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We further develop a new framework, called PDE Acceleration, by applying itto calculus of variations problems defined for general functions on$\mathbb{R}^n$, obtaining efficient numerical algorithms to solve the resultingclass of optimization problems based on simple discretizations of theircorresponding accelerated PDE's. While the resulting family of PDE's andnumerical schemes are quite general, we give special attention to theirapplication for regularized inversion problems, with particular illustrativeexamples on some popular image processing applications. The method is ageneralization of momentum, or accelerated, gradient descent to the PDEsetting. For elliptic problems, the descent equations are a nonlinear dampedwave equation, instead of a diffusion equation, and the acceleration isrealized as an improvement in the CFL condition from $\Delta t\sim \Deltax^{2}$ (for diffusion) to $\Delta t\sim \Delta x$ (for wave equations). We workout several explicit as well as a semi-implicit numerical schemes, togetherwith their necessary stability constraints, and include recursive updateformulations which allow minimal-effort adaptation of existing gradient descentPDE codes into the accelerated PDE framework. We explore these schemes morecarefully for a broad class of regularized inversion applications, with specialattention to quadratic, Beltrami, and Total Variation regularization, where theaccelerated PDE takes the form of a nonlinear wave equation. Experimentalexamples demonstrate the application of these schemes for image denoising,deblurring, and inpainting, including comparisons against Primal Dual, SplitBregman, and ADMM algorithms.

Accelerated PDE's for efficient solution of regularized inversion problems