A Stream Function Approach to Optical Flow with Applications to Fluid Transport Dynamics

Optical flow is one of the classical problems in computer vision, but it has recently also been adapted to applications from other fields, such as fluid mechanics and dynamical systems. If the goal is to analyze the dynamics of system whose evolution is governed by a flow field that is the gradient of a potential function – which describes many flows in fluid dynamics – it is natural to approach the optical flow problem by reconstructing the potential function, also called the stream function, rather than reconstructing the components of the flow directly. This alternate approach allows one to impose scientific priors, via regularization, directly on the flow itself rather than on its components independently. We demonstrate the stream function formulation of optical flow and its application to reconstructing an oceanic fluid flow driven by satellite measurements. It is also shown how these flow fields can be used to analyze mixing and mass transport in the fluid system being imaged. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic Geometric Models with Non-stationary Spatial Correlations in Lagrangian Fluid Flows

Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration’s “Global Drifter Program”, this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie–Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD.     

Localization and Coherent Structures in Wave Dynamics via Multiresolution

We apply variational‐wavelet approach for constructing multiscale high‐localized eigenmodes expansions in different models of nonlinear waves. We demonstrate appearance of coherent localized structures and stable pattern formation in different collective dynamics models.     

The Endogenous Analysis of Flows,with Applications to Migrations,Social Mobility and Opinion Shifts

For exponential random graph models, under quite general conditions, it is proved that induced subgraphs on node sets disconnected from the other nodes still have distributions from an exponential random graph model. This can help in the theoretical interpretation of such models. An application is that for saturated snowball samples from a potentially larger graph which is a realization of an exponential random graph model, it is possible to do the analysis of the observed snowball sample within the framework of exponential random graph models without any knowledge of the larger graph.     

Perturbed Augmented Lagrangian Method Framework with Applications to Proximal and Smoothed Variants

Journal of Optimization Theory and Applications - We introduce a perturbed augmented Lagrangian method framework, which is a convenient tool for local analyses of convergence and rates of...     

Augmented Lagrangian Methods for $p$-Harmonic Flows with the Generalized Penalization Terms and Application to Image Processing

In this paper, we propose a generalized penalization technique and a convex constraint minimization approach for the $p$-harmonic flow problem following the ideas in [Kang & March, IEEE T. Image Process., 16 (2007), 2251-2261]. We use fast algorithms to solve the subproblems, such as the dual projection methods, primal-dual methods and augmented Lagrangian methods. With a special penalization term, some special algorithms are presented. Numerical experiments are given to demonstrate the performance of the proposed methods. We successfully show that our algorithms are effective and efficient due to two reasons: the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately (even 2 inner iterations of the subproblem are enough). It is also observed that better PSNR values are produced using the new algorithms.     

A Geometric Inequality with Applications to the Kakeya Problem in Three Dimensions

((Without abstract)) Submitted: May 1997, Revised version: August 1997     

Geometric Properties and Coincidence Theorems with Applications to Generalized Vector Equilibrium Problems

The present paper is divided into two parts. In the first part, we derive a Fan-KKM type theorem and establish some Fan type geometric properties of convex spaces. By applying our results, we also obtain some coincidence theorems and fixed-point theorems in the setting of convex spaces. The second part deals with the applications of our coincidence theorem to establish some existence results for a solution to the generalized vector equilibrium problems.     

On the Relation between Gradient Flows and the Large-Deviation Principle,with Applications to Markov Chains and Diffusion

Motivated by the occurrence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ? that induce a flow, given by \(\mathcal{L} (\rho _{t},\dot \rho _{t})=0\) . We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ? is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.     

On the Stability and the Approximation of Branching Distribution Flows,with Applications to Nonlinear Multiple Target Filtering

We analyze the exponential stability properties of a class of measure-valued equations arising in nonlinear multi-target filtering problems. We also prove the uniform convergence properties w.r.t. the time parameter of a rather general class of stochastic filtering algorithms, including sequential Monte Carlo type models and mean field particle interpretation models. We illustrate these results in the context of the Bernoulli and the Probability Hypothesis Density filter, yielding what seems to be the first results of this kind in this subject.     

