achieves accurate uncertainty quantification in complex vibroacoustics models
- Demonstrates effectiveness of adaptive refinement strategies in stochastic inversion

## Abstract

Stochastic inverse problems are generally solved by some form of finite sampling of a space of uncertain parameters. For computationally expensive models, surrogate response surfaces are often employed to increase the number of samples used in approximating the solution. The result is generally a trade off in errors where the stochastic error is reduced at the cost of an increase in deterministic/discretization errors in the evaluation of the surrogate. Such stochastic errors pollute predictions based on the stochastic inverse. In this work, we formulate a method for adaptively creating a special class of surrogate response surfaces with this stochastic error in mind. Adjoint techniques are used to enhance the local approximation properties of the surrogate allowing the construction of a higher-level enhanced surrogate. Using these two levels of surrogates, appropriately derived local error indicators are computed and used to guide refinement of both levels of the surrogates. Three types of refinement strategies are presented and combined in an iterative adaptive surrogate construction algorithm. Numerical examples, including a complex vibroacoustics application, demonstrate how this adaptive strategy allows for accurate predictions under uncertainty for a much smaller computational cost than uniform refinement.