A Least Squares Approach to Constructing High Order Whitney Forms

Core Concepts

This work describes and tests a least squares approach to constructing high order Whitney forms, which provide a finite dimensional representation of differential forms. The least squares method is proposed as an alternative to the challenging task of determining unisolvence of sets of simplicial supports for interpolation of Whitney forms.

Abstract

The paper introduces the concept of Whitney forms, which are a finite dimensional space of differential forms that capture the underlying geometry of the domain. Traditional interpolation of Whitney forms requires finding unisolvent sets of simplicial supports, which is a complex task.
The authors propose a least squares approach as an alternative. This involves finding the Whitney form that minimizes the weighted 2-norm distance from the given data over a rich set of supports. The authors observe several interesting properties of this least squares approach:

When the total polynomial degree is fixed, the least squares and interpolation approaches perform similarly, both for easy-to-capture and Runge-type differential forms.

Increasing the number of observations beyond a certain threshold does not improve the least squares error, which stabilizes below the interpolation error. This is consistent with nodal observations.

Carefully increasing both the polynomial degree and the number of observations can defeat the Runge phenomenon, even for uniform distributions of supports.

The paper provides details on the computational implementation, including the use of quadrature rules to efficiently compute the required integrals. Numerical experiments on segments in 2D and faces in 3D are presented, demonstrating the effectiveness of the least squares approach.

Stats

The error in the 0-norm for the least squares approximation of the 1-form ω1 = ex+y dx + sin(xy) dy is comparable to the error of interpolation on minimal supports.
The error in the 0-norm for the least squares approximation of the Runge-type 1-form (20) can be reduced by increasing both the polynomial degree and the number of supports.
The error in the 0-norm for the least squares approximation of the 2-form ω2 = ez+xy dx ∧ dy + sin(xy) sqrt(z+1) dx ∧ dz + cos(x+y) dy ∧ dz is comparable to the error of interpolation on minimal supports.
The error in the 0-norm for the least squares approximation of the Runge-type 2-form can be reduced by increasing both the polynomial degree and the number of supports.

Quotes

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Key Insights Distilled From

A least squares approach to Whitney forms

by Ludovico Bru... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15727.pdf

Deeper Inquiries

How can the least squares approach be combined with interpolation to obtain a hybrid method that leverages the strengths of both techniques?

In the context of Whitney forms, the least squares approach and interpolation can be combined to create a hybrid method that capitalizes on the benefits of each technique.
One way to achieve this is by using the least squares approach to handle noisy or uncertain data points that may not fit well with traditional interpolation methods. By minimizing the overall error between the observed data and the approximated values, the least squares method can provide a more robust and stable solution in the presence of noise.
On the other hand, interpolation can be used to capture the general trend or structure of the data, providing a smooth and continuous representation of the underlying function. By combining the two approaches, the hybrid method can leverage the accuracy of interpolation while benefiting from the stability and robustness of the least squares approach.
This hybrid method can be particularly useful in scenarios where the data is noisy or when there are outliers that may disrupt the interpolation process. By incorporating the strengths of both techniques, the hybrid method can offer a more reliable and accurate approximation of the Whitney forms.

How can the impact of noise or uncertainty in the input data on the stability and accuracy of the least squares approximation of Whitney forms?

Noise or uncertainty in the input data can have a significant impact on the stability and accuracy of the least squares approximation of Whitney forms.

Stability:

Noise Sensitivity: The presence of noise in the input data can lead to instability in the least squares approximation. Small perturbations in the data can result in large variations in the computed solution, affecting the stability of the method.
Ill-Conditioning: Noisy data can lead to ill-conditioned systems of equations, making it challenging to obtain a reliable solution. This can result in numerical instability and inaccuracies in the approximation.

Accuracy:

Bias and Variance Trade-off: Noise in the data can introduce bias or variance in the least squares approximation. Balancing the bias-variance trade-off becomes crucial to ensure the accuracy of the approximation.
Overfitting: In the presence of noise, there is a risk of overfitting the data, where the model captures the noise rather than the underlying pattern. This can lead to a decrease in the accuracy of the approximation.

To mitigate the impact of noise on the stability and accuracy of the least squares approximation of Whitney forms, techniques such as regularization, robust optimization methods, and data preprocessing (e.g., noise reduction, outlier detection) can be employed. These strategies can help improve the robustness and reliability of the approximation in the presence of noisy or uncertain data.

How can the computational efficiency of the least squares approach be further improved, for example by exploiting the structure of the space of Whitney forms or by developing specialized numerical linear algebra techniques?

Improving the computational efficiency of the least squares approach for Whitney forms can be achieved through various strategies that leverage the structure of the space of Whitney forms and specialized numerical linear algebra techniques:

Exploiting Structure:

Low-Rank Approximations: Utilizing low-rank approximations or exploiting the inherent structure of Whitney forms can reduce the computational complexity of the least squares problem.
Sparse Representations: If the Whitney forms exhibit sparsity, techniques such as sparse matrix factorization or iterative solvers tailored for sparse systems can enhance efficiency.
Hierarchical Approximations: Hierarchical bases or decomposition methods can be employed to represent Whitney forms in a more computationally efficient manner.

Specialized Numerical Linear Algebra Techniques:

Preconditioning: Designing effective preconditioners specific to the structure of Whitney forms can accelerate the convergence of iterative solvers used in the least squares approach.
Parallel Computing: Leveraging parallel computing techniques to distribute the computational workload and exploit multi-core architectures for faster computations.
GPU Acceleration: Utilizing GPU acceleration for matrix operations and computations can significantly speed up the least squares approximation process.

By combining these strategies and tailoring them to the specific characteristics of Whitney forms, the computational efficiency of the least squares approach can be further improved, leading to faster and more scalable computations.