Local Gradient Estimates Research Articles

Some Gradient Estimates for Nonlinear Heat-Type Equations on Smooth Metric Measure Spaces with Compact Boundary

In this paper we prove some Hamilton type and Li–Yau type gradient estimates on positive solutions to generalized nonlinear parabolic equations on smooth metric measure space with compact boundary. The geometry of the space in terms of lower bounds on the weighted Bakry–Émery Ricci curvature tensor and weighted mean curvature of the boundary are key in proving generalized local and global gradient estimates. Various applications of these gradient estimates in terms of parabolic Harnack inequalities and Liouville type results are discussed. Further consequences to some specific models informed by the nature of the nonlinearities are highlighted.

A Particle Based Approach For Improved Resolution PIV And TOMO-PIV Based On The Coherent Point Drift And The Affine Least-Squares Transformation

Particle Image Velocimetry (PIV) is a widely used method for flow diagnostics, but there is still potential for improvement, particularly in terms of velocity gradient estimation and computational cost when considering three-dimensional problems. This paper presents a framework that combines a Particle Tracking Velocity (PTV) approach with local gradient-based Eulerian reconstruction to improve PIV performance. The approach uses the Coherent Point Drift (CPD) method for particle pairing and introduces the Affine Least-Squares Transformation (ALST) for local deformation gradient estimation. The CPD method consists of pairing particles whose positions at two successive instants have been obtained from the images. The ALST estimates local deformation gradients, allowing the Eularian reconstruction of the velocity field. The effectiveness of the proposed method compared to traditional PIV algorithms is demonstrated by synthetic test cases in both 2D and 3D configurations. In 2D cases, the CPD+ALST approach outperforms standard PIV methods, especially in capturing local velocity gradients. In 3D cases, comparisons with TOMO-PIV show the improved performance of CPD+ALST in the context of locally large velocity gradients. However, the linear nature of ALST makes it less efficient than quadratic binning techniques. The study demonstrates the potential of CPD+ALST to improve velocity field reconstruction in complex flows.

Local gradient estimates for a type of fully nonlinear equations

Local gradient estimates for a type of fully nonlinear equations

Local Hamilton type Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds

Local Hamilton type Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds

Elliptic gradient estimates for a nonlinear equation with Dirichlet boundary condition

We prove local elliptic type gradient estimates for a nonlinear parabolic equation (including the fast diffusion equation and the porous medium equation) with Dirichlet boundary condition on Riemannian manifolds with compact boundary when the Ricci curvature and the mean curvature of boundary are bounded below. As an application, we obtain Liouville type results for certain ancient solutions to the nonlinear parabolic equation on Riemannian manifolds with compact boundary.

Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications

AbstractThis article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville-type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed.

Gradient estimates for a nonlinear elliptic equation on a smooth metric measure space

Let (M, g, e −f dv) be a smooth metric measure space. We consider local gradient estimates for positive solutions to the following elliptic equation ∆ f u + au log u + bu = 0 where a, b are two real constants and f be a smooth function defined on M. As an application, we obtain a Liouville type result for such equation in the case a < 0 under the m-dimensions Bakry-Émery Ricci curvature.

Central difference information filtering phase unwrapping algorithm based on deep learning

This paper presents a deep learning-based central difference information filtering phase unwrapping algorithm. First, the central difference information filter (CDIF) is introduced into phase unwrapping for wrapped phase images for the first time to our knowledge, and is combined with an efficient local phase gradient estimation technique, and an efficient path-following strategy based on heap sorting to construct a CDIF phase unwrapping procedure. Second, a region segmentation model based on deep learning is constructed to divide the wrapped phase images into a series of segmented regions along the edge of interferometric fringes, which is beneficial to improve the accuracy and efficiency of subsequent phase unwrapping. Finally, each segmented region of the wrapped phase images is unwrapped separately using the CDIF phase unwrapping procedure, and the unwrapping results of each segmented region are combined according to the phase consistency criterion of adjacent regions of the wrapped phase images to obtain the unwrapped phase of the whole wrapped phase images. Results delivered by the experiments on synthetic data and experimental data demonstrate the robustness and acceptable efficiency of the proposed method in phase unwrapping for the wrapped phase images, compared with some commonly used algorithms.

An Edge Detection Method Based on Local Gradient Estimation: Application to High-Temperature Metallic Droplet Images

Edge detection is a fundamental step in many computer vision systems, particularly in image segmentation and feature detection. There are a lot of algorithms for detecting edges of objects in images. This paper proposes a method based on local gradient estimation to detect metallic droplet image edges and compare the results to a contour line obtained from the active contour model of the same images, and to results from crowdsourcing to identify droplet edges at specific points. The studied images were taken at high temperatures, which makes the segmentation process particularly difficult. The comparison between the three methods shows that the proposed method is more accurate than the active contour method, especially at the point of contact between the droplet and the base. It is also shown that the reliability of the data from the crowdsourcing is as good as the edge points obtained from the local gradient estimation method.

Kernel-based gradient evolution optimization method

A new hybrid metaheuristic method is proposed which is inspired from the recently proposed gradient-evolution method. This study introduces for the first time the concept of reproducing kernel in order to correctly estimate the numerical gradient which is used as an updating rule. This proposed method is called Kernel-based Gradient Evolution (GE) which inherits good exploration and exploitation abilities as well as non localization through several operators taken from the original gradient-evolution algorithm. In the proposed KGE algorithm the gradient vector is computed using a kernel reproducing function. Thus the local gradient estimation is numerically computed considering a local Taylor series expansion. The IEEE CEC 2019 test suite is considered in order to evaluate the numerical performance of the proposed KGE algorithm. A numerical comparison between some competitive methods and the one proposed in this work is reported. The obtained numerical results suggested that in terms of convergence, KGE algorithm performs in most cases significantly better than the original GE algorithm and gets respectable results against other considered methods.

