Interior Gradient Research Articles

Interior Gradient and Epsilon-Subgradient Descent Methods for Constrained Convex Minimization

We extend epsilon-subgradient descent methods for unconstrained nonsmooth convex minimization to constrained problems over polyhedral sets, in particular over ℝp+. This is done by replacing the usual squared quadratic regularization term used in subgradient schemes by the logarithmic-quadratic distancelike function recently introduced by the authors. We then obtain interior ε-subgradient descent methods, which allow us to provide a natural extension of bundle methods and Polyak's subgradient projection methods for nonsmooth convex minimization. Furthermore, similar extensions are considered for smooth constrained minimization to produce interior gradient descent methods. Global convergence as well as an improved global efficiency estimate are established for these methods within a unifying principle and minimal assumptions on the problem's data.

Global boundedness, interior gradient estimates, and boundaryregularity for the mean curvature equation with boundaryconditions

We obtain global estimates for the modulus, interior gradient estimates, and boundary Hölder continuity estimates for solutions u to the capillarity problem and to the Dirichlet problem for the mean curvature equation merely in terms of the mean curvature, together with the boundary contact angle in the capillarity problem and the boundary values in the Dirichlet problem.

Vegetation and microclimatic edge effects in two mixed-mesophytic forest fragments

Forest edges are known to consist of microenvironments that may provide habitat for a different suite of species than forest interiors. Several abiotic attributes of the microenvironment may contribute to this change across the edge to center gradient (e.g., light, air temperature, soil moisture, humidity). Biotic components, such as seed dispersal, may also give rise to changes in species composition from forest edge to interior. We predicted that abiotic and biotic measures would correlate with distance from forest edge and would differ among aspects. To test these predictions, we measured abiotic and biotic variables on twelve 175 m transects in each of two 24 ha forest fragments in east-central Illinois that have remained in continuous isolation for upwards of 100 years. Both univariate and multivariate techniques were used to best describe the complex relationships among abiotic factors and between abiotic and biotic factors. Results indicate that microclimatic variables differ in the degree to and distance over which they show an edge effect. Relative humidity shows the widest edge, while light and soil moisture have the steepest gradients. Aspect influences are evidenced by the existence of more pronounced edge effects on south and west edges, except when these edges are protected by adjacent habitat. Edges bordered by agricultural fields have more extreme changes in microclimate than those bordered by trees. According to PCA results, species richness correlates well with microclimatic variation, especially light and soil moisture; however, in many cases species richness had a different depth of edge influence than either of these variables. The herbaceous plant community is heavily dominated by three species. Distributions of individual species as well as changes in plant community composition, estimated with a similarity index, indicate that competition may be influencing the response of the vegetation to the edge to interior gradient. This study indicates that edge effects must be considered when the size and potential buffering habitat of forest preserves are planned.

Interior gradient estimates for mean curvature equations

The interior gradient estimate for the prescribed mean curvature equation has been extensively studied, see [9] and the references therein. For high order mean curvature equations it has also been obtained in [11, 18]. In most articles such estimates were obtained by carrying out analysis on the graphs of solutions and so the arguments depend on the invariance of the equations under rigid motion. From the view point of partial differential equations such estimate should hold for equations with similar structural conditions. In [13, 6] the gradient estimate was obtained for certain fully nonlinear elliptic equations under various conditions. Different proofs for the gradient estimate for mean curvature equations have been given in [1, 3]. In this note we show that for curvature and Hessian equations the interior gradient estimate can be obtained very easily. Our proof, which is based on suitable choice of auxiliary functions, is elementary and avoids geometric computations on the graph of solutions. The technique in this note has actually been widely used in literature, see, e.g., [9].

