From gas dynamics with large friction to gradient flows describing diffusion theories

We study the emergence of gradient flows in Wasserstein distance as high friction limits of an abstract Euler flow generated by an energy functional. We develop a relative energy calculation that connects the Euler flow to the gradient flow in the diffusive limit regime. We apply this approach to prove convergence from the Euler–Poisson system with friction to the Keller–Segel system in the regime that the latter has smooth solutions. The same methodology is used to establish convergence from the Euler–Korteweg theory with monotone pressure laws to the Cahn–Hilliard equation.     

Isotopy invariants for smooth tori in 4-manifolds

In this paper we define invariants under smooth isotopy for certain two-dimensional knots using some refined Cerf theory. One of the invariants is the knot type of some classical knot generalizing the string number of closed braids. The other invariant is a generalization of the unique invariant of degree 1 for classical knots in 3-manifolds. Possibly, these invariants can be used to distinguish smooth embeddings of tori in some 4-manifolds but which are equivalent as topological embeddings.     

A gradient bound for free boundary graphs

We prove an analogue for a one‐phase free boundary problem of the classical gradient bound for solutions to the minimal surface equation. It follows, in particular, that every energy‐minimizing free boundary that is a graph is also smooth. The method we use also leads to a new proof of the classical minimal surface gradient bound. © 2010 Wiley Periodicals, Inc.     

An analysis of the Perona‐Malik scheme

We investigate how the Perona‐Malik scheme evolves piecewise smooth initial data in one dimension. By scaling a natural parameter that appears in the scheme in an appropriate way with respect to the grid size, we obtain a meaningful continuum limit. The resulting evolution can be seen as the gradient flow for an energy, just as the discrete evolutions are gradient flows for discrete energies. It involves, except at special isolated times, solving a system of heat equations coupled to each other through nonlinear boundary conditions. At the special times, the solutions experience gradient blowup; nevertheless, there is a natural continuation for the solutions beyond these singular times. © 2001 John Wiley & Sons, Inc.     

Contact interaction inside knot connections of cables

Knots as a method for the fastening of ropes and other linear materials are widely appearing in practical applications ­ in sailing, in surgery, in textile and rope structures etc. The mechanics of knots, however, appears to be not sufficiently covered neither by analytical methods, nor by computational methods. From a computational mechanics point of view a knot is a perfect example requiring both a robust smooth cable element and a robust curve-to-curve contact algorithm. The current contribution is aimed on the development this combination ­ the isogeometric approach for curvilinear beams and the robust curve-to-curve contact algorithm for curvilinear cables. The developed model is applied studying the mechanics of various knots. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A curve flow evolved by a fourth order parabolic equation

We study a fourth order curve flow, which is the gradient flow of a functional describing the shapes of human red blood cells. We prove that for any smooth closed initial curve in ℝ², the flow has a smooth solution for all time and the solution subconverges to a critical point of the functional.     

Global nonlinear stability of convection in a heat generating fluid filled channel with a moving boundary

The energy method is employed to investigate the stability of a steady convective flow in a heat generating fluid arising due to the combined effect of buoyancy, shear and pressure gradient. By introducing a suitable generalized energy functional and using energy inequalities sufficient conditions for the existence of such a flow are found. An analysis through the variational principles is then made to find sharper limits for nonlinear stability. Comparisons are made with linear results in the literature and it is shown that the linear theory fails to capture the physics of the onset of secondary flow.     

积分算子方程的连续型配置方法

1．引言由于对积分算子方程来说，配置法比Galerkin法具计算量小的优点（少算一重积分），故配置法更受人们重视．但已有的文献几乎都是将配置空间取作非连续的分片多项式样条空间，以得到某种超收敛结果（如［1，2］）．这种方法存在下列不足：（a）光滑核Volterra积分方程与光滑核Fredholm积分方程具完全不同的收敛性质［1］，且需用不同的方法获得其加速收敛结果（比较［31与［4］），尽管Volterra积分方程在理论上被看作是Fredholm积分方程的特殊情形；（b）光滑核Volterra积分方程的配置解不具任何超收敛性，其迭代配置解也只在结点…     

Coverage Maximization with Autonomous Agents in Fast Flow Environments

This work examines the cooperative motion of a group of autonomous vehicles in a fast flow environment. The magnitude of the flow velocity is assumed to be greater than the available actuation to each agent. Collectively, the agents wish to maximize total coverage area defined as the set of points reachable by any agent within T time. The reachable set of an agent in a fast flow is characterized using optimal control techniques. Specifically, this work addresses the complementary cases where the static flow field is smooth, and where the flow field is piecewise constant. The latter case arises as a proposed approximation of a smooth flow that remains analytically tractable. Furthermore, the techniques used in the piecewise constant flow case enable treatment for obstacles in the environment. In both cases, a gradient ascent method is derived to maximize the total coverage area in a distributed fashion. Simulations show that such a network is able to maximize the coverage area in a fast flow.     

