3. Examples

This directory includes applications of specular differentiation. Each subdirectory is based on a paper.

3.1. Ordinary differential equation

• Directory: examples/ode/

2026-Jung

• Directory: examples/ode/2026-Jung/
• Keywords: generalized differentiation, Mean Value Theorem, Fermat's Theorem, explicit Euler scheme, implicit Euler scheme, Crank-Nicolson scheme
• Reference: K. Jung. Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations. arXiv preprint arXiv:2601.09900, 2026

3.2. Optimization

• Directory: examples/optimization/
• Requirements:
• scipy >= 1.10.0
• torch >= 2.0.0

2026-Jung

• Directory: examples/optimization/2026-Jung/
• Keywords: generalized differentiation, Mean Value Theorem, Fermat's Theorem, subgradient method, stochastic subgradient method
• Reference: K. Jung. Specular differentiation in normed vector spaces and its application to nonsmooth convex optimization. arXiv preprint arXiv:2601.10950, 2026

2024-Jung-Oh

• Directory: examples/optimization/2024-Jung-Oh/
• Keywords: nonsmooth convex optimization, subgradient methods, non-differentiable convex functions, generalization of derivatives, convergence rate
• Reference: K. Jung and J. Oh. Nonsmooth convex optimization using the specular gradient method with root-linear convergence. arXiv preprint arXiv:2210.06933, 2024