# Rank adaptive tensor recovery based model reduction for partial differential equations with high-dimensional random inputs

## TL;DR

This paper introduces a rank adaptive tensor recovery method for reducing high-dimensional PDE models with random inputs, utilizing kernel PCA and novel initialization strategies to improve efficiency and stability.

## Contribution

It develops a new tensor recovery approach with adaptive rank adjustment and efficient initialization for high-dimensional stochastic PDE model reduction.

## Key findings

- Effective reduction of high-dimensional PDE models demonstrated.
- Improved stability and efficiency shown through numerical experiments.
- Kernel PCA enhances stochastic collocation approximation.

## Abstract

This work proposes a systematic model reduction approach based on rank adaptive tensor recovery for partial differential equation (PDE) models with high-dimensional random parameters. Since the standard outputs of interest of these models are discrete solutions on given physical grids which are high-dimensional, we use kernel principal component analysis to construct stochastic collocation approximations in reduced dimensional spaces of the outputs. To address the issue of high-dimensional random inputs, we develop a new efficient rank adaptive tensor recovery approach to compute the collocation coefficients. Novel efficient initialization strategies for non-convex optimization problems involved in tensor recovery are also developed in this work. We present a general mathematical framework of our overall model reduction approach, analyze its stability, and demonstrate its efficiency with numerical experiments.