Sums of Regular Cantor Sets,Dynamics and Applications to Number Theory

Periodica Mathematica Hungarica -     

On a Moving Boundary Solution to the Fokker-Planck Equation for Particle Transport in Turbulent Flows with Absorbing Boundaries

A Fokker—Planck equation describing the transport of particlesin turbulent flows is considered. The initial value problemwith perfectly absorbing boundary conditions on the wall issolved by introducing a characteristic line in phase space.It is found that the solution domain in phase space decomposesinto three regions which are connected by two moving boundaries,and the moving boundaries can be found explicitly. For Gaussian-typeinitial conditions, the solution of the Fokker—Planckequation can be obtained in each of the three regions by solvinga diffusion equation. An approximation procedure for the generalinitial value problem is established and the approximate solutionsequence is shown to be convergent in a certain Hilbert space.     

Formation of Coherent Structures in Kolmogorov Flow with Stratification and Drag

We study a weakly stratified Kolmogorov flow under the effect of a small linear drag. We perform a linear stability analysis of the basic state. We construct the finite dimensional dynamical system deriving from the truncated Fourier mode approximation. Using the Reynolds number as bifurcation parameter we build the corresponding diagram up to Re=100. We observe the coexistence of three coherent structures.     

Life-Span of Classical Solutions to Hyperbolic Geometric Flow in Two Space Variables with Slow Decay Initial Data

In this paper we investigate the life-span of classical solutions to the hyperbolic geometric flow in two space variables with slow decay initial data. By establishing some new estimates on the solutions of linear wave equations in two space variables, we give a lower bound of the life-span of classical solutions to the hyperbolic geometric flow with asymptotic flat initial Riemann surfaces.     

Geometric quadrisection in linear time, with application to VLSI placement

We consider the following problem: given a set of points in the plane, each with a weight, and capacities of the four quadrants, assign each point to one of the quadrants such that the total weight of points assigned to a quadrant does not exceed its capacity, and the total distance is minimized.

This problem is most important in placement of VLSI circuits and is likely to have other applications. It is NP-hard, but the fractional relaxation always has an optimal solution which is “almost” integral. Hence for large instances, it suffices to solve the fractional relaxation. The main result of this paper is a linear-time algorithm for this relaxation. It is based on a structure theorem describing optimal solutions by so-called “American maps” and makes sophisticated use of binary search techniques and weighted median computations.

This algorithm is a main subroutine of a VLSI placement tool that is used for the design of many of the most complex chips.     

Simple Asymptotics for a Generalized Wave Equation with Degenerating Velocity and Their Applications in the Linear Long Wave Run-Up Problem

Asymptotic solutions of the wave equation degenerating on the boundary of the domain (where the wave propagation velocity vanishes as the square root of the distance from the boundary) can be represented with the use of a modified canonical operator on a Lagrangian submanifold, invariant with respect to theHamiltonian vector field, of the nonstandard phase space constructed by the authors in earlier papers. The present paper provides simple expressions in a neighborhood of the boundary for functions represented by such a canonical operator and, in particular, for the solution of the Cauchy problem for the degenerate wave equation with initial data localized in a neighborhood of an interior point of the domain.     

Almost-Normed Spaces of Functions with Given Asymptotics, Lagrangian Asymptotics, and Their Application to Ordinary Differential Equations

The concept of almost-normed spaces is introduced. It is proved that the space of sufficiently smooth functions asymptotically approximating to polynomials (of degrees no higher than a given one) as their argument tends to infinity is an almost-normed space. It is demonstrated that this space is a complete metric space with respect to the metrics generated by the almost-norm introduced. The space of functions strongly asymptotically approximating to polynomials is defined, and its embedding into the space of functions asymptotically approximating to polynomials is proved. The results obtained give a new approach to studying boundary-value problems with asymptotic initial value data at singular points of ordinary differential equations.     

A Green's Function Approach to Wave Propagation and Potential Flow around Obstacles in Infinite Cylindrical Channels

We study (a) acoustic waves generated by a time-harmonic force distribution and (b) the potential flow with prescribed velocity at infinity in an infinite cylinder Ω₀ = Ω′×ℝ with bounded cross-section Ω′⊂ℝ² in the presence of m embedded obstacles B₁,…,B_m. By using Green's function G_κ(x,y) of the Neumann problem for the reduced wave equation ΔU+κ²U = 0 in the unperturbed domain Ω₀, both problems can be reduced to integral equations over the boundaries of the obstacles. The main properties of G_κ(x,y), which are required for this approach, are derived in the first part of this paper.