Least squares approximation by splines with free knots

Suppose we are given noisy data which are considered to be perturbed values of a smooth, univariate function. In order to approximate these data in the least squares sense, a linear combination of B-splines is used where the tradeoff between smoothness and closeness of the fit is controlled by a smoothing term which regularizes the least squares problem and guarantees unique solvability independent of the position of knots. Moreover, a subset of the knot sequence which defines the B-splines, the so-calledfree knots, is included in the optimization process.The resulting constrained least squares problem which is linear in the spline coefficients but nonlinear in the free knots is reduced to a problem that has only the free knots as variables. The reduced problem is solved by a generalized Gauss-Newton method. The method developed can be combined with a knot removal strategy in order to obtain an approximating spline with as few parameters as possible.Dedicated to Professor Dr.-Ing. habil. Dr. h.c. Helmut Heinrich on the occasion of his 90th birthdayResearch of the second author was partly supported by Deutsche Forschungsgemeinschaft under grant Schm 968/2-1.     

Penalized Estimation of Free-Knot Splines

Abstract

Polynomial splines are often used in statistical regression models for smooth response functions. When the number and location of the knots are optimized, the approximating power of the spline is improved and the model is nonparametric with locally determined smoothness. However, finding the optimal knot locations is an historically difficult problem. We present a new estimation approach that improves computational properties by penalizing coalescing knots. The resulting estimator is easier to compute than the unpenalized estimates of knot positions, eliminates unnecessary “corners” in the fitted curve, and in simulation studies, shows no increase in the loss. A number of GCV and AIC type criteria for choosing the number of knots are evaluated via simulation.     

Rational-Slice Knots via Strongly Negative-Amphicheiral Knots

We show that certain satellite knots of every strongly negative-amphicheiral rational knot are rational-slice knots. This proof also shows that the 0-surgery manifold of a certain strongly negative amphicheiral knot such as the figure-eight knot bounds a compact oriented smooth 4-manifold homotopy equivalent to the 2-sphere such that a second homology class of the 4-manifold is represented by a smoothly embedded 2-sphere if and only if the modulo two reduction of it is zero.     

Stability analysis of a difference scheme for the Navier-Stokes equations

An application of stability analysis by the energy method is made to a practical problem of unsteady viscous flow solved numerically by Fromm. The practical stability criterion for the scheme is determined and a rigorous proof of convergence to a smooth solution is given. It is also shown how to construct an energy for any 3-level scalar difference equation.     

Comparative Analysis of Cubic Spline and Kernel Estimation of a Probit Function

The least-squares cubic spline and the kernel estimators produce comparable mean squared errors, although the kernel produces smaller mean squared errors when the variable increases away from 0. Mean squared error increases with an increase in the number of knots (for the cubic spline) or reduced band width (for the kernel estimator). The cubic spline produces smaller mean squared errors when all observations are made at knots than when they are spaced out between knots. Irrespective of the exact form of the probit function g(x), the cubic spline estimator is asymptotically unbiased, while the kernel estimator only converges to g(x) under certain conditions. Moreover, the cubic spline is a smooth function, which is twice differentiable on the interval [0,1].     

Intersection Alexander polynomials

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander polynomials that are defined for non-locally-flat knots. We also investigate the relations between the intersection Alexander polynomials of a knot and the intersection and classical Alexander polynomials of the link knots around the singular strata. To facilitate some of these investigations, we introduce spectral sequences for the computation of the intersection homology of certain stratified bundles.     

The gradient flow motion of boundary vortices

We consider the gradient flow of a family of energy functionals describing the formation of boundary vortices in thin magnetic films. We obtain motion laws for the singularities in all time scalings by using the method of Γ-convergence of gradient flows.     

Rational-slice Knots via Strongly Negative-amphicheiral Knots

We show that certain satellite knots of every strongly negative-amphicheiral rational knot are rational-slice knots. This proof also shows that the O-surgery manifold of a certain strongly negative amphicheiral knot such as the figure-eight knot bounds a compact oriented smooth 4-manifold homotopy equivalent to the 2-sphere such that a second homology class of the 4-manifold is represented by a smoothly embedded 2-sphere if and only if the modulo two reduction of it is zero.     

Uniform approximation of curvature for smooth classes of plane curves

We consider problems of approximating the curvature of plane curves from smooth classes by the curvature of elements of smooth finite-dimensional function spaces (trigonometric polynomials, splines with equidistant knots) in the uniform norm.     

Regularity of minimizing extrinsic polyharmonic maps in the critical dimension

Regularity of extrinsic polyharmonic maps is proved with a method relying on the gradient flow for the underlying functional. This technique permits more general functionals than studied previously, but is restricted to energy minimizers.     

Divergence-L q and divergence-measure tensor fields and gradient flows for linear growth functionals of maps into the unit sphere

Divergence-L ^q and divergence-measure tensor fields are introduced and their properties are analyzed. These measure-theoretic tools are then used to study the gradient flow of linear growth functionals of maps into the unit sphere. A BV-solution concept is introduced and the existence of such a solution is established for the flow using the energy method together with a regularization and a penalization technique. Based on the analytical results, practical fully discrete finite element methods are then proposed for computing weak solutions of the gradient flow, and the convergence of the proposed numerical method is also proved.