A Sketched Finite Element Method for Elliptic Models (1907.09852v1)

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.

• The paper proposes a sketched finite element method that accelerates solving high-dimensional elliptic partial differential equations using randomized linear algebra.
• The method involves projecting the high-dimensional problem onto a low-dimensional subspace and solving the reduced system via leverage score-based random sampling.
• Numerical experiments demonstrate that this approach achieves significant speedup over standard FEM while maintaining acceptable accuracy, often within a 10% error.

The paper "A sketched finite element method for elliptic models" proposes an innovative approach to solving elliptic partial differential equations (PDEs) using the Finite Element Method (FEM). The authors, Robert Lung, Yue Wu, Dimitris Kamilis, and Nick Polydorides, introduce a technique that leverages randomised linear algebra to expedite FEM computations, especially in the high-dimensional case, which is critical for applications in digital manufacturing twins and real-time simulation in robotics.

The principal contribution of this work is the development of a sketched FEM approach that effectively reduces computational complexity while maintaining an acceptable level of accuracy. Here's a detailed examination of the methodology and results:

Methodology

1. Projection and Sketching: The methodology involves two main steps - projecting the high-dimensional FEM solution onto a low-dimensional subspace and then solving the reduced system using a sketching technique based on randomised sampling.
2. Leverage Score-Based Sampling: Leveraging the statistical leverage scores of matrices, particularly those associated with the Laplacian operator, the authors define a sampling scheme that optimises the sketching process. This choice helps in achieving nearly optimal performance by focusing computational resources on the most informative parts of the system matrix.
3. Complexity Analysis: An expression is derived linking the complexity of the algorithm to the number of samples necessary to achieve a predefined error tolerance. The method ensures that both the dimensionality is reduced and the error due to sketching is regularised.

Theoretical Insights

• Regularisation through Projection: The projection not only serves as a dimensionality reduction mechanism but also provides regularisation to stabilise the solution against sketching errors.
• Probabilistic Error Bounds: The authors provide probabilistic bounds for the quality of the approximation, showing that the sketched approach achieves significant speed improvements without substantial accuracy loss.

Numerical Experiments

The paper presents three benchmark tests evaluating the performance of the proposed algorithm:

1. Uniformly Random Parameter Field: The tests show that with a modest increase in computational effort (e.g., 1 million samples), solutions could be obtained within a 10% error margin, significantly faster than traditional FEM solvers.
2. Smooth Parameter Field: In this case, the projection error is effectively controlled, and results are achieved with minimal sketching error, demonstrating the method's robustness for smoothly varying coefficients.
3. Non-Smooth Parameter Field: This scenario poses more challenges due to potential discontinuities in the parameter field. The results suggest a larger basis is needed, increasing the computational burden but still providing results faster than standard methods.

Conclusions

The proposed sketched FEM method offers a viable and efficient alternative for solving high-dimensional PDEs, particularly in applications demanding speed, such as real-time simulations. The method's efficacy hinges on the clever use of leverage scores to guide sampling, ensuring that computational resources are utilised where they are most impactful. This paper lays a solid foundation for future exploration into further optimising FEM techniques using randomised linear algebra, particularly in contexts where model reduction and computational efficiency are paramount.