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In this article, we address the solution of advection-dominated flow problems by stabilised methods, by means of least-squares computed stabilised coefficients. As main methodological tool, we introduce a data-driven off-line/on-line…

We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder,…

van Holten's algorithm is used to construct Runge-Lenz-type conserved quantities, induced by Killing tensors, on curved manifolds. For the generalized Taub-NUT metric, the most general external potential such that the combined system admits…

We consider the (sub-Riemannian type) control problem of finding a path going from an initial point $x$ to a target point $y$, by only moving in certain admissible directions. We assume that the corresponding vector fields satisfy the…

Image inpainting techniques have shown significant improvements by using deep neural networks recently. However, most of them may either fail to reconstruct reasonable structures or restore fine-grained textures. In order to solve this…

We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…

Optical flow estimation is a widely known problem in computer vision introduced by Gibson, J.J(1950) to describe the visual perception of human by stimulus objects. Estimation of optical flow model can be achieved by solving for the motion…

We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and $\mathbb{R}$. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve…

Reconstructing a finite set of curves from an unordered set of sample points is a well studied topic. There has been less effort that considers how much better the reconstruction can be if tangential information is given as well. We show…

One-dimensional blood flow models take the general form of nonlinear hyperbolic systems but differ greatly in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class…

We proved a Bernstein theorem for ancient solution to symplectic mean curvature flow via the complex phase map .

We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point,…

We have recently proposed a new method of flow analysis, based on a cumulant expansion of multiparticle azimuthal correlations. Here, we describe the practical implementation of the method. The major improvement over traditional methods is…

In this article we study Chen's flow of curves from theoreical and numerical perspectives. We investigate two settings: that of closed immersed $\omega$-circles, and immersed lines satisfying a cocompactness condition. In each of the…

We construct an ancient solution to planar curve shortening. The solution is at all times compact and embedded. For $t\ll0$ it is approximated by the rotating Yin-Yang soliton, truncated at a finite angle $\alpha(t) = -t$, and closed off by…

We have developed a coupled level set and volume of fluid-based computational fluid dynamics model to analyze the droplet formation mechanism in a square flow-focusing microchannel. We demonstrate a flexible manipulation of droplet…

Generating a 3D point cloud from a single 2D image is of great importance for 3D scene understanding applications. To reconstruct the whole 3D shape of the object shown in the image, the existing deep learning based approaches use either…

Optical flow estimation is a well-studied topic for automated driving applications. Many outstanding optical flow estimation methods have been proposed, but they become erroneous when tested in challenging scenarios that are commonly…

Most of current Convolution Neural Network (CNN) based methods for optical flow estimation focus on learning optical flow on synthetic datasets with groundtruth, which is not practical. In this paper, we propose an unsupervised optical flow…

Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time…