Directional Derivatives Research Articles

On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at -2

Let d(δ) denote the Hausdorff dimension of the Julia set of the polynomial fδ(z)=z2−2+δ.In this paper, we will study the directional derivative of the function δ↦d(δ) along directions landing at the parameter 0, which corresponds to −2 in the case of family pc(z)=z2+c. We will consider all directions except the one δ∈R+, which is inside the Mandelbrot set.We will prove an asymptotic formula for the directional derivative of d(δ). Moreover, we will see that the derivative is negative for all directions in the closed left half-plane. Computer calculations show that it is negative except for a cone (with an opening angle of approximately 74∘) around R+.

Matrix splitting preconditioning based on sine transform for solving two-dimensional space-fractional diffusion equations

Finite difference discretization of the two-dimensional space-fractional diffusion equations derives a complicated linear system consisting of identity matrix and four scaled block-Toeplitz with Toeplitz block (BTTB) matrices resulted from the left and right Riemann–Liouville fractional derivatives in different directions. Incorporating with the diffusion coefficients and the symmetric parts of the BTTB matrices, we construct a diagonal and symmetric splitting (DSS) iteration method, which is demonstrated to be convergent conditionally when the considered space-fractional diffusion equations have sufficiently close diffusion coefficients. By further replacing the symmetric Toeplitz matrices involved in the BTTB matrices with τ matrices, an approximated DSS (ADSS) preconditioner based on two-dimensional fast sine transform is designed to accelerate the convergence rates of the Krylov subspace iteration methods. In this way, the total computational complexity of the ADSS-preconditioned GMRES method will be of O(n2logn), where n2 represents the dimension of the corresponding discrete linear system. In addition, theoretical analysis demonstrates that the eigenvalues of the ADSS-preconditioned matrix are weakly clustered around a complex disk centered at 1 with the radius less than 1. Numerical experiments show that the ADSS-preconditioned GMRES method is much more efficient than the other existing methods, and can show h-independent convergence behavior.

Directional derivatives in set optimization with the set order defined by Minkowski difference

This paper focuses on the directional derivative of a set optimization problem with a set order relation defined by the Minkowski difference. Firstly, we investigate a nonlinear scalar function related to the Minkowski difference and propose a Hausdorff-type distance of sets. Some properties of Hausdorff-type distance are given. Secondly, a directional derivative of set-valued mappings is introduced, and some properties are examined. Finally, the obtained results are applied to establish optimality conditions for set optimization problems.

Money-metrics in local welfare analysis: Pareto improvements and equity considerations

We identify local Pareto improvements from a valuation equilibrium, and extend the results of Hirshleifer, Arrow-Lind, Milleron, and Radner on the evaluation of small projects to behavioral or nonstandard choice models. We use the sign of directional derivative of the sum of McKenzie-Samuelson money metrics to evaluate small projects, but, rather than assume its differentiability, furnish preference conditions that guarantee it. Our methods yield, as an unintended consequence, (i) a refutation of Samuelson's (1974) conjecture that the money metric is locally concave in a neighborhood of a demand point, thereby settling an issue open for five decades; and (ii) a substantive extension of the 1988 Blackorby-Donaldson theorem that the money metric is concave in consumption only if preferences are quasihomothetic. We explain some equity implications of our local-welfare result, and as part of the rehabilitation of money metrics, suggest a case for using a second-order approximation to a money metric for local welfare. We illustrate when our results hold and don't hold with several non-standard choice models.

