Akash Anand

HighFIE Lab is an open research platform designed to accelerate innovation in high-order Fourier interpolation and extension methods. The platform provides an interactive environment where researchers can develop, test, and benchmark new extension algorithms against established baselines.

Key Capabilities

• Algorithm Testing: Upload and test your own Fourier extension strategies with custom functions and parameters
• Performance Benchmarking: Compare your methods against implemented baselines including Hermite-FD (additional methods being added)
• Real-Time Analysis: Interactive visualization of convergence, and approximation errors
• Method Comparison: Side-by-side evaluation of different extension techniques on identical test cases
• Customizable Testing: Define custom test functions, grid parameters, and convergence metrics

For Researchers

HighFIE Lab serves as a community tool for advancing the field of spectral methods. Researchers developing new Fourier continuation techniques can:

• Rapidly prototype and validate algorithmic ideas
• Identify optimal parameter choices through interactive exploration
• Benchmark against state-of-the-art methods with standardized test problems
• Generate publication-ready convergence plots and performance comparisons
• Share reproducible computational experiments with colleagues

Current Methods & Ongoing Development

The platform currently implements Hermite-FD and related Fourier continuation methods. We are actively expanding the library to include:

• Hermite-FCG (Modified FC-Gram) extension scheme
• Further generalization of the modified FC-Gram scheme
• Methods optimized for specific problem classes

Our goal is to create a comprehensive benchmarking suite that becomes a standard reference for the Fourier Continuation/Extension methods community.

Applications

The methods demonstrated and benchmarked in HighFIE Lab are essential for:

• High-order PDE solvers using spectral and pseudospectral methods
• Fast convolution algorithms for wave scattering and propagation problems
• Numerical integration with singular or oscillatory kernels
• Signal processing requiring high-fidelity Fourier representations
• Scientific computing applications demanding high-order accuracy

Related Publications

• A. Anand and P. Nainwal, A modified FC-Gram approximation algorithm with provable error bounds , Journal of Scientific Computing, Volume 105, Article Number 8 (2025).
• A. Anand and A. K. Tiwari, A Fourier extension based numerical integration scheme for fast and high-order approximation of convolutions with weakly singular kernels , SIAM Journal on Scientific Computing, Volume 41, Issue 5, Pages A2772–A2794 (2019).
• A. Anand, A Fourier continuation framework for high-order approximations , Preprint (2018).

Community Contributions

We welcome contributions from the research community. If you have developed a Fourier extension method you would like to see included in the platform, or have suggestions for test cases and benchmarking scenarios, please contact us.

Note for Researchers: HighFIE Lab is under active development. We are continuously adding new methods, test cases, and analytical tools. Check back regularly for updates, or subscribe to notifications about new features.

An interactive platform developed for the course Ordinary Differential Equations (MTH114M). It provides hands-on visualizations and interactive tools to help students explore and build intuition for concepts in differential equations.