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We consider nonlinear, nonhomogeneous elliptic Dirichlet equations driven by the sum of a p‐Laplacian and a Laplacian (so‐called (p, 2)‐equation). We are concerned with both cases 1 2. In the first one, the reaction f(z,x) is linear grow near ±∞ and resonant with respect to a nonprincipal nonnegative eigenvalue. In the second case, the reaction f(z,·) is (p−1)‐superlinear near ±∞ and has z‐dependent zeros of constant sign. Using variational methods together with flow invariance arguments, we establish the existence of nodal solutions.

The bending of bilayer plates is a mechanism that allows for large deformations via small externally induced lattice mismatches of the underlying materials. Its mathematical modeling, discussed herein, consists of a nonlinear fourth-order problem with a pointwise isometry constraint. A discretization based on Kirchhoff quadrilaterals is devised and its Γ-convergence is proved. An iterative method that decreases the energy is proposed, and its convergence to stationary configurations is investigated. Its performance, as well as reduced model capabilities, are explored via several insightful numerical experiments involving large (geometrically nonlinear) deformations.© 2015 Wiley Periodicals, Inc.

Summary This paper is concerned with the optimal time-weighted H2 model reduction problem for discrete Markovian jump linear systems (MJLSs). The purpose is to find a mean square stable MJLS of lower order such that the time-weighted H2 norm of the corresponding error system is minimized for a given mean square stable discrete MJLSs. The notation of time-weighted H2 norm of discrete MJLS is defined for the first time, and then a computational formula of this norm is given, which requires the solution of two sets of recursive discrete Markovian jump Lyapunov-type linear matrix equations. Based on the time-weighted H2 norm formula, we propose a gradient flow method to solve the optimal time-weighted H2 model reduction problem. A necessary condition for minimality is derived, which generalizes the standard result for systems when Markov jumps and the time-weighting term do not appear. Finally, numerical examples are used to illustrate the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.

We present variational approaches (developed in [3,4,11]) to the study of statics and dynamics of screw dislocations in crystals. We model the crystal as a cubic lattice and we give the asymptotic Γ-convergence expansion of the elastic energy induced by a finite family of screw dislocations as the lattice spacing goes to zero. We show that the effective energy associated to the presence of a finite system of screw dislocations coincides with the renormalized energy, studied within the Ginzburg-Landau framework and ruling the interactions between the dislocations. As a byproduct of this analysis, we show the existence of many metastable configurations of dislocations pinned by energy barries. Using the minimizing movement approach a la De Giorgi, we introduce a discrete-in-time variational dynamics, referred to as Discrete Gradient Flow, which allows to overcome these energy barriers. More precisely, we show that lettting first the lattice spacing and then the time step of minimizing movements tend to zero, dislocations move accordingly with the gradient flow of the renormalized energy. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Recently, dissipation elements have been gaining popularity as a mechanism for measurement of fundamental properties of turbulent flow, such as turbulence length scales and zonal partitioning. Dissipation elements segment a domain according to the source and destination of streamlines in the gradient flow field of a scalar function f: M → R. They have traditionally been computed by numerically integrating streamlines from the center of each voxel in the positive and negative gradient directions, and grouping those voxels whose streamlines terminate at the same extremal pair. We show that the same structures map well to combinatorial topology concepts developed recently in the visualization community. Namely, dissipation elements correspond to sets of cells of the Morse-Smale complex. The topology-based formulation enables a more exploratory analysis of the nature of dissipation elements, in particular, in understanding their stability with respect to small scale variations. We present two examples from combustion science that raise significant questions about the role of small scale perturbation and indeed the definition of dissipation elements themselves.

In this paper, we study the numerical approximations of a gradient flow associated with a phase field bending elasticity model of a vesicle membrane with prescribed volume and surface area. A spatially semi-discrete scheme based on a mixed finite element formulation and a fully discrete in space and time scheme are analyzed. Optimal order error estimates are rigorously derived for these numerical schemes without any a priori assumption. Copyright © 2013 John Wiley & Sons, Ltd.

