# A distributed active subspace method for scalable surrogate modeling of function valued outputs

## TL;DR

This paper introduces a scalable surrogate modeling approach combining active subspaces and Karhunen-Loève expansions for high-dimensional, function-valued outputs, enabling efficient approximation of complex physical processes.

## Contribution

It presents a novel distributed active subspace method integrated with KL expansions, along with an error analysis and an application example demonstrating its effectiveness.

## Key findings

- Effective dimension reduction for high-dimensional inputs.
- Accurate surrogate models for function-valued outputs.
- Scalable gradient computation via adjoint methods.

## Abstract

We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen-Lo\`eve (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport.