Gradient Constraint Research Articles

Non-convex Hamilton–Jacobi equations with gradient constraints

We study non-convex Hamilton–Jacobi equations in the presence of gradient constraints and produce new, optimal, regularity results for the solutions. A distinctive feature of those equations regards the existence of a lower bound to the norm of the gradient; it competes with the elliptic operator governing the problem, affecting the regularity of the solutions. This class of models relates to various important questions and finds applications in several areas; of particular interest is the modeling of optimal dividends problems for multiple insurance companies in risk theory and singular stochastic control in reversible investment models.

Exploring underlying drivers of genotype by environment interactions in feed efficiency traits for dairy cattle with a mechanistic model involving energy acquisition and allocation

Feed efficiency (FE) is a complex phenotype made up of multiple traits for which there is potential for substantial genotype by environment interaction (G × E). The objective of this study is to evaluate the extent of G × E for FE traits with a simulation approach. We used a mechanistic model of the dairy cow that simulates trajectories of phenotypes throughout lifetime, depending on trajectories of resource acquisition and allocation, driven by 4 genetic scaling parameters, and depending on the nutritional environment (quantity and quality of feed resources). The cow model, calibrated for a grass-based farming system and seasonal calving, was combined with a genetic module. This simulated genetic variation in the 4 genetic scaling parameters related to resource acquisition and allocation, based on a simple balanced pedigree structure (200 paternal half-sib groups each of 100 daughters). The population of 20,000 cows generated was simulated in 4 nutritional environment scenarios, representing a gradient of feeding constraints. In each scenario, 6 traits derived from the model outputs were analyzed to obtain population genetic parameters. Genetic correlations between second-lactation production and FE were positive and high in all scenarios and increased as the nutritional environment became more constraining. A measure of lifetime FE was positively correlated with second-lactation production under a less constrained environment, but these correlations decreased as the environment became more constraining. The genetic correlation between body reserves at second calving, and lifetime FE was positive and low in the least constraining scenario and increased as the environment became more constraining. In addition to genetic parameters, we looked at the distributions of acquisition and allocation parameters among the best performing cows for lactation and life FE, in the 2 most contrasted scenarios. The 4 subpopulations of best cows had acquisition and allocation strategies different from the whole population. In conclusion, this simulation study identifies the potential underlying biological basis for important G × E in FE traits. This highlights the importance of having a balanced breeding goal when undertaking selection that should also be based on phenotypes relevant to the target performance environment.

L₀-l₁ Hybrid Total Variation Regularization and its Applications on Hyperspectral Image Mixed Noise Removal and Compressed Sensing

The total variation (TV) regularization has been widely used in various applications related to hyperspectral (HS) signal and image processing due to its potential in modeling the underlying smoothness of HS data. However, most existing TV norms usually tend to generate spatial oversmoothing or artifacts. To this end, we propose a novel l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> - l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> hybrid TV ( l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> - l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> HTV) regularization with the applications to HS mixed noise removal and compressed sensing (CS). More specifically, l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> - l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> HTV can be regarded as a globally and locally integrated TV regularizer, where the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> gradient constraint is incorporate into the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> spatial-spectral TV ( l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -SSTV). l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -SSTV is capable of exploiting the local structure information across both spatial and spectral domains, while the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> gradient can promote a globally spectral-spatial smoothness by directly controlling the number of nonzero gradients of HS images. This efficient combination considers more comprehensive prior knowledge of HS images, yielding sharper edge preservation and resolving the above drawbacks of existing pure TV norms. More significantly, l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> - l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> HTV can be easily injected into HS-related processing models, and an effective algorithm based on the alternating direction method of multipliers (ADMM) is developed to solve the optimization problems. Extensive experiments conducted on several HS data sets substantiate the superiority and effectiveness of the proposed method in comparison with many state-of-the-art methods.

Topological gradient in structural optimization under stress and buckling constraints

Structural topology optimization aims to design mechanical structures by seeking the optimal material layout within a given design space. Within this framework, this paper addresses the minimization of the structural mass under stress and buckling constraints, formulated as a nonlinear combinatorial optimization problem. An algorithm is proposed for such a problem, that follows a topological gradient-based approach. The adjoint method is applied to efficiently compute the constraint gradients. An iterative algorithm for buckling analysis, featuring low memory requirements, is also proposed. Numerical results, including a real application arising in the aeronautical field, illustrate the efficiency of the two proposed algorithms.

An Improved Algorithm Robust to Illumination Variations for Reconstructing Point Cloud Models from Images

Reconstructing 3D point cloud models from image sequences tends to be impacted by illumination variations and textureless cases in images, resulting in missing parts or uneven distribution of retrieved points. To improve the reconstructing completeness, this work proposes an enhanced similarity metric which is robust to illumination variations among images during the dense diffusions to push the seed-and-expand reconstructing scheme to a further extent. This metric integrates the zero-mean normalized cross-correlation coefficient of illumination and that of texture information which respectively weakens the influence of illumination variations and textureless cases. Incorporated with disparity gradient and confidence constraints, the candidate image features are diffused to their neighborhoods for dense 3D points recovering. We illustrate the two-phase results of multiple datasets and evaluate the robustness of proposed algorithm to illumination variations. Experiments show that ours recovers 10.0% more points, on average, than comparing methods in illumination varying scenarios and achieves better completeness with comparative accuracy.

Global optimal regularity for variational problems with nonsmooth non-strictly convex gradient constraints

We prove the optimal W2,∞ regularity for variational problems with convex gradient constraints. We do not assume any regularity of the constraints; so the constraints can be nonsmooth, and they need not be strictly convex. When the domain is smooth enough, we show that the optimal regularity holds up to the boundary. In this process, we also characterize the set of singular points of the viscosity solutions to some Hamilton-Jacobi equations. Furthermore, we obtain an explicit formula for the second derivative of these viscosity solutions; and we show that the second derivatives satisfy a monotonicity property.

Double obstacle problems and fully nonlinear PDE with non-strictly convex gradient constraints

We prove the optimal W2,∞ regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be C1 or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal W2,∞ regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the second derivative of obstacles, which we have obtained in a previous work.

Hierarchical global search for fuel-optimal impulsive transfers between lunar libration orbits

The attention to the periodic orbit in the Earth-Moon restricted three-body system continues to grow due to its special environment and locations. This research investigates the feasibility of constructing fuel-optimal single and multiple impulse transfers between unstable periodic orbits at L1 and L2 points. Invariant manifolds, which could provide the appropriate initial trajectories for optimization, are analyzed deeply to enable previously unknown orbit options and potentially to reduce mission cost. A global search strategy based on comparing the orbital state of the unstable and stable manifolds, incorporated with low-thrust techniques, is performed to seek a suitable matching point for maneuver application. Then the sequential quadratic programming (SQP) is adopted to further optimize the velocity increment and obtain the single/multiple impulse optimal transfers. The associated constraint gradients are derived to achieve higher accuracy and rapidity of the algorithm. To highlight the effectivity of the transfer scheme, three-dimensional low-energy transfers between different types and spatial regions of performing single and multiple impulses are explored. The total Delta-V required varies between a few meters per second and tens of meters per second, and the related flight time is about several weeks, mainly depending on the energy of periodic orbits and the invariant manifold structure. The results obtained in this paper can provide a useful reference for the selection of escape and capture site along the manifolds, maneuver magnitude and transfer time.

A Free Boundary Problem of Liquidity Management for Optimal Dividend and Insurance in Finite Horizon

In this paper, we devote our attention to the liquidity and risk management of a firm who faces two types of risks: a Poisson risk that can be insured for a fair premium and a Brownian risk that cannot be hedged nor insured. We apply a PDE method to solve this problem; the associated HJB equation is a Barenblatt equation with a gradient constraint in finite horizon. We not only show the existence of a classical solution to the problem, but also characterize the properties of the free boundaries arising from the HJB equation; especially, we are able to give a sufficient and necessary condition for the existence of the insurance free boundary.

