Fixed-Time Gradient Flows for Solving Constrained Optimization: A Unified Approach

The accelerated method in solving optimization problems has always been an absorbing topic.Based on the fixedtime(FxT)stability of nonlinear dynamical systems,we provide a unified approach for designing FxT gradient flows(FxTGFs).First,a general class of nonlinear functions in designing FxTGFs is provided.A unified method for designing first-order FxTGFs is shown under Polyak-Łjasiewicz inequality assumption,a weaker condition than strong convexity.When there exist both bounded and vanishing disturbances in the gradient flow,a specific class of nonsmooth robust FxTGFs with disturbance rejection is presented.Under the strict convexity assumption,Newton-based FxTGFs is given and further extended to solve time-varying optimization.Besides,the proposed FxTGFs are further used for solving equation-constrained optimization.Moreover,an FxT proximal gradient flow with a wide range of parameters is provided for solving nonsmooth composite optimization.To show the effectiveness of various FxTGFs,the static regret analyses for several typical FxTGFs are also provided in detail.Finally,the proposed FxTGFs are applied to solve two network problems,i.e.,the network consensus problem and solving a system linear equations,respectively,from the perspective of optimization.Particularly,by choosing component-wisely sign-preserving functions,these problems can be solved in a distributed way,which extends the existing results.The accelerated convergence and robustness of the proposed FxTGFs are validated in several numerical examples stemming from practical applications.

Flow pattern analysis of linear gradient flow distribution

This paper uses the Oseen transformation to solve the differential equations governing motion of the vertical linear gradient flow distribution close to a wall surface. The Navier-Stokes equations are used to consider the inertia term along the flow direction. A novel contour integral method is used to solve the complex Airy function. The boundary conditions of linear gradient flow distribution for finite problems are determined. The vorticity function, the pressure function, and the turbulent velocity profiles are provided, and the stability of particle trajectories is studied. An Lx-function form of the third derivative circulation is used to to simplify the solution. Theoretical results are compared with the experimental measurements with satisfactory agreement.

Stratified Convexity &Concavity of Gradient Flows on Manifolds with Boundary

As has been observed by Morse [1], any generic vector field v on a compact smooth manifold X with boundary gives rise to a stratification of the boundary by compact submanifolds , where . Our main observation is that this stratification re-flects the stratified convexity/concavity of the boundary ?with respect to the ?v-flow. We study the behavior of this stratification under deformations of the vector field v. We also investigate the restrictions that the existence of a convex/concave traversing ?v-flow imposes on the topology of X. Let be the orthogonal projection of on the tangent bundle of . We link the dynamics of theon the boundary with the property of in X being convex/concave. This linkage is an instance of more general phenomenon that we call “holography of traversing fields”—a subject of a different paper to follow.

A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure.These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential.The proposed scheme is able to cope with non-smooth stationary states,different time scales including metastability,as well as concentrations and self-similar behavior induced by singular nonlocal kernels.We use the scheme to explore properties of these equations beyond their present theoretical knowledge.

An SAV Method for Imaginary Time Gradient Flow Model in Density Functional Theory

In this paper,based on the imaginary time gradient flow model in the density functional theory,a scalar auxiliary variable(SAV)method is developed for the ground state calculation of a given electronic structure system.To handle the orthonormality constraint on those wave functions,two kinds of penalty terms are introduced in designing the modified energy functional in SAV,i.e.,one for the norm preserving of each wave function,another for the orthogonality between each pair of different wave functions.A numerical method consisting of a designed scheme and a linear finite element method is used for the discretization.Theoretically,the desired unconditional decay of modified energy can be obtained from our method,while computationally,both the original energy and modified energy decay behaviors can be observed successfully from a number of numerical experiments.More importantly,numerical results show that the orthonormality among those wave functions can be automatically preserved,without explicitly preserving orthogonalization operations.This implies the potential of our method in large-scale simulations in density functional theory.

