T14 — Gradient Flow Convergence (Łojasiewicz)

Open source document →

1. #1

   The projected gradient flow on the constraint manifold ,

   $$\dot{u}=-{\Pi }_{{\Sigma }_m}\nabla \mathcal{E}(u),$$
2. #2

   Exponential rate via Łojasiewicz. For analytic functions on compact semi-algebraic sets (which includes ), the Łojasiewicz–Simon inequality holds: for some and ,

   $$|\mathcal{E}(u)-\mathcal{E}(u^{\ast }){|}^{1-\theta }\le C\Vert {\Pi }_{{\Sigma }_m}\nabla \mathcal{E}(u)\Vert$$