Vegetation responses along edge-to-interior gradients in a high altitude tropical forest in peninsular India

A study was conducted in a high altitude tropical forest (shola forest) in peninsular India to examine the changes in floristic composition (mainly woody species regeneration) along an edge to interior gradient in relation to changes in edaphic and microenvironmental factors. Species inventory was taken in 25 m 2 plots, established at 10 m intervals along edge to interior transects. The measured soil variables included pH, organic carbon, total nitrogen, available phosphorus, and soil moisture. Microenvironmental factors including light transmittance, relative humidity, and air and soil temperatures also were monitored. Both edaphic and microenvironmental factors exhibited strong patterns along the edge to interior gradient. Forest edges were characterized by higher light transmittance, higher air and soil temperatures, and lower relative humidity. Soil variables including pH, organic carbon, total nitrogen, available phosphorus, and moisture increased toward the forest interior. Significant increases in organic carbon (53.9%), total nitrogen (47%), and soil moisture (55%) indicated a relatively fertile forest interior compared with the forest edge. A definite floristic compositional pattern also was observed along the edge to interior gradient which was correlated to the edaphic and microenvironmental variables as revealed by canonical correspondence analysis. It appears that edge effects in these high altitude forests penetrate to a distance of 15–30 m. Further, edaphic factors have an important influence on woody species regeneration, perhaps much more than microenvironmental factors. This indicates that any disturbance that significantly exposes theforest floor, thus lowering soil moisture and altering soil nutrient status, can adversely affect the regeneration of many of the shola species.

Interior gradient blow-up in a semilinear parabolic equation

We present a one-dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions we show that gradient blow-up occurs at a single point, and we study the shape of the singularity. Our argument for gradient blow-up also provides a pair of "naive viscosity sub- and super-solutions" which violate the comparison principle.

Multiplicative Interior Gradient Methods for Minimization over the Nonnegative Orthant

We introduce a new class of multiplicative iterative methods for solving minimization problems over the nonnegative orthant. The algorithm is akin to a natural extension of gradient methods for unconstrained minimization problems to the case of nonnegativity constraints, with the special feature that it generates a sequence of iterates which remain in the interior of the nonnegative orthant. We prove that the algorithm combined with an appropriate line search is weakly convergent to a saddle point of the minimization problem, when the minimand is a differentiable function with bounded level sets. If the function is convex, then weak convergence to an optimal solution is obtained. Moreover, by using an appropriate regularized line search, we prove that the level set boundeness hypothesis can be removed, and full convergence of the iterates to an optimal solution is established in the convex case.

Optimal Growth Rates in the Quasigeostrophic Initial Value Problem

A large class of wave structures in quasigeostrophic flow have instantaneous growth rates significantly larger than normal-mode growth rates. Since energy and potential enstrophy growth rates can be defined as functions of the perturbation structure, this structure can be varied in order to maximize the growth rates. Green's model is used as an illustration of the method. The only essential constraint in the inviscid case is the conservation of wave action. When friction is included, an appropriate modification of this constraint can be found. The problem reduces to a fourth-order nonlinear ordinary differential equation for the perturbation structure function. An analytic solution is obtained for the continuous inviscid problem with no interior potential vorticity gradient (i.e., β = 0) and yields structures that include the classical Eady wave as a special case. For the more general problem, the equation is discretized and solved numerically.A general feature of the fastest-growing baroclinic waves is that their phase tilts are more uniform with height than normal modes. As the waves evolve toward (unstable) normal-mode structures, the phase tilts concentrate at lower levels and the potential vorticity fluxes become shallower. There is no long-wave or short-wave cutoff in the initial value problem, even though the corresponding normal-mode instability may have long-wave or short-wave cutoffs or both. Indeed, the largest instantaneous growth rates in both the energy and potential enstrophy norms are obtained for the longest waves.While the variational principle requires no information about normal modes in order to find the fastest-growing structures, the principle may also be used to obtain normal modes as particular cases.

Interior gradient estimates for solutions of certain fully nonlinear elliptic equations

Korevaar has given a new proof for interior gradients estimates for solutions of prescribed mean curvature equations in [ N. J. Korevaar, Proc. Sympos. Pure Math. 45, Part 2 (1986) ]. Interior gradient estimates for elliptic Weingarten equations are also obtained in [ N. J. Korevaar, Ann. Inst. H. Poincaré 43 (1987) , 1–17]. In this paper we apply the normal perturbation method introduced in [ N. J. Korevaar, Proc. Sympos. Pure Math. 45, Part 2 (1986) ; N. J. Korevaar, Ann. Inst. H. Poincaré 43 (1987) , 1–17] to generalize the result in [ N. J. Korevaar, Ann. Inst. H. Poincaré 43 (1987) , 1–17] to more general fully nonlinear elliptic equations.