A Convergence Analysis of a Structure-Preserving Gradient Flow Method for the All-Electron Kohn-Sham Model

In[Dai et al.,Multi.Model.Simul.18(4)(2020)],a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory,based on which a linearized method was developed in[Hu et al.,EAJAM.13(2)(2023)]for further improving the numerical efficiency.In this paper,a complete convergence analysis is delivered for such a linearized method for the all-electron Kohn-Sham model.Temporally,the convergence,the asymptotic stability,as well as the structure-preserving property of the linearized numerical scheme in the method is discussed following previous works,while spatially,the convergence of the h-adaptive mesh method is demonstrated following[Chen et al.,Multi.Model.Simul.12(2014)],with a key study on the boundedness of the Kohn-Sham potential for the all-electron Kohn-Sham model.Numerical examples confirm the theoretical results very well.

GRADIENT FLOW FINITE ELEMENT DISCRETISATIONS WITH ENERGY-BASED ADAPTIVITY FOR EXCITED STATES OF SCHRÖDINGER'S EQUATION

The purpose of this paper is to verify that the computational scheme from[Heid et al.,Gradient flow finite element discretizations with energy-based adaptivity for the Gross–Pitaevskii equation,J.Comput.Phys.436(2021)]for the numerical approximation of the ground state of the Gross–Pitaevskii equation can equally be applied for the effective approximation of excited states of Schr¨odinger’s equation.That procedure employs an adaptive interplay of a Sobolev gradient flow iteration and a novel local mesh refinement strategy,and yields a guaranteed energy decay in each step of the algorithm.The computational tests in the present work highlight that this strategy is indeed able to approximate excited states,with(almost)optimal convergence rate with respect to the number of degrees of freedom.

Gradient Flow of the L_(β)-Functional

In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic property is preserved along the gradient flow.Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form.When β=1,we derive a monotonicity formula for the flow.As applications,we show that the l-tangent cone of the flow consists of the finite flat planes.

Active contours with normally generalized gradient vector flow external force

Gradient vector flow （GVF） is an effective external force for active contours, but its iso- tropic nature handicaps its performance. The recently proposed gradient vector flow in the normal direction （NGVF） is anisotropic since it only keeps the diffusion along the normal direction of the isophotes; however, it has difficulties forcing a snake into long, thin boundary indentations. In this paper, a novel external force for active contours called normally generalized gradient vector flow （NGGVF） is proposed, which generalizes the NGVF formulation to include two spatially varying weighting functions. Consequently, the proposed NGGVF snake is anisotropic and would improve ac- tive contour convergence into long, thin boundary indentations while maintaining other desirable properties of the NGVF snake, such as enlarged capture range, initialization insensitivity and good convergence at concavities. The advantages on synthetic and real images are demonstrated.

Centerline Extraction for Image Segmentation Using Gradient and Direction Vector Flow Active Contours

In this paper, we propose a fast centerline extraction method to be used for gradient and direction vector flow of active contours. The gradient and direction vector flow is a recently reported active contour model capable of significantly improving the image segmentation performance especially for complex object shape, by seamlessly integrating gradient vector flow and prior directional information. Since the prior directional information is provided by manual line drawing, it can be inconvenient for inexperienced users who might have difficulty in finding the best place to draw the directional lines to achieve the best segmentation performance. This paper describes a method to overcome this problem by automatically extracting centerlines to guide the users for providing the right directional information. Experimental results on synthetic and real images demonstrate the feasibility of the proposed method.