Implementation of incremental yield conditions in spherical coordinates

The application of incremental plasticity conditions in the axisymmetric problem of a spherical tank loaded with internal and external pressure is considered. Inhomogeneous stress distribution leads to increased limiting gradient stresses, which depend on the internal cavity size. For the tanks of real dimensions, the magnification limits are established for different laws of approximation of the deformations’ constraint gradient effect.

A modified Nelder-Mead barrier method for constrained optimization

&lt;p style='text-indent:20px;'&gt;An interior point modified Nelder Mead method for nonlinearly constrained optimization is described. This method neither uses nor estimates objective function or constraint gradients. A modified logarithmic barrier function is used. The method generates a sequence of points which converges to KKT point(s) under mild conditions including existence of a Slater point. Numerical results are presented that show the algorithm performs well in practice.&lt;/p&gt;

Equivalence of Viscosity Solutions between Obstacle and Gradient Constraint Problems

Equivalence of Viscosity Solutions between Obstacle and Gradient Constraint Problems

The interplay between two Euler\u2013Lagrange operators relating to the nonlinear elliptic system Sigma [(u, {mathscr {P}}), varOmega

We establish the existence of multiple whirling solutions to a class of nonlinear elliptic systems in variational form subject to pointwise gradient constraint and pure Dirichlet type boundary conditions. A reduced system for certain mathbf{SO}(n)-valued matrix fields, a description of its solutions via Lie exponentials, a structure theorem for multi-dimensional curl free vector fields and a remarkable explicit relation between two Euler–Lagrange operators of constrained and unconstrained types are the underlying tools and ideas in proving the main result.

Some class of nonlinear inequalities with gradient constraints in Orlicz spaces

Abstract In the present paper, we show the existence of solutions of some nonlinear inequalities of the form 〈Au + g(x, u,∇ u), v −u〉 ≥〈 f, v −u〉 with gradient constraint that depend on the solution itself, where A is a Leray-Lions operator defined on Orlicz spaces, g is a nonlinearity and f ∈ L 1.

Gradient-Limiting Shape Control for Efficient Aerodynamic Optimization

Local shape control methods, such as B-spline surfaces, are well-conditioned such that they allow high-fidelity design optimization; however, this comes at the cost of degraded optimization convergence rate as control fidelity is refined due to the resulting exponential increase in the size of the design space. Moreover, optimizations in higher-fidelity design spaces become ill-posed due to high-frequency shape components being insufficiently bounded; this can lead to nonsmooth and oscillatory geometries that are invalid in both physicality (shape) and discretization (mesh). This issue is addressed here by developing a geometrically meaningful constraint to reduce the effective degrees of freedom and improve the design space, thereby improving optimization convergence rate and final result. A new approach to shape control is presented using coordinate control () to recover shape-relevant displacements and surface gradient constraints to ensure smooth and valid iterates. The new formulation transforms constraints directly onto design variables, and these bound the out-of-plane variations to ensure smooth shapes as well as the in-plane variations for mesh validity. Shape gradient constraints approximating a continuity condition are derived and demonstrated on a challenging test case: inviscid transonic drag minimization of a symmetric NACA0012 airfoil. Significantly, the regularized shape problem is shown to have an optimization convergence rate independent of both shape control fidelity and numerical mesh resolution, while still making use of increased control fidelity to achieve improved results. Consequently, a value of 1.6 drag counts is achieved on the test case, the lowest value achieved by any method.

