Directional Derivatives Research Articles

Determination of the degree of crystallinity of polyphenylene sulfide composited with crystalline and non-crystalline fillers by applying the direct derivation method. Corrigendum

An erroneous equation and some values of related parameters in the paper by Toraya [J. Appl. Cryst. (2024), 57, 1115–1126] are corrected.

Imaging Enhancement of Gas Reservoir via Full-Waveform Inversion Imaging: Case Study, Gas Field

Abstract Seismic data are essential for analyzing and characterizing reservoir distributions in oil and gas exploration. While many advances have been made to obtain accurate seismic data, conventional methods still have limitations. The main limitations include the inability to use the full wavefield and the time-consuming sequential process of preprocessing, velocity model building, and migration. These challenges have highlighted the need for more efficient and accurate data processing. Full-waveform inversion (FWI) and FWI imaging have been developed to address these drawbacks. FWI uses the full wavefield, including diving and multi-scattered waves, to predict subsurface properties. It uses an optimization that iteratively updates the model to minimize the residual between the observed and the modeled data. Moreover, FWI imaging derives seismic images in a short time by taking the directional derivative of the FWI results. Compared with conventional methods, FWI imaging, since it is based on FWI, improves the signal-to-noise ratio and mitigates migration artifacts. In this study, a case study was conducted using 3D streamer data from a deepwater gas field to demonstrate the effectiveness of FWI imaging. As the gas field contains shallow gas, conventional methods have limitations in restoring subgas zones. To overcome this, FWI imaging was introduced, and to verify its effectiveness, reverse time migration (RTM) was used for comparison. After updating the velocity model with FWI up to 25 Hz, FWI imaging and RTM were applied using the same velocity model. Compared with RTM, FWI imaging exhibited significant improvements in resolution and structural detail at various depths. Additionally, it improved lateral resolution and structural continuity and effectively suppressed noise. Furthermore, FWI imaging significantly reduces processing time by eliminating the need for preprocessing and migration. These findings suggest that FWI imaging is a promising alternative to conventional methods by providing high-quality subsurface information while reducing the processing time.

Viscosity solutions of centralized control problems in measure spaces

This work focuses on a control problem in the Wasserstein space of probability measures over $\mathbb{R}^d$. Our aim is to link this control problem to a suitable Hamilton-Jacobi-Bellman (HJB) equation. We explore a notion of viscosity solution using test functions that are locally Lipschitz and locally semiconvex or semiconcave functions. This regularity allows to define a notion of viscosity and a Hamiltonian function relying on directional derivatives. Using a generalization of Ekeland's principle, we show that the corresponding HJB equation admits a comparison principle, and deduce that the value function is the unique solution in this viscosity sense. The PDE tools are developed in the general framework of Measure Differential Equations.

NECESSARY CONDITIONS FOR OPTIMALITY IN ONE NONSMOOTH OPTIMAL CONTROL PROBLEM FOR GOURSAT-DARBOUX SYSTEMS

We consider a nonsmooth optimal control problem described by a system of second-order hyperbolic equations with Goursat boundary conditions. A number of necessary optimality conditions are proved in terms of directional derivatives.

Multi-axial fatigue of high-strength concrete: Model-enabled interpretation of punch-through shear test response

This paper aims to fill the gap in multi-axial fatigue characterization of concrete by presenting a comprehensive numerical analysis of data from the punch-through shear test (PTST), which allows control over combined shear and compression loads in the tested ligament. To accurately reproduce the degradation process in this multi-axial configuration, a material model must capture two key effects: (i) dissipative behavior at the inter-aggregate level during subcritical pulsating loading and (ii) fatigue-induced tri-axial stress redistribution in the test ligament. The recently published MS1 model by the authors effectively incorporates these features, thereby enabling a deeper interpretation of the experimental data. The former effect is seamlessly integrated into the model using the Lemaitre-type fatigue hypothesis, facilitating the direct derivation of general evolution equations. The latter effect is efficiently represented through the microplane homogenization framework. The model’s predictions for PTST response under monotonic shear loading with varying levels of radial confinement align well with experimental results. Similarly, the simulated fatigue response accurately reproduces the experimentally observed S-N (Wöhler) curves. Further studies demonstrate the model’s ability to predict the effect of fatigue loading sequence. A thermodynamically-based formulation provides an energetic interpretation of this effect, offering a quantitative breakdown of energy dissipation attributed to individual dissipative mechanisms over the lifetime of the material. The studies show that damage-induced dissipation remains almost constant regardless of different fatigue loading histories.