Gradient estimates for weighted $p$-Laplacian equations on Riemannian manifolds with a Sobolev inequality and integral Ricci bounds

In this paper, we consider the non-linear general $p$-Laplacian equation $\Delta_{p,f}u+F(u)=0$ for a smooth function $F$ on smooth metric measure spaces. Assume that a Sobolev inequality holds true on $M$ and an integral Ricci curvature is small, we first prove a local gradient estimate for the equation. Then, as its applications, we prove several Liouville type results on manifolds with lower bounds of Ricci curvature. We also derive new local gradient estimates provided that the integral Ricci curvature is small enough.

Gradient estimates for the weighted Lichnerowicz equation on smooth metric measure spaces

We consider the weighted Lichnerowicz equation △fu+cuσ=0 on smooth metric measure spaces, where c≥0,σ≤1 are real constants. A local gradient estimate for positive solutions to this equation is derived, and as an application we give a corresponding Harnack inequality.

Blind Federated Learning at the Wireless Edge With Low-Resolution ADC and DAC

We study collaborative machine learning systems where a massive dataset is distributed across independent workers which compute their local gradient estimates based on their own datasets. Workers send their estimates through a multipath fading multiple access channel with orthogonal frequency division multiplexing to mitigate the frequency selectivity of the channel. We assume that there is no channel state information (CSI) at the workers, and the parameter server (PS) employs multiple antennas to align the received signals. To reduce the power consumption and the hardware costs, we employ complex-valued low-resolution digital-to-analog converters (DACs) and analog-to-digital converters (ADCs), at the transmitter and the receiver sides, respectively, and study the effects of practical low-cost DACs and ADCs on the learning performance. Our theoretical analysis shows that the impairments caused by low-resolution DACs and ADCs, including those of one-bit DACs and ADCs, do not prevent the convergence of the federated learning algorithms, and the multipath channel effects vanish when a sufficient number of antennas are used at the PS. We also validate our theoretical results via simulations, and demonstrate that using low-resolution, even one-bit, DACs and ADCs causes only a slight decrease in the learning accuracy.

Local gradient estimates for second-order nonlinear elliptic and parabolic equations by the weak Bernstein’s method

In the theory of second-order, nonlinear elliptic and parabolic equations, obtaining local or global gradient bounds is often a key step for proving the existence of solutions but it may be even more useful in many applications, for example to singular perturbations problems. The classical Bernstein’s method is a well-known tool to obtain these bounds but, in most cases, it has the defect of providing only a priori estimates. The “weak Bernstein’s method”, based on viscosity solutions’ theory, is an alternative way to prove the global Lipschitz regularity of solutions together with some estimates but it is not so easy to perform in the case of local bounds. The aim of this paper is to provide an extension of the “weak Bernstein’s method” which allows to prove local gradient bounds with reasonable technicalities.

Gradient estimates for a nonlinear elliptic equation under geometric flow

We investigate the parabolic Allen-Cahn equation ∂u∂t=Δu+(1−u2)u under general geometric flow on complete noncompact manifolds. We first prove a local gradient estimate for the positive solution u to this equation, then we obtain a Harnack inequality as an application.

Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian

Abstract In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a ( k , m ) {(k,m)} -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results.

A robust modifier adaptation method via Hessian augmentation using model uncertainties

Typical model-based optimization approaches cannot handle plant-model mismatch, therefore the use of real-time optimization (RTO) schemes which take advantage of measurements from the plant is required. Modifier adaptation (MA) uses the measurements to add a bias to the model which iteratively matches the model with the local gradient estimates of the plant, leading to satisfaction of the Karush–Kuhn–Tucker (KKT) conditions of the plant upon convergence. Whilst feasibility of the convergence solution is guaranteed, there is no such promise of the feasibility of the iterates before convergence. Some methods have been proposed which can guarantee feasibility of the iterates, however all proposed methods suffer from being extremely conservative with long convergence times and are not readily applicable without global information of the plant. This article proposes an alternative approach which uses model uncertainties to avoid the use of unobtainable information whilst removing the overly conservative iterates of previous methods. This new approach is illustrated on the Williams-Otto CSTR, illustrating rapid convergence.

Conormal derivative problem for elliptic equations in divergence form with partial Dini mean oscillation coefficients

We provide detailed proofs for local gradient estimates for weak solutions to elliptic equations in divergence form with partial Dini mean oscillation coefficients subject to conormal derivative boundary conditions.

Elliptic gradient estimates for a nonlinear f-heat equation on weighted manifolds with evolving metrics and potentials

Abstract We develop local elliptic gradient estimates for a basic nonlinear f-heat equation with a logarithmic power nonlinearity and establish pointwise upper bounds on the weighted heat kernel, all in the context of weighted manifolds, where the metric and potential evolve under a Perelman-Ricci type flow. For the heat bounds use is made of entropy monotonicity arguments and ultracontractivity estimates with the bounds expressed in terms of the optimal constant in the logarithmic Sobolev inequality. Some interesting consequences of these estimates are presented and discussed.

A Class of Fully Nonlinear Equations Arising in Conformal Geometry

Abstract In this paper, we consider the equations of Krylov type in conformal geometry on closed smooth Riemannian manifolds, which can be viewed as an extension of $\sigma _k$-Yamabe equation. Moreover, we prove local gradient and 2nd-derivative estimates for solutions to these equations and establish an existence result.