A gradient flow-based explicit finite element method (L2GF) for reconstructing the 3D density function from a set of 2D electron micrographs has been proposed in recently published papers. The experimental results showed that the proposed method was superior to the other classical algorithms, especially for the highly noisy data. However, convergence analysis of the L2GF method has not been conducted. In this paper, we present a complete analysis on the convergence of L2GF method for the case of using a more general form regularization term, which includes the Tikhonov-type regularizer and modified or smoothed total variation regularizer as two special cases. We further prove that the L2-gradient flow method is stable and robust. These results demonstrate that the iterative variational reconstruction method derived from the L2-gradient flow approach is mathematically sound and effective and has desirable properties. Copyright © 2013 John Wiley & Sons, Ltd.

The linearized stability of stationary solutions for surface diffusion is studied. We consider hypersurfaces that lie inside a fixed domain, touch its boundary with a right angle and fulfill a no-flux condition. We formulate the geometric evolution law as a partial differential equation with the help of a parametrization from Vogel [Vog00], which takes care of a possible curved boundary. For the linearized stability analysis we identify as in the work of Garcke, Ito and Kohsaka [GIK05] the problem as an H

Phase contrast velocimetry can be utilized to measure complex flow for both quantitative and qualitative assessment of vascular hemodynamics. However, phase contrast requires that a maximum measurable velocity be set that balances noise and phase aliasing. To efficiently reduce noise in phase contrast images, several investigators have proposed extended velocity encoding schemes that use extra encodings to unwrap phase aliasing; however, existing techniques can lead to significant increases in echo and scan time, limiting their clinical benefits. In this work, we have developed a novel five-point velocity encoding scheme that efficiently reduces noise with minimal increases in scan and echo time. Investigations were performed in phantoms, demonstrating a 63% increase in velocity-to-noise ratio compared to standard four-point encoding schemes. Aortic velocity measurements were performed in healthy volunteers, showing similar velocity-to-noise ratio improvements. In those volunteers, it was also demonstrated that, without sacrificing accuracy, low-resolution images can be used for the fifth encoding point, reducing the scan time penalty from 25% down to less than 1%.

In this article, we propose to solve numerically the problem of finding the smallest eigenvalues of a Hermitian operator (and the space spanned by the corresponding eigenvectors) by a gradient flow technique. This method is then applied to the Hartree-Fock problem. Improvements are also proposed in two directions: preconditioning of the dynamical system and development of a specific flow that enables to compute directly the eigenvectors. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

We study an evolution equation proposed by Chapman, Rubinstein, and Schatzman as a mean-field model for the evolution of the vortex density in a superconductor. We treat the case of a bounded domain where vortices can exit or enter the domain. We show that the equation can be derived rigorously as the gradient flow of some specific energy for the Riemannian structure induced by the Wasserstein distance on probability measures. This leads us to some existence and uniqueness results and energy-dissipation identities. We also exhibit some “entropies” that decrease through the flow and allow us to get regularity results (solutions starting in Lp, p > 1, remain in Lp). © 2007 Wiley Periodicals, Inc.

The J-flow of S. K. Donaldson and X. X. Chen is a parabolic flow on Kahler manifolds with two Kahler metrics. It is the gradient flow of the J-functional that appears in Chen's formula for the Mabuchi energy. We find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the J-flow to a critical metric. We use this result to show that on manifolds with ample canonical bundle, the Mabuchi energy is proper on all Kahler classes in an open neighborhood of the canonical class defined by a positivity condition. This improves previous results of Chen and of the second author. We discuss the implications of this for the problem of the existence of constant-scalar-curvature Kahler metrics. We also study the singularities of the J-flow and, under certain conditions (which always hold for dimension 2) derive some estimates away from a subvariety. We discuss the conjectural remark of Donaldson that if the J-flow does not converge on a Kahler surface, then it should blow up over some curves of negative self-intersection. © 2007 Wiley Periodicals, Inc.