Generalizing J_2 flow theory: Fundamental issues in strain gradient plasticity

It has not been a simple matter to obtain a sound extension of the classical J2 flow theory of plasticity that incorporates a dependence on plastic strain gradients and that is capable of capturing size-dependent behaviour of metals at the micron scale. Two classes of basic extensions of classical J2 theory have been proposed： one with increments in higher order stresses related to increments of strain gradients and the other characterized by the higher order stresses themselves expressed in terms of increments of strain gradients. The theories proposed by Muhlhans and Aifantis in 1991 and Fleck and Hutchinson in 2001 are in the first class, and, as formulated, these do not always satisfy thermodynamic requirements on plastic dissipation. On the other hand, theories of the second class proposed by Gudmundson in 2004 and Gurtin and Anand in 2009 have the physical deficiency that the higher order stress quantities can change discontinuously for bodies subject to arbitrarily small load changes. The present paper lays out this background to the quest for a sound phenomenological extension of the rateindependent J2 flow theory of plasticity to include a de- pendence on gradients of plastic strain. A modification of the Fleck-Hutchinson formulation that ensures its thermo- dynamic integrity is presented and contrasted with a comparable formulation of the second class where in the higher or- der stresses are expressed in terms of the plastic strain rate. Both versions are constructed to reduce to the classical J2 flow theory of plasticity when the gradients can be neglected and to coincide with the simpler and more readily formulated J2 deformation theory of gradient plasticity for deformation histories characterized by proportional straining.

Determination of Curie Point Depth, Heat Flow and Geothermal Gradient from High Resolution Aeromagnetic Data around Lamurde Area, Adamawa State, North-Eastern Nigeria

Analysis of high resolution of aeromagnetic data was carried out over Lamurde, Adamawa state north-eastern Nigeria to determine the Curie point depth (CPD), heat flow and geothermal gradient. The aeromagnetic data used for this work was obtained at Nigerian geological survey agency, the total magnetic intensity was processed to produce the residual magnetic map which was divided into 4 overlapping blocks, each block was subjected to spectral analyses to obtain depths to the top boundary and centroid, while depth to bottom of the magnetic sources was calculated using empirical formula. The depths values obtained were then used to assess the CPD, heat flow and geothermal gradient in the area. The result shows that the CPD varies between 9.62 and 10.92 km with an average of 10.45 k, the heat flow varies between 150.73 and 132.78 mWm−20⋅°C−1 with an average of 139.12 mWm−20⋅°C−1 and the geothermal gradient in the study area varies between 12.16 and 15.67 °C/km with an average of 13.39 °C/km. In view of the above results, the high heat flow may be responsible for maturation of hydrocarbon in Benue Trough as well as responsible for the lead Zinc Mineralization. Again by implication, Lamurde area can be a good area for geothermal reservoir exploration for an alternative source for power generation.

Paradoxical Low Flow Low Gradient Severe Aortic Stenosis—An Under-Recognized Entity

Some patients with severe aortic stenosis (AS), due to restrictive cardiac physiology, paradoxically have relatively low flow and low gradients across stenotic aortic valves despite preserved left ventricular (LV) systolic function. It results in symptoms and reduced quality of life and carries a high mortality. Whilst this form of severe AS, termed paradoxical low flow low gradient (pLFLG), is well reported, patients with this diagnosis experience inappropriate barriers to aortic valve replacement (AVR), the only efficacious treatment. We present the case of an 88-year-old female with 12 months of exertional dyspnoea on a background of hypothyroidism and hypercholesterolemia. Transthoracic echocardiogram (TTE) revealed LV hypertrophy, with a small LV cavity size and reduced stroke volume, yet normal systolic function. A heavily calcified aortic valve was identified with severe aortic stenosis, based on valve area, yet with incongruous mean transvalvular gradient of 25 mmHg (severe ≥ 50 mmHg). Following exclusion of other differential diagnoses, her symptoms were attributed to paradoxical LFLG severe AS. She was however declined definitive transcatheter aortic valve implantation (TAVI) due to her paradoxically low mean aortic gradient. Following further deterioration in her symptoms and supportive quantification of poor exercise performance, she was ultimately re-referred, accepted, and underwent TAVI. Following her AVR, the patient experiences significant improvement in both symptoms and quality of life after only one month. Paradoxical LFLG severe AS remains a well-documented yet under recognized disease. It carries high morbidity and mortality if untreated, yet is significantly less likely to be referred and accepted for intervention. With its prevalence expected to rise with an ageing population, this case serves as a timely reminder for clinicians to address the under recognition of important pathology.