A class of projected-search methods for bound-constrained optimization

Projected-search methods for bound-constrained optimization are based on performing a search along a piecewise-linear continuous path obtained by projecting a search direction onto the feasible region. A potential benefit of a projected-search method is that many changes to the active set can be made at the cost of computing a single search direction. As the objective function is not differentiable along the search path, it is not possible to use a projected-search method with a step that satisfies the Wolfe conditions, which require the directional derivative of the objective function at a point on the path. For this reason, methods based in full or in part on a simple backtracking procedure must be used to give a step that satisfies an ‘Armijo-like’ sufficient decrease condition. As a consequence, conventional projected-search methods are unable to exploit sophisticated safeguarded polynomial interpolation techniques that have been shown to be effective for the unconstrained case. This paper describes a new framework for the development of a general class of projected-search methods for bound-constrained optimization. At each iteration, a descent direction is computed with respect to a certain extended active set. This direction is used to specify a search direction that is used in conjunction with a step length computed by a quasi-Wolfe search. The quasi-Wolfe search is designed specifically for use with a piecewise-linear search path and is similar to a conventional Wolfe line search, except that a step is accepted under a wider range of conditions. These conditions take into consideration steps at which the restriction of the objective function on the search path is not differentiable. Standard existence and convergence results associated with a conventional Wolfe line search are extended to the quasi-Wolfe case. In addition, it is shown that under a standard nondegeneracy assumption, any method within the framework will identify the optimal active set in a finite number of iterations. Computational results are given for a specific projected-search method that uses a limited-memory quasi-Newton approximation of the Hessian. The results show that, in this context, a quasi-Wolfe search is substantially more efficient and reliable than an Armijo-like search based on simple backtracking. Comparisons with a state-of-the-art bound-constrained optimization package are also presented.

Solving nonlinear equations with the convex combination of two positive spectral coefficients

We present a spectral algorithm based on the convex combination of two modified spectral coefficients for solving systems of nonlinear equations. The proposed algorithm does not require the exact or approximated directional derivative for its implementation. By employing a derivative-free line search, the global convergence of the sequence generated by the algorithm is supported. Numerical experiments are given to demonstrate the performance of the algorithm compared with a similar algorithm in the literature for solving nonlinear equations problems.

Nitrogen oxides emissions from selected cities in North America, Europe, and East Asia observed by the TROPOspheric Monitoring Instrument (TROPOMI) before and after the COVID-19 pandemic

Abstract. Nitrogen oxides (NOx=NO+NO2) emissions are estimated in three regions in the Northern Hemisphere, generally located in North America, Europe, and East Asia, by calculating the directional derivatives of NO2 column amounts observed by the TROPOspheric Monitoring Instrument (TROPOMI) with respect to the horizontal wind vectors. We present monthly averaged emissions from 1 May 2018 to 31 January 2023 to capture variations before and after the COVID-19 pandemic. We focus on a diverse collection of 54 cities, 18 in each region. A spatial resolution of 0.04∘ resolves intracity emission variations and reveals NOx emission hotspots at city cores, industrial areas, and sea ports. For each selected city, post-COVID-19 changes in NOx emissions are estimated by comparing monthly and annually averaged values to the pre-COVID-19 year of 2019. While emission reductions are initially found during the first outbreak of COVID-19 in early 2020 in most cities, the cities' paths diverge afterwards. We group the selected cities into four clusters according to their normalized annual NOx emissions in 2019–2022 using an unsupervised learning algorithm. All but one of the selected North American cities fall into cluster 1 characterized by weak emission reduction in 2020 (−7 % relative to 2019) and an increase in 2022 by +5 %. Cluster 2 contains mostly European cities and is characterized by the largest reduction in 2020 (−31 %), whereas the selected East Asian cities generally fall into clusters 3 and 4, with the largest impacts in 2022 (−25 % and −37 %). This directional derivative approach has been implemented in object-oriented, open-source Python and is available publicly for high-resolution and low-latency emission estimation for different regions, atmospheric species, and satellite instruments.

Canonical momenta in digitized Su(2) lattice gauge theory: definition and free theory

Hamiltonian simulations of quantum systems require a finite-dimensional representation of the operators acting on the Hilbert space mathcal {H}. Here we give a prescription for gauge links and canonical momenta of an SU(2) gauge theory, such that the matrix representation of the former is diagonal in mathcal {H}. This is achieved by discretising the sphere S_3 isomorphic to SU(2) and the corresponding directional derivatives. We show that the fundamental commutation relations are fulfilled up to discretisation artefacts. Moreover, we directly construct the Casimir operator corresponding to the Laplace–Beltrami operator on S_3 and show that the spectrum of the free theory is reproduced again up to discretisation effects. Qualitatively, these results do not depend on the specific discretisation of SU(2), but the actual convergence rates do.