A new scheme of shape optimization is applied to the design of a flow guide in flat-die extrusion processes. In general, tremendous time is inevitably required for the optimization of large-scale three-dimensional extrusion processes. This is because the finite element analysis requires large computation time owing to the complexity of the die geometry and flow behaviour. The proposed scheme effectively reduces the computation time for the optimization process by approximating the objective function. This is achieved by introducing a transformed equation of the state variables. The scheme is then applied to the practical extrusion processes to produce '1', 'H' and 'L' sections. The objective of the optimization is to make a balanced flow of the material in the exit region. Control points of a Bezier curve describing the outline of the flow guide are regarded as the design variables. Through application to large-scale problems, the effectiveness and usefulness of the proposed scheme is demonstrated.

We study the action minimization problem which is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharp-interface limit is related to (but di erent from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization problem, we find distinct “most likely switching pathways,” depending on the relative costs of nucleation and propagation of interfaces. This competition is captured by the limit of the action functional, which we derive formally and propose as the natural candidate for the -limit. Guided by the reduced functional, we prove upper and lower bounds for the minimal action which agree on the level of scaling. (This is a preprint of an article accepted for publication in Comm. Pure App. Math, October 2005.)

Model reduction problem was investigated for singular systems. To solve the problem, the covariance for singular systems was defined. Then, a model reduction method based on covariance approximation was presented for obtaining stable and impulse controllable models for singular systems. Thirdly, the error criterion was explicitly derived via a free parameter and the optimization procedure was presented in terms of gradient flow. Finally, illustrative examples were given to show the effectiveness of the proposed approach.

We present a method to prove convergence of gradient flows of families of energies that Γ-converge to a limiting energy. It provides lower-bound criteria to obtain the convergence that correspond to a sort of C1-order Γ-convergence of functionals. We then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the Ginzburg-Landau energy in dimension 2, retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field. © 2004 Wiley Periodicals, Inc.

In a non-compact context the first natural step in the search for periodic orbits of a hamiltonian flow is to detect bounded ones. In this paper we show that, in a non-compact setting, certain algebraic topological constraints imposed to a gradient flow of the hamiltonian function $f$ imply the existence of bounded orbits for the hamiltonian flow of $f$. Once the existence of bounded orbits is established, under favorable circumstances, application of the $C^{1}$-closing lemma leads to periodic ones., Comment: 54 pages

This paper discusses the balanced circulation polytope, that is, the convex hull of balanced circulations of a given balanced flow network. The LP description of this polytope is the LP description of ordinary circulations plus some odd-set constraints. The paper starts with an exposition of several classes of odd-set inequalities. These inequalities are described in terms of balanced network flows as well as matchings and put into relation to each other. Step by step, the problem of finding a cost minimum balanced circulation can be reduced to the b-matching problem. We present an LP characterization of the b-matching polytope by blossom inequalities. With a moderate effort, these odd sets are lifted to the setting of balanced-network flows. We finish with the dualization of the derived LP formulation, an introduction of reduced-cost labels, and a corresponding optimality condition. © 2001 John Wiley & Sons, Inc.

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous Hessian equations. To achieve this we derive uniform, first- and second-order a priori estimates for the elliptic and parabolic Hessian equations. Our results generalize well-known results for semilinear elliptic equations and the Monge-Ampere equation. c 2001 John Wiley & Sons, Inc.

Time-discrete variational schemes are introduced for both the Vlasov-Poisson-Fokker-Planck (VPFP) system and a natural regularization of the VPFP system. The time step in these variational schemes is governed by a certain Kantorovich functional (or scaled Wasserstein metric). The discrete variational schemes may be regarded as discretized versions of a gradient flow, or steepest descent, of the underlying free energy functionals for these systems. For the regularized VPFP system, convergence of the variational scheme is rigorously established.

