Accelerated Variational PDEs for Efficient Solution of Regularized Inversion Problems.

Area of Science:

• Numerical analysis
• Optimization
• Image processing

Background:

• Calculus of variation problems are crucial for optimization.
• Existing methods often rely on diffusion equations for solving inverse problems.
• Gradient descent methods can be slow for complex optimization tasks.

Purpose of the Study:

• To develop a new framework called PDE acceleration.
• To create efficient numerical algorithms for optimization problems.
• To apply this framework to regularized inversion and image processing.

Main Methods:

• Generalizing momentum (accelerated) gradient descent to the PDE setting.
• Deriving nonlinear damped wave equations from elliptic problems.
• Developing explicit and semi-implicit numerical schemes with stability constraints.

Main Results:

• Achieved improved CFL conditions (Δt ~ Δx) compared to diffusion equations (Δt ~ Δx²).
• Demonstrated applicability to quadratic, Beltrami, and total variation regularization.
• Showcased effectiveness in image denoising, deblurring, and inpainting.

Conclusions:

• PDE acceleration offers a generalized and efficient approach for solving optimization problems.
• The framework provides significant improvements for regularized inversion and image processing tasks.
• The proposed numerical schemes are adaptable and effective compared to existing algorithms.