# A Sketched Finite Element Method for Elliptic Models

## TL;DR

This paper introduces a sketched finite element method for elliptic PDEs that uses random sampling based on leverage scores to significantly speed up computations while maintaining accuracy.

## Contribution

It proposes a novel algorithm combining low-dimensional projection and randomized sketching with leverage score sampling for efficient high-dimensional elliptic PDE solutions.

## Key findings

- Achieves nearly optimal performance with leverage score sampling
- Provides theoretical bounds on error and complexity
- Demonstrates two orders of magnitude speedup in simulations

## Abstract

We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that involves projecting the finite element solution onto a low-dimensional subspace and sketching the reduced equations using randomised sampling. We show that a sampling distribution based on the leverage scores of a tall matrix associated with the discrete Laplacian operator, can achieve nearly optimal performance and a significant speedup. We derive an expression of the complexity of the algorithm in terms of the number of samples that are necessary to meet an error tolerance specification with high probability, and an upper bound for the distance between the sketched and the high-dimensional solutions. Our analysis shows that the projection not only reduces the dimension of the problem but also regularises the reduced system against sketching error. Our numerical simulations suggest speed improvements of two orders of magnitude in exchange for a small loss in the accuracy of the prediction.