The problem of global optimal static output feedback control for linear time-invariant systems with linear quadratic index is investigated. The contributions of this paper are two-fold. One is to investigate the dependence of the global optimal output feedback gain on the system initial conditions. The other is to construct a globally optimal feedback under a certain output measurement structure. Copyright © 2000 John Wiley & Sons, Ltd.

This paper deals with the problem of model reduction based on an optimization technique. The objective function being minimized in the impulse energy of the overall system with unity, single-sided and double-sided weightings. A number of properties of the gradient flows associated with the objective function are obtained. Two examples are presented to illustrate the effectiveness of the proposed method, and results are compared with unweighted and weighted balanced truncation methods. © 1998 John Wiley & Sons, Ltd.

This paper provides a computational procedure for the robust partial pole-placement problem. The algorithm is based on a gradient flow formulation on a differentiable potential function which provides a minimizing solution for the Frobenius condition number of the closed-loop state matrix. The algorithm faces no singularity problem with the resulting eigenvector matrix. Convergence properties of the algorithm are discussed. A numerical example is employed to illustrate the technique and comparison to other existing methods is made.

In this paper we consider LQ cost optimization for the simultaneous stabilization problem. The objective is to find a single simultaneously stabilizing feedback gain matrix such that all closed-loop systems exhibit good transient behaviour. The cost function used is a quadratic function of the system states and the control vector. This paper proposes to seek an optimization solution by solving an ordinary differential equation which is a gradient flow associated with the cost function. Two examples are presented to illustrate the effectiveness of the proposed procedure.

Balanced equations of motion based on potential vorticity evolution and inversion for the shallow water and stratified primitive equations are derived and, in some shallow-water cases, numerically tested. The schemes are based on asymptotic expansions in Rossby or Froude number, or rational scaling-based truncations of the equations of motion, assuming that the dynamics are determined by the advection of potential vorticity. Thus, regimes of validity are rapidly rotating and/or highly stratified flow. Both new and familiar results are straightforwardly obtained, in a unified framework in both height and isentropic coordinates. For both shallow-water and stratified equations, Rossby number expansions schemes give quasi-geostrophy at lowest order. Both gradient-wind balance and the nonlinear terms in the potential vorticity enter at next order. A low Froude number expansion for non-rotating flow gives two-dimensional flow, uncoupled in the vertical at lowest order. A single consistent inversion scheme can be derived that is valid at lowest order in Froude number for all Rossby numbers, for both shallow-water and the stratified equations. It may be a particularly appropriate model for the atmospheric mesoscale and oceanic submesoscale, where rotation and stratification can both be important in defining balanced motion. A model is also proposed that is valid at both planetary and synoptic scales, combining the familiar planetary geostrophic and quasi-geostrophic equations. Most of the models derived require the solution only of linear or near linear elliptic equations, possibly with varying coefficients. Numerical experiments indicate that a higher-order inversion can be quantitatively better than quasigeostrophy, if Rossby number and divergence are sufficiently small. In some other cases, no noticeable improvement over quasi-geostrophy is found, even when the Rossby number is quite small. However, the balanced model valid for both planetary and synoptic scales shows a significant qualitative and quantitative improvement over both planetary geostrophy and quasi-geostrophy for large-scale flows, and its evolution is in good agreement with a primitive equation model.

International audience; This paper deals with a class of macroscopic models for cell migration in a saturated medium for two-species mixtures. Those species tend to achieve some motion according to a desired velocity, and congestion forces them to adapt their velocity. This adaptation is modelled by a correction velocity which is chosen minimal in a least-square sense. We are especially interested in two situations: a single active species moves in a passive matrix (cell migration) with a given desired velocity, and a closed-loop Keller-Segel type model, where the desired velocity is the gradient of a self-emitted chemoattractant. We propose a theoretical framework for the open-loop model (desired velocities are defined as gradients of given functions) based on a formulation in the form of a gradient flow in the Wasserstein space. We propose a numerical strategy to discretize the model, and illustrate its behaviour in the case of a prescribed velocity, and for the saturated Keller-Segel model.