Gradient estimates for porous medium equations under the Ricci flow

A local gradient estimate for positive solutions of porous medium equations on complete noncompact Riemannian manifolds under the Ricci flow is derived. Moreover, a global gradient estimate for such equations on compact Riemannian manifolds is also obtained.

Geothermal Gradient and Heat Flow in the Nigeria Sector of the Chad Basin, Nigeria

Information on geothermal gradient and heat flow within the subsurface is critical in the quest for geothermal energy exploration. In a bid to ascertain the thermal potential of Nigeria sector of the Chad Basin for energy generation, subsurface temperature information from 19 oil wells, 24 water boreholes drilled to depths beyond 100 metres and atmospheric temperature from the Chad basin were utilized in calculating geothermal gradient of the area. Selected ditch cuttings from the wells were subjected to thermal conductivity test using Thermal Conductivity Scanner (TCS) at the Polish Geological Institute Laboratory in Warsaw. The terrestrial heat flow was calculated according to the Fourier’s law as a simple product of the geothermal gradient and the mean thermal conductivity. Results obtained indicated geothermal gradient range of 2.81 °C/100 m to 5.88 °C/100 m with an average of 3.71 °C/100 m. The thermal conductivity values from the different representative samples range from 0.58 W/m*K to 4.207 W/m*K with an average of 1.626 W/m*K. The work presented a heat flow value ranging from 45 mW/m2 to about 90 mW/m2 in the Nigerian sector of the Chad Basin.

Effect of slope gradient on the subsurface water flow velocity of sand layer profile

Subsurface water flow velocity influences the hydrodynamic characteristics of soil seepage and the interaction between subsurface water flow and surface runoff during soil erosion and sediment transport.A visualized method and equipment was adopted in this study to observe the subsurface water flow.Quartz sand was used as the test material of subsurface water flow and fluorescent dye was used as the indicator for tracing subsurface water flow.Water was supplied at the same flow discharge to the three parts at the bottom of the test flume,and the subsurface water flow were determined with four slope gradients(4°,8°,10°,and 12°).The results showed that the seepage velocity gradually increased with increasing slope gradient.The pore water velocity at different depths of sand layer profile increased with increasing slope gradient,whereas the thickness of the flow front gradually decreased.For the same slope gradient,the pore water velocity in the lower layer was the largest,whereas the thickness of the flow front was the smallest.Comparative analysis of the relationship between seepage velocity and pore water velocity at different depths of sand layer profile showed that the maximum relative difference between the measured pore water velocity and the computational pore water velocity at different depths of sand profile in the experiment was 4.38%.Thus,the test method for measuring the subsurface water flow velocity of sand layer profile adopted in this study was effective and feasible.The development of this experiment and the exploration of research methods would lay a good test foundation for future studies on the variation law of subsurface water flow velocity and the determination of flow velocity in purple soils,thus contributing to the improvement of the hydrodynamic mechanism of purple soils.

Converging Parallel Plate Flow Chambers for Studies on the Effect of the Spatial Gradient of Wall Shear Stress on Endothelial Cells

Many in vitro studies focus on effects of wall shear stress (WSS) and wall shear stress gradient (WSSG) on endothelial cells, which are linked to the initiation and progression of atherosclerosis in the arterial system. Limitation in available flow chambers with a constant WSSG in the testing region makes it difficult to quantify cellular responses to WSSG. The current study proposes and characterizes a type of converging parallel plate flow chamber (PPFC) featuring a constant gradient of WSS. A simple formula was derived for the curvature of side walls, which relates WSSG to flow rate (Q), height of the PPFC (h), length of the convergent section (L), its widths at the entrance (w0) and exit (w1). CFD simulation of flow in the chamber is carried out. Constant WSSG is observed in most regions of the top and bottom plates except those in close proximity of side walls. A change in Q or h induces equally proportional changes in WSS and WSSG whereas an alteration in the ratio between w0 and w1 results in a more significant change in WSSG than that in WSS. The current design makes possible an easy quantification of WSSG on endothelial cells in the flow chamber.