Well-posedness and global error bound for generalized mixed quasi-variational-hemivariational inequalities via regularized gap functions

The aim of this paper is to establish new results on a class of generalized mixed quasi-variational-hemivariational inequalities (GMQVHVI, for short) via regularized gap functions. First, we introduce the new regularized gap function of GMQVHVI. Then we establish the criterion of the Levitin-Polyak well-posedness for GMQVHVI under suitable conditions. Further, we provide the equivalence between the Levitin-Polyak well-posedness in the generalized sense for GMQVHVI and that for a quasi-optimization problem using the new regularized gap function. Finally, the global error bound for GMQVHVI in term of the regularized gap function is derived by employing some the properties of the Clarke generalized directional derivative and strong monotonicity conditions.

Poland as a Player in the Middle East in the First Two Decades of the 20th Century. Selected Military and Non-Military Dilemmas

This article aims to show that Poland’s foreign policy towards the Middle East is important, although not a priority in the pursuit of its interests. However, the significance of the Middle East in the Polish raison d’État has been increasing continuously and this trend will continue. Dilemmas related to the issue of military involvement in the region were among the most important for the Polish ruling class over the last three decades. Of equal importance were implications of the Arab Spring, the refugee crisis and the threats of lone wolf terrorist attacks. Also problematic was the position of Polish authorities regarding Iran, or rather the dilemmas connected with unthinking support for the United States in this matter, and the simultaneous cooling of relations with Tehran and the European Union’s states. The policy towards the Middle East is a derivative of the Euro-Atlantic direction, which has been a matter of priority to Poland. Presenting Poland as a loyal ally as regards policy towards the Middle East is an outcome of Polish elites’ endeavours to ensure security in Poland by strengthening the alliance with the USA and NATO, which are to constitute a security provider against the neo-imperial policy of Russia.

Hilbert and Poincaré problems for semi-linear equations in rectifiable domains

The study of the boundary value problem with arbitrary measurable data originated in the dissertation of Luzin where he investigated the Dirichlet problem for harmonic functions in the unit disk. Recently, in \cite{R7}, we studied the Hilbert, Poincaré and Neumann boundary value problems with arbitrary measurable data for generalized analytic and generalized harmonic functions and provided applications to relevant problems in mathematical physics. The present paper is devoted to the study of the boundary value problem with arbitrary measurable boundary data in a domain with rectifiable boundary corresponding to semi-linear equation with suitable nonlinear source. We construct a completely continuous operator and generate nonclassical solutions to the Hilbert and Poincaré boundary value problems with arbitrary measurable data for Vekua type and Poisson equations, respectively. Based on that, we prove the existence of solutions of the Hilbert boundary value problem for the nonlinear Vekua type equation with arbitrary measurable data in a domain with rectifiable boundary. It is necessary to point out that our approach differs from the classical variational approach in PDE as it is based on the geometric interpretation of boundary values as angular (along non-tangential paths) limits. The latter makes it possible to also obtain a theorem on the boundary value problem for directional derivatives, and, in particular, of the Neumann problem with arbitrary measurable data for the Poisson equation with nonlinear sources in any Jordan domain with rectifiable boundary. As a result we arrive at applications to some problems of mathematical physics.

A clustering heuristic to improve a derivative-free algorithm for nonsmooth optimization

In this paper we propose an heuristic to improve the performances of the recently proposed derivative-free method for nonsmooth optimization CS-DFN. The heuristic is based on a clustering-type technique to compute an estimate of Clarke’s generalized gradient of the objective function, obtained via calculation of the (approximate) directional derivative along a certain set of directions. A search direction is then calculated by applying a nonsmooth Newton-type approach. As such, this direction (as it is shown by the numerical experiments) is a good descent direction for the objective function. We report some numerical results and comparison with the original CS-DFN method to show the utility of the proposed improvement on a set of well-known test problems.

