This article concerns the construction of high-order energy-decaying numerical methods for gradient flows of evolving surfaces with curvature-dependent energy functionals. The semidiscrete evolving surface finite element method is derived based on the calculus of variation of the semidiscrete surface energy functional. This makes the semidiscrete problem naturally inherit the energy decay structure. With this property, the semidiscrete problem is furthermore formulated as a gradient flow system of ODEs. The averaged vector-field collocation method is used for time discretization of the ODEs to preserve energy decay at the fully discrete level while achieving high-order accuracy in time. Extensive numerical examples are provided to illustrate the accuracy and energy diminishing property of the proposed method, as well as the effectiveness of the method in capturing singularities in the evolution of closed surfaces.

A fundamental task in phase retrieval is to recover an unknown signal $x\in\mathbb{R}^n$ from a set of magnitude-only measurements $y_i=|\langle a_i,x\rangle|,$ $ i=1,\cdots,m$. In this paper, we propose two novel perturbed amplitude models (PAMs) which have a non-convex and quadratic-type loss function. When the measurements $ a_i \in \mathbb{R}^n$ are Gaussian random vectors and the number of measurements $m\ge Cn$, we rigorously prove that the PAMs admit no spurious local minimizers with high probability, i.e., the target solution $ x$ is the unique local minimizer (up to a global phase) and the loss function has a negative directional curvature around each saddle point. Thanks to the well-tamed benign geometric landscape, one can employ the vanilla gradient descent method to locate the global minimizer $x$ (up to a global phase) without spectral initialization. We carry out extensive numerical experiments to show that the gradient descent algorithm with random initialization outperforms  state-of-the-art algorithms with spectral initialization in empirical success rate and convergence speed.

We study the global well-posedness of large-data solutions to the Cauchy problem of the energy critical Cahn-Hilliard-Brinkman equations in $\mathbb{R}^4$. By developing delicate energy estimates, we show that for any given initial datum in $H^5(\mathbb{R}^4)$, there exists a unique global-in-time classical solution to the Cauchy problem. As a special consequence of the result, the global well-posedness of large-data solutions to the energy critical Cahn-Hilliard equation in $\mathbb{R}^4$ follows, which has not been established since the model was first developed over 60 years ago. The proof is constructed based on extensive applications of Gagliardo-Nirenberg type interpolation inequalities, which provides a unified approach for establishing the global well-posedness of large-data solutions to the energy critical Cahn-Hilliard and Cahn-Hilliard-Brinkman equations for spatial dimension up to four.