Interior gradient bounds for the mean curvature equation by viscosity solutions methods

Interior gradient bounds for the mean curvature equation by viscosity solutions methods

A priori interior gradient bounds for solutions to elliptic Weingarten equations

In this paper a maximum principle approach is used to derive a priori interior gradient bounds for smooth solutions to the Weingarten equationsf(λ)=∑i1<i2<…<ikλi1…λik=ψ(x,u,v).Here λ = (λ1, …, λn) is the vector of principal curvatures of the graph of u at a point (x, u(x)) on the graph, with downward normal v. One requires a one-sided height bound (u < 0), natural structure conditions on the prescribed function ψ, and the restriction that all λ lie in a certain cone of eigenvalues for which f is elliptic. The result generalizes what is known to be true for the prescribed mean curvature equation (k = 1).

An interior gradient bound for solutions of equations of capillary type

An interior gradient bound for solutions of equations of capillary type

Quasilinear elliptic equations with small boundary data

It is shown that if the order of non-uniformity of a quasilinear elliptic equation is h, 1<h≤2, then the critical norm separating existence and non-existence of a solution to the Dirichlet problem with small boundary data is the\(C^{o,\frac{{2(h - 1)}}{h}}\) norm. For 0≤h≤1, existence of a solution is guaranteed without any smallness assumption on the given boundary data, provided that the usual a priori interior gradient bound for solution is available.

The temperature and moisture content in lumber during preheating and drying

The time required to thoroughly preheat/presteam green 8/4 (51 mm) thick red pine and white birch lumber was determined. Thermocouples imbedded in the centre of samples were used to determine presteaming/preheating times for experimental material. These data were then compared to time determined by analytical methods (using formulae developed by Kollmann). No significant differences were found. During the subsequent drying period data were collected on interior wood temperature, mousture content and gradient. Results of the drying tests indicated: i) interior temperature increases in direct relation to elapsed time, ii) moisture content drops exponentially with elapsed time, and iii) an exponential relationship exists between average moisture content and the core temperature of the lumber. Results indicate that monitoring of the lumber drying process through measurements of interior wood temperature may be feasible.

External Rossby Modes on a β–Plane with Realistic Vertical Wind Shear

Abstract Free oscillations of small amplitude and low frequency in an isothermal atmosphere at rest have the vertical structure of an edge wave, energy density decaying exponentially away from the ground. In this paper we treat these external Rossby modes on a middle latitude β-plane to see how their vertical structure and phase speed are changed by the presence of a winter zonal wind with realistic vertical structure. We find that for planetary-scale modes the basic state interior potential vorticity gradient and the temperature gradient at the lower boundary produce roughly equal but opposite changes in phase speed. Energy density decays less rapidly with altitude when the basic state temperature gradient is included in the lower boundary condition. As wavenumber is increased, the phase speed of the external Rossby mode is found to approach the speed of the zonal wind at the surface, and there is a limiting wavenumber beyond which the mode does not exist. At this limiting value of wavenumber, the mode b...

The Generalized Inverse in Linear Programming—Interior Gradient Projection Methods

Constructive realizations of necessary and sufficient conditions for optimality in linear programming yield sequences of points, interior to the convex set of feasible solutions, converging to optimal solutions.

A New Proof of the Interior Gradient Bound for the Minimal Surface Equation in n Dimensions

An interior gradient bound for classical solutions of the minimal surface equation in n variables was established by Bombieri, De Giorgi, and Miranda in 1968. We provide a new and simpler derivation of this estimate and partly develop in the process some new techniques applicable to the study of hypersurfaces in general.

Interior gradient estimates for surface $z=f(x,\,y)$ with prescribed mean curvature

Interior gradient estimates for surface $z=f(x,\,y)$ with prescribed mean curvature

Representations for the Generalized Inverse of a Partitioned Matrix

Representations for the Generalized Inverse of a Partitioned Matrix