Differential Geometry Using Quaternions

This paper establishes the basis of the quaternionic differential geometry (HDG) initiated in a previous article. The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and torsion concepts, differential forms, directional derivatives and the structural equations. The analogy between the quaternionic and the real geometries were obtained using a matrix representation of quaternions. The results evidences the quaternionic formalism as a suitable language to differential geometry that can be useful in various directions of future investigation.

Design of Input Signal for System Identification of a Generic Fighter Configuration

This article investigates the design of time-accurate input signals in the angle-of-attack and pitch rate space to identify the aerodynamic characteristics of a generic triple-delta wing configuration at subsonic speeds. Regression models were created from the time history of signal simulations in DoD HPCMP CREATETM-AV/Kestrel software. The input signals included chirp, Schroeder, pseudorandom binary sequence (PRBS), random, and sinusoidal signals. Although similar in structure, the coefficients of these regression models were estimated based on the specific input signals. The signals covered a wide range of angle-of-attack and pitch rate space, resulting in varying regression coefficients for each signal. After creating and validating the models, they were used to predict static aerodynamic data at a wide range of angles of attack but with zero pitch rate. Next, slope coefficients and dynamic derivatives in the pitch direction were estimated from each signal. These predictions were compared with each other as well as with the ONERA wind tunnel data and some CFD calculations from the DLR TAU code provided by the NATO Science and Technology Organization research task group AVT-351. Subsequently, the models were used to predict different pitch oscillations at various mean angles of attack with given amplitudes and frequencies. Again, the model predictions were compared with wind tunnel data. Final predictions involved responses to new signals from different models. A feed-forward neural network was then used to model pressure coefficients on the upper surface of the vehicle at different spanwise sections for each signal and the validated models were used to predict pressure data at different angles of attack. Overall, the models predict similar integrated forces and moments, with the main discrepancies appearing at higher angles of attack. All models failed to predict the stall behavior observed in the measurements and CFD data. Regarding the pressure data, the PRBS signal provided the best accuracy among all the models.

Second law and Liu relations: the no-reversible-direction axiom—revisited

AbstractA thermodynamic process is governed by balance equations in field-formulated thermodynamics. Especially the balance equation of entropy takes a prominent role: it introduces the Second Law in the form of a dissipation inequality via the non-negative entropy production. Balance equations and dissipation inequality are independent of the considered material which is described by additional constitutive equations which need the introduction of a state space which is spanned by the state space variables. Inserting these constitutive equations into the balance equations results in the balance equations on state space which include the first order time and position derivatives of the state space variables, called “higher derivatives” wich are directional derivatives in a mathematicle sense. Why do not appear the latter in the Liu Relations which pretend to describe material as well as the equations on state space do ? The answer is that the Liu Relations describe materials whose entropy production does not depend on the higher derivatives. Consequently, the Liu Relations are more specific than the balance equations on state space. A toy example concerning heat conduction in compressible fluids is in two different versions added for elucidation.

Primal characterizations of stability of error bounds for semi-infinite convex constraint systems in Banach spaces

This article focuses on the stability of error bounds, both local and global, for semi-infinite convex constraint systems in Banach spaces. We present primal characterizations of the stability of these error bounds under small perturbations. These characterizations are expressed through the directional derivatives of the functions that define the systems. It is shown that ensuring stability of the error bounds is closely tied to verifying that the optimal values of several minimax problems, formulated using the directional derivatives, remain outside a certain neighbourhood of zero. Furthermore, this stability condition only requires that all component functions of the system share the same linear perturbation. When applied to the sensitivity analysis of Hoffman's constants for semi-infinite linear systems, these stability results yield primal criteria that guarantee the uniform boundedness of Hoffman's constants under perturbations in the problem data.

A new approach to the directional derivative of fractional order

The fractional derivative approximations offer many approaches to understanding real‐world problems. The conformable fractional derivative operator that is one of the fractional derivative operators has recently attracted a lot of interesting. In this paper, a new approach to the directional derivative obtained with the help of the conformal fractional derivative is presented. Considering this approach, the definition of the fractional partial derivative is reformulated. In addition, a new definition of fractional gradient, fractional curl, and fractional divergence is given, and the properties of these new concepts are examined.