Local search versus linear programming to detect monotonicity in simplicial branch and bound

AbstractThis study focuses on exhaustive global optimization algorithms over a simplicial feasible set with simplicial partition sets. Bounds on the objective function value and its partial derivative are based on interval automatic differentiation over the interval hull of a simplex. A monotonicity test may be used to decide to either reject a simplicial partition set or to reduce its simplicial dimension to a relative border (at the boundary of the feasible set) facet (or face) by removing one (or more) vertices. A monotonicity test is more complicated for a simplicial sub-set than for a box, because its orientation does not coincide with the components of the gradient. However, one can focus on directional derivatives (DD). In a previous study, we focused on either basic directions, such as centroid to vertex or vertex to vertex directions, or finding the best directional derivative by solving an LP or MIP. The research question of this paper refers to using local search (LS) based sampling of directions from vertex to facet. Results show that most of the monotonic DD found by LP are also found by LS, but with much less computational cost. Notice that finding a monotone direction does not require to find the direction in which a derivative bound is the steepest.

Convergence of least energy sign-changing solutions for logarithmic Schrödinger equations on locally finite graphs

In this paper, we study the following logarithmic Schrödinger equation −Δu+λa(x)u=ulogu2inVon a connected locally finite graph G=(V,E), where Δ denotes the graph Laplacian, λ>0 is a constant, and a(x)≥0 represents the potential. Using variational techniques in combination with the Nehari manifold method based on directional derivative, we can prove that, there exists a constant λ0>0 such that for all λ≥λ0, the above problem admits a least energy sign-changing solution uλ. Moreover, as λ→+∞, we prove that the solution uλ converges to a least energy sign-changing solution of the following Dirichlet problem −Δu=ulogu2inΩ,u(x)=0on∂Ω,where Ω={x∈V:a(x)=0} is the potential well.

Stability of the approximate solution sets for set optimization problems with the perturbations of feasible set and objective mapping

The aim of this paper is to study stability of the sets of l-minimal approximate solutions and weak l-minimal approximate solutions for set optimization problems with respect to the perturbations of feasible sets and objective mappings. We introduce a new metric between two set-valued mappings by utilizing a Hausdorff-type distance proposed by Han [A Hausdorff-type distance, the Clarke generalized directional derivative and applications in set optimization problems. Appl Anal. 2022;101:1243–1260]. The new metric between two set-valued mappings allows us to discuss set optimization problems with respect to the perturbation of objective mappings. Then, we establish semicontinuity and Lipschitz continuity of l-minimal approximate solution mapping and weak l-minimal approximate solution mapping to parametric set optimization problems by using the scalarization method and a density result. Finally, our main results are applied to stability of the approximate solution sets for vector optimization problems.

A novel infrared spectral preprocessing method based on self-deconvolution and differentiation in the frequency domain

Spectral preprocessing is a crucial step before establishing a multivariate calibration model for Fourier transform infrared (FTIR) spectroscopic analysis. However, a single preprocessing method has limited effects, such as insufficient noise reduction or limited resolution improvement. As two classical preprocessing methods, Fourier self-deconvolution (FSD) and Fourier filtering and differentiation (FFD) have their benefits and drawbacks. FSD has low noise but limited resolution, while FFD could improve resolution and eliminate baseline simultaneously, but is susceptive to noise. By combining FSD and FFD, we design a novelty resolution enhancement method, termed Fourier self-deconvolution differentiation (FSDD), which can significantly improve spectral resolution with good noise and baseline robustness. Partial least squares regression (PLS) is combined with five preprocessing methods, FSD, FFD, direct derivation (DD), Savitsky-Golay smoothing and derivation (SGSD), and FSDD, to analyze the albiflorin dataset and the γ-polyglutamic acid (γ-PGA) fermentation dataset. Compared with other preprocessing methods, FSDD obtained the optimum performance merits, including root mean squared error of prediction (RMSEP), determination coefficient (R2), and Mean Absolute Error (MAE).