On study of horizontal thin film flow of Sisko fluid due to surface tension gradient

The present paper is concerned with the steady thin film flow of the Sisko fluid on a horizontal moving plate, where the surface tension gradient is a driving mechanism. The analytic solution for the resulting nonlinear ordinary differential equation is obtained by the Adomian decomposition method （ADM）. The physical quantities are derived including the pressure profile, the velocity profile, the maximum residue time, the stationary points, the volume flow rate, the average film velocity, the uniform film thickness, the shear stress, the surface tension profile~ and the vorticity vector. It is found that the velocity of the Sisko fluid film decreases when the fluid behavior index and the Sisko fluid parameter increase, whereas it increases with an increase in the inverse capillary number. An increase in the inverse capillary number results in an increase in the surface tension which in turn results in an increase in the surface tension gradient on the Sisko fluid film. The locations of the stationary points are shifted towards the moving plate with the increase in the inverse capillary number, and vice versa locations for the stationary points are found with the increasing Sisko fluid parameter. Furthermore, shear thinning and shear thickening characteristics of the Sisko fluid are discussed. A comparison is made between the Sisko fluid film and the Newtonian fluid film.

基于三周期极小曲面晶胞梯度支架的设计及力学性能

背景:现阶段骨软骨一体化支架的弹性模量与天然骨软骨相差较大,植入人体后会引起应力遮蔽现象,从而导致植入物松动变形,影响骨软骨组织修复。轴向三周期极小曲面晶胞梯度支架可与人体骨软骨组织孔隙率及弹性模量相匹配,为骨软骨支架设计提供一种新的思路。目的:研究不同晶胞种类和孔径结构对晶胞梯度支架力学性能的影响。方法:使用Gyroid(G)型、Diamond(D)型Primitive(P)型3种基础晶胞,通过三周期极小曲面数学建模,在梯度区域使用不同尺寸、不同种类晶胞,共计构建6种晶胞梯度支架(G-2P-4D、P-2D-4G、D-2P-4D、G-2D-4P、P-2G-4D、D-2G-4P),进行力学实验及仿真模拟实验,评估支架的力学性能,通过计算流体动力学仿真得到了支架内流体的流动性能参数。结果与结论:有限元力学仿真和轴向压缩实验表明,P-2G-4D型与P-2D-4G型梯度支架的弹性模量相较高,分别为148.67 MPa和152.1 MPa,能够承受较高的轴向载荷,提高植入物的力学稳态;D-2P-4G型梯度支架应力分布最为均匀,可有效减少应力集中,使得连接函数区域均能够有效传递应力,减少应力遮蔽;G-2D-4P型梯度支架流动速率变化最小,为0.10-0.48 mm/s,渗透率较高,有利于植入后体液在支架内部流动。基于三周期极小曲面的晶胞梯度支架设计为骨软骨支架设计提供了新思路,仿真分析结果也为支架植入人体后的骨整合预测提供了参考。

INFLUENCES OF SLOPE GRADIENT ON SOIL EROSION

The main factors influencing soil erosion include the net rain excess, the water depth, the velocity, the shear stress of overland flows, and the erosion-resisting capacity of soil. The laws of these factors varying with the slope gradient were investigated by using the kinematic wave theory. Furthermore, the critical slope gradient of erosion was driven. The analysis shows that the critical slope gradient of soil erosion is dependent on grain size, soil bulk density, surface roughness, runoff length, net rain excess, and the friction coefficient of soil, etc. The critical slope gradient has been estimated theoretically with its range between 41.5 degrees similar to 50 degrees.