NumPy’s fft.rff2 function computes real-valued spectral derivatives in 2D.

Real-valued spectral derivatives in 2D with the rfft saves half the computation compared to the classical fft. Though straightforward in 1D, it gets tricky in 2D. Wave numbers need modification, and derivative operator multiplication with rfft enables correct computation without losing any information. Real-valued fft is the way to go for spectral derivatives in 2D! 📊

💡 Introduction

In this follow-up video, I will demonstrate the usage of real-valued Fourier transformation (r fft) in 2D. This method enables us to achieve computational efficiency when taking spectral derivatives in comparison to the classical Fourier transformation.

Key Takeaways:

• Real valued Fourier transformation (r fft) saves computational resources when working with real-valued inputs.
• Applying r fft in 2D involves some additional complexities compared to its application in 1D.

The rft is a modification of the classical fft. If our function returns a real value, r fft enables us to save about half of the computation since the classical fft operates on complex signals. However, when it comes to 2D, using r fft becomes more intricate.

🔄 Transforming to Spectral Space

In the previous video, we explored spectral derivatives in 2D using NumPy and Matplotlib. The process involved setting up wave numbers, creating derivative operators, and performing transformations using the fft2 method and inverse transformations. The savings in computation come from discarding half of the Fourier components.

Example of Discretization:

• We achieved a computational advantage by discarding the imaginary components.

When applying real-valued Fourier transformation in higher dimensions, we observe that only one axis changes, necessitating adjustments in the wave numbers to account for this change.

🛠 Handling Shape Mismatch

Naively attempting to take the spectral derivative using r fft results in a shape mismatch error. In 1D, we could simply provide the number of degrees of freedom in state space. However, in 2D, we need to specify the shape to resolve this issue.

Solution:

• We modified the wave number creation to account for two types of wave numbers related to both the full Fourier transformation and the real Fourier transformation.
• This modification successfully resolved the shape mismatch error.

🔀 Application and Validation

By using real-valued Fourier transformation and modifying the wave number creation, we were able to effectively compute the correct derivative values, demonstrating the usefulness of r fft in handling real-valued inputs.

Conclusion:

• Real-valued Fourier transformation offers computational efficiency and accuracy when performing spectral derivatives.

This demonstrated the practical application of real-valued Fourier transformation when dealing with real-valued inputs, showcasing its benefits in terms of computational savings and accurate results.

I hope you enjoyed this practical insight into real-valued spectral derivatives in 2D and how to effectively utilize the advantages of r fft. For more insightful videos on similar topics, consider subscribing to the channel.

FAQ

• What are the advantages of using real-valued Fourier transformation in 2D?

  • Real valued Fourier transformation offers computational efficiency and accurately computes spectral derivatives for real-valued inputs.

• How does the shape mismatch error occur when applying r fft for spectral derivatives?

  • The shape mismatch error arises due to the differences in dimensionality when applying real-valued Fourier transformation, requiring adjustments in the wave numbers.

• What are the key considerations when working with real-valued inputs in Fourier transformation?

  • Modifying the wave number creation and accurate specification of shape are crucial considerations to successfully handle real-valued inputs in Fourier transformation.