Monocular Depth Estimation Using Information Exchange Network

Depth estimation from single monocular image attracts increasing attention in autonomous driving and computer vision. While most existing approaches regress depth values or classify depth labels based on features extracted from limited image area, the resulting depth maps are still perceptually unsatisfying. Neither local context nor low-level semantic information is sufficient to predict depth. Learning based approaches suffer from inherent defects of supervision signals. This paper addresses monocular depth estimation with a general information exchange convolutional neural network. We maintain a high-resolution prediction throughout the network. Meanwhile, both low-resolution features capturing long-range context and fine-grained features describing local context can be refined with information exchange path stage by stage. Mutual channel attention mechanism is applied to emphasize interdependent feature maps and improve the feature representation of specific semantics. The network is trained under the supervision of improved log-cosh and gradient constraints so that the abnormal predictions have less impacts and the estimation can be consistent in high order. The results of ablation studies verify the efficiency of every proposed components. Experiments on the popular indoor and street-view datasets show competitive results compared with the recent state-of-the-art approaches.

Role of Materials Science in Increasing the Efficiency of a Rock-Crushing Tool Equipped with WC–Co Cemented Carbide Inserts: A Review

A concept is proposed for increasing the efficiency of a rock-crushing tool. The concept uses basics of the thermodynamics of open systems. Based on the proposed concept, the following promising approaches to increasing the efficiency of a rock-crushing tool are identified: Co-phase doping, creating gradient constraints in the volume of the rock-breaking component, and formation of mesostructures in sintered hard alloys.

A regularized magnetotelluric inversion with a minimum support gradient constraint*

Magnetotelluric (MT) inversion is an ill-posed problem and the standard way to address it is through regularization, by adding a stabilizing functional to the data objective functional in order to obtain a stable solution. The traditional stabilizing functionals, in which a low-order differential operator is used, yield a smooth solution that may not be appropriate when anomalies occur in block patterns. In some cases the focused imaging of a sharp electrical boundary is necessary. Even though various experiments have used stabilizing functionals that are suitable to obtain a clear and sharp boundary, such as the minimum support (MS) and the minimum gradient support (MGS) functionals, there are still some limitations in practice. In this paper, the minimum support gradient (MSG) is proposed as the stabilizing functional. Under the uniform regularization framework, a regularized inversion with a variety of stabilizing functionals is performed and the inversion results are compared. This study shows that MSG inversion can not only obtain a clearly focused inversion but also a quite stable and robust one.

Local low-rank matrix recovery for hyperspectral image denoising with ℓ0 gradient constraint

Hyperspectral image (HSI) acquisition often suffers from the mixed noise, which greatly limits its subsequent applications. This paper proposes a novel HSI denoising method by using local low-rank matrix recovery and ℓ0 gradient, which can simultaneously identify the low-rank structures of the clean HSI and the sparse components of the mixed noise. Specifically, the HSI is modeled locally and a scheme of rank-fixed low-rank matrix recovery is employed to separate the latent clean HSI patches from the noisy counterpart. Meanwhile, the ℓ0 gradient constraint mechanism is utilized to characterize the piecewise smooth structure along both the spectral and spatial dimensions of the reconstructed image from the patches. Moreover, the ℓ1 norm regularization is adopted to suppress the sparse noise, such as stripes, deadlines, impulse noise, and so on. Also, an efficient iterative schema is developed based on an augmented Lagrange algorithm. The corresponding closed-form solutions of the subproblems are derived for calculating an approximate result of the nonconvex optimization problem. Extensive experiments on both simulated and real HSIs prove that the ℓ0 gradient constraint can preserve the edge details well and the proposed method can yield better performance than the state-of-the-art methods for HSI denoising.

Necessary Second-Order Conditions for a Strong Local Minimum in a Problem with Endpoint and Control Constraints

The method of sliding modes (relaxation) was originally invented in optimal control in order to give a transparent proof of the maximum principle (a first-order necessary condition for a strong local minimum) using the local maximum principle (a first-order necessary condition for a weak local minimum). In the present work, we use this method to derive second-order necessary conditions for a strong local minimum on the base of such conditions for a weak local minimum. For simplicity, we confine ourselves to the consideration of the Mayer problem with endpoint equality and inequality constraints and control inequality constraints given by a finite number of twice smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide a rather short proof of second-order necessary conditions for a strong local minimum.