QRFODD: Quaternion Riesz fractional order directional derivative for color image edge detection

Edge detection is the prominent method to determine the discontinuities present in an image. There exists an issue of loss of information and correlation during the extraction of edges in color images. To resolve it, a novel quaternion domain based Riesz fractional order directional derivative approach is presented in this paper to extract more edge details in color images. The Riesz fractional masks obtained by utilizing Aitken interpolation are applied on color images in the quaternion domain for yielding the edge map as there is no need for recalculation of basis polynomials in case of the addition of any data point. The performance of the proposed quaternion based technique is analyzed on the benchmark BSDS300 and BSDS500 datasets in the terms of Figure of Merit and F-Score. The results obtained exhibit the efficiency of proposed technique over the classical and state-of-the-art edge detection approaches. The robustness of this approach is further established by taking into account uncontrolled conditions such as noise, JPEG compression, and varying illumination as it provided more features than the classical integer-order detectors. Moreover, the superiority of the proposed quaternion based technique is validated for medical images as well as its application in image enhancement.

A general differential quasi variational–hemivariational inequality: Well-posedness and application

In the paper we analyze a general differential quasi variational-hemivariational inequality in Banach spaces which couples the Cauchy problem for an ordinary differential equation with a quasi variational-hemivariational inequality considered with a unilateral constraint and history-dependent operators. The latter appear in all of the data: the governing nonlinear operator, the generalized directional derivative of a locally Lipschitz potential, a convex potential, and the ordinary differential equation. The results on the well-posedness and regularity of solutions are proved by exploiting a fixed point approach. The theory is illustrated by an application to a quasistatic generalized Signorini unilateral frictional contact problem for viscoplastic materials with a nonsmooth multivalued contact condition.

Lattice Boltzmann finite-difference-based model for fully nonlinear electrohydrodynamic deformation of a liquid droplet.

Electric field-induced flows involving multiple fluid components with a range of different electrical properties are described by the coupled Taylor-Melcher leaky-dielectric model. We present a lattice Boltzmann (LB)-finite difference (FD) method-based hybrid framework to solve the complete Taylor-Melcher leaky-dielectric model considering the nonlinear surface charge convection effects. Unlike the existing LB-based models, we treat the interfacial discontinuities using direction-specific continuous gradients, which prevents the miscalculation arising due to volumetric gradients without directional derivatives, simultaneously maintaining the electroneutrality of the bulk. While fluid transport is recovered through the LB method using a multiple relaxation time (MRT) scheme, the FD method with a central difference scheme is applied to discretize the charge transport equationat the interface, in addition to the electric field governing equationsin the bulk and at the interface. We apply the developed numerical model to study the different regimes of droplet deformation due to an external electric field. Similar to the existing analytical and other numerical models, excluding the surface charge convection (SCC) term from the charge transport equation, the present methodology has shown excellent agreement with the existing literature. In addition, the effect of SCC in each of the regimes is analyzed. With the present numerical model, we observe a strong presence of SCC in the oblate deformation regime, contrary to the weak effect on prolate deformations. We further discuss the reason behind such differences in the magnitude of nonlinearity induced by the SCC in all the regimes of deformation.

A 3D face interpolated discretisation method for simulating anisotropic diffusive processes on meshes coming from wood morphology

In this paper, a new 3D numerical discretisation method for solving the anisotropic, steady-state diffusion problem is developed and analysed. The scheme is constructed on hexahedral meshes using the geometrical properties of the cells. Indeed, each cell provides a local basis formed using the centres of its lateral faces. The discrete cell gradient approximation is then obtained by using three discrete directional derivatives resolved in terms of this basis, and by invoking the consistent relationships between opposite faces of the same cell. The face degrees of freedom are interpolated to reduce the complexity of the numerical scheme, resulting in the main unknowns being entirely nodal based. The scheme is unconditionally coercive and admits a unique solution. Various numerical experiments are performed to highlight the accuracy and the robustness of the method with respect to the mesh and anisotropy. An important outcome is that second order convergence is observed for all problems considered, even for highly deformed meshes. After this validation process, the method is applied to the prediction of the effective thermal conductivity of wood from its real 3D morphology. The property is estimated in the radial tangential and longitudinal directions. The solver is robust, efficient and yields to similar results compared to recent contributions.