Control of harmonic map heat flow with an external field

We investigate the control problem of harmonic map heat flow by means of an external magnetic field. In contrast to the situation of a parabolic system with internal or boundary control, the magnetic field acts as the coefficients of the lower order terms of the equation. We show that for initial data whose image stays in a hemisphere, with one control acting on a subset of the domain plus a spatial-independent control acting on the whole domain, the state of the system can be steered to any ground state, i.e. any given unit vector, within any short time. To achieve this, in the first step a spatial independent control is applied to steer the solution into a small neighborhood of the peak of the hemisphere. Then under stereographic projection, the original system is reduced to an internal parabolic control system with initial data sufficiently close to $0$ such that the existing method for local controllability can be applied. The key process in this step is to give an explicit solution of an underdetermined algebraic system such that the affine type control can be converted into an internal control.Comment: a few details are added in this new versio

Patchwork: A Patch-wise Attention Network for Efficient Object Detection and Segmentation in Video Streams

Recent advances in single-frame object detection and segmentation techniques have motivated a wide range of works to extend these methods to process video streams. In this paper, we explore the idea of hard attention aimed for latency-sensitive applications. Instead of reasoning about every frame separately, our method selects and only processes a small sub-window of the frame. Our technique then makes predictions for the full frame based on the sub-windows from previous frames and the update from the current sub-window. The latency reduction by this hard attention mechanism comes at the cost of degraded accuracy. We made two contributions to address this. First, we propose a specialized memory cell that recovers lost context when processing sub-windows. Secondly, we adopt a Q-learning-based policy training strategy that enables our approach to intelligently select the sub-windows such that the staleness in the memory hurts the performance the least. Our experiments suggest that our approach reduces the latency by approximately four times without significantly sacrificing the accuracy on the ImageNet VID video object detection dataset and the DAVIS video object segmentation dataset. We further demonstrate that we can reinvest the saved computation into other parts of the network, and thus resulting in an accuracy increase at a comparable computational cost as the original system and beating other recently proposed state-of-the-art methods in the low latency range.Comment: ICCV 2019 Camera Ready + Supplementar

Phase-field approximation of the Willmore flow

We investigate the phase-field approximation of the Willmore flow. This is a fourth-order diffusion equation with a parameter $\epsilon>0$ that is proportional to the thickness of the diffuse interface. We show rigorously that for well-prepared initial data, as $\epsilon$ trends to zero the level-set of solution will converge to motion by Willmore flow before the singularity of the later occurs. This is done by constructing an approximate solution from the limiting flow via matched asymptotic expansions, and then estimating its difference with the real solution. The crucial step and also the major contribution of this work is to show a spectrum condition of the linearized operator at the optimal profile. This is a fourth-order operator written as the sum of the squared Allen-Cahn operator and a singular linear perturbation.Comment: convergence rate is improved and more details are provided for the matched asymptotical expansion

The Oseen-Frank limit of Onsager's molecular theory for liquid crystals

We study the relationship between Onsager's molecular theory and the Oseen-Frank theory for nematic liquid crystals. Under the molecular setting, we consider the free energy that includes the effects of nonlocal molecular interactions. By imposing the strong anchoring boundary condition on the second moment of the number density function, we prove the existence of global minimizers for the free energy. Moreover, when the re-scaled interaction distance tends to zero, the corresponding global minimizers will converge to a uniaxial distribution whose orientation is described by a minimizer of Oseen-Frank energy.Comment: 24 page

The small Deborah number limit of the Doi-Onsager equation without hydrodynamics

We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals without hydrodynamics. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position $x\in \mathbb{R}^d(d=2,3)$ and orientation vector $m\in\mathbb{S}^2$ (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into $\mathbb{S}^2$. This flow is a special case of the gradient flow to the Oseen-Frank energy functional for nematic liquid crystals. The key ingredient is to show the strong compactness of the family of number density functions and the proof relies on the strong compactness of the corresponding second moment (or the $Q$-tensor), a spectral decomposition of the linearized operator near the limit local equilibrium distribution, as well as the energy dissipation estimate.Comment: 38 pages. Comments are welcom

Well-posedness of a fully-coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data

We prove short-time well-posedness and existence of global weak solutions of the Beris--Edwards model for nematic liquid crystals in the case of a bounded domain with inhomogeneous mixed Dirichlet and Neumann boundary conditions. The system consists of the Navier-Stokes equations coupled with an evolution equation for the $Q$-tensor. The solutions possess higher regularity in time of order one compared to the class of weak solutions with finite energy. This regularity is enough to obtain Lipschitz continuity of the non-linear terms in the corresponding function spaces. Therefore the well-posedness is shown with the aid of the contraction mapping principle using that the linearized system is an isomorphism between the associated function spaces.Comment: 25 page

Global Well-posedness of the Two Dimensional Beris-Edwards System with General Laudau-de Gennes Free Energy

In this paper, we consider the Beris-Edwards system for incompressible nematic liquid crystal flows. The system under investigation consists of the Navier-Stokes equations for the fluid velocity $\mathbf{u}$ coupled with an evolution equation for the order parameter $Q$-tensor. One important feature of the system is that its elastic free energy takes a general form and in particular, it contains a cubic term that possibly makes it unbounded from below. In the two dimensional periodic setting, we prove that if the initial $L^\infty$-norm of the $Q$-tensor is properly small, then the system admits a unique global weak solution. The proof is based on the construction of a specific approximating system that preserves the $L^\infty$-norm of the $Q$-tensor along the time evolution

Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions

Existence and uniqueness of local strong solution for the Beris--Edwards model for nematic liquid crystals, which couples the Navier-Stokes equations with an evolution equation for the Q-tensor, is established on a bounded domain in the case of homogeneous Dirichlet boundary conditions. The classical Beris--Edwards model is enriched by including a dependence of the fluid viscosity on the Q-tensor. The proof is based on a linearization of the system and Banach's fixed-point theorem.Comment: 31 page

An algorithm to find the spectral radius of nonnegative tensors and its convergence analysis

In this paper we propose an iterative algorithm to find out the spectral radius of nonnegative tensors. This algorithm is an extension of the smoothing method for finding the largest eigenvalue of a nonnegative matrix \cite{s14}. For nonnegative irreducible tensors, we establish the converges of the algorithm. Finally we report some numerical results and conclude this paper with some remarks

Deep Approximately Orthogonal Nonnegative Matrix Factorization for Clustering

Nonnegative Matrix Factorization (NMF) is a widely used technique for data representation. Inspired by the expressive power of deep learning, several NMF variants equipped with deep architectures have been proposed. However, these methods mostly use the only nonnegativity while ignoring task-specific features of data. In this paper, we propose a novel deep approximately orthogonal nonnegative matrix factorization method where both nonnegativity and orthogonality are imposed with the aim to perform a hierarchical clustering by using different level of abstractions of data. Experiment on two face image datasets showed that the proposed method achieved better clustering performance than other deep matrix factorization methods and state-of-the-art single layer NMF variants.Comment: 5 page