The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded

Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions,provided that the initial data are compactly supported and sufficiently small in Sobolev norm.In this work,Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions.A natural question then arises whether the time-growth is a true phenomenon,despite the possible conservation of basic energy.In the present paper,we establish that the top-order energy of the solutions in Alinhac theorem remains globally bounded in time.

The Global Landscape of Phase Retrieval Ⅱ:Quotient Intensity Models

A fundamental problem in phase retrieval is to reconstruct an unknown signal from a set of magnitude-only measurements.In this work we introduce three novel quotient intensity models(QIMs) based on a deep modification of the traditional intensity-based models.A remarkable feature of the new loss functions is that the corresponding geometric landscape is benign under the optimal sampling complexity.When the measurements ai∈Rn are Gaussian random vectors and the number of measurements m≥Cn,the QIMs 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.Such benign geometric landscape allows the gradient descent methods to find the global solution x(up to a global phase) without spectral initialization.

The Global Landscape of Phase Retrieval I:Perturbed Amplitude Models

A fundamental task in phase retrieval is to recover an unknown signal x∈R^(n) from a set of magnitude-only measurements y_(i)=|〈a_(i),x〉|,i=1,…,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)∈R^(n) are Gaussian random vectors and the number of measurements m≥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.