Enhancing seismic feature orientations: A novel approach using directional derivatives and Hilbert transform of gradient structure tensor

Seismic dip calculation serves as a widely employed technique in the realms of seismic interpretation and reservoir characterization, strategically employed to highlight faults and attributes within the seismic volume. Among the various methodologies utilized for estimating structural dip and azimuth, the Gradient Structure Tensor (GST) stands out. This approach involves leveraging the dominant eigenvector of the positive definite GST matrix to ascertain the inline and crossline dip of seismic data.In the initial phase of our innovative proposal, we employed the spectral balancing technique to enhance the fidelity of seismic data. Subsequently, leveraging this groundwork, we introduced an Analytical Directional Gradient Structure Tensor technique, a distinctive adaptation of GST. This novel approach involves the calculation of directive derivatives in both perpendicular and parallel directions to seismic features. By incorporating directive derivatives, our method excels in capturing subtle stratigraphic nuances, particularly in the dipping direction of interest. To validate the accuracy and effectiveness of our approach, we present compelling evidence through the examination of synthetic and real-field seismic volume outcomes. This underscores the robustness and reliability of our proposed method in enhancing the precision of seismic dip calculations and providing valuable insights into subsurface geological features.

Vector Field Model of CKD Stage and Its Directional Derivative Mathematically Enable Accurate Kidney Prognosis Prediction

Vector Field Model of CKD Stage and Its Directional Derivative Mathematically Enable Accurate Kidney Prognosis Prediction

A nonequilibrium statistical mechanics derivation of the hydrodynamic equations of simple fluids using a noncanonical form of the Poisson bracket

The continuum level hydrodynamic equations of a simple fluid were derived by Irving and Kirkwood directly from discrete particle dynamics using statistical mechanics almost 75 years ago. Their elegant derivation demonstrated the fundamental molecular basis of macroscopic fluid flow and culminated in molecular expressions for the stress tensor and heat current density that have since been employed in countless molecular simulations to date. In this article, an alternative derivation is presented, which leads to more general expressions for the fundamental transport relationships and which arrives at them in a more straightforward chain of consistency that ensues directly from the Principle of Least Action. The main point of departure from the Irving–Kirkwood derivation is the application of a transformation mapping of the total momentum of each individual particle onto the sum of its peculiar momentum and its momentum relative to the local velocity field. This mapping provides a phase-space distribution function applicable in the space of particle positions and peculiar momentum, from which a noncanonical Poisson bracket can be derived in terms of the same set of microscopic variables. For a given dynamic variable, expressed in terms of particle positions and peculiar momenta, the expectation value of the noncanonical Poisson bracket of the dynamic variable is shown to correspond to the evolution equation of the expectation value of the dynamic variable. This allows for a direct derivation of all macroscopic density evolution equations (mass, momentum, and energy density fields) using a systematic procedure free of assumptions concerning the macroscopic state of the system. Furthermore, an explicit expression of the time evolution of the entropy density at the hydrodynamic level is derived following the same procedure. Finally, in the limit of short-range interparticle interactions, a molecular-based expression for the local stress tensor as properly defined from continuum mechanics is developed at the hydrodynamic level that elucidates the continuum mechanics connection of the general stress expression of Irving and Kirkwood.

A Combined Quasi-Newton-Type Method Using 4th-Order Directional Derivatives for Solving Degenerate Problems of Unconstrained Optimization

Introduction. Methods of unconstrained optimization play a significant role in machine learning [1–6]. When solving practical problems in machine learning, such as tuning nonlinear regression models, the extremum point of the chosen optimality criterion is often degenerate, which significantly complicates its search. Therefore, degenerate problems are the most challenging in optimization. Known numerical methods for solving the general unconstrained optimization problem, up to the second order, have very low convergence rates when solving degenerate problems [7, 8]. This is explained by the fact that for a significant improvement in convergence rate in this case, it is necessary to use higher-order derivatives in the method than the second order [10]. The purpose of the paper is to develop an efficient quasi-Newton method for solving degenerate unconstrained optimization problems, the idea of which (unlike regularization) involves dividing the entire space into the sum of two orthogonal subspaces. This idea was introduced in [23]. The space division at each iteration of the method is based on the spectral decomposition of the matrix approximating the Hessian of the objective function using the BFGS formula [3]. Each subspace exhibits its own behavior of the objective function, and therefore, an appropriate minimization method is applied on it. Results. A combined quasi-Newton method is presented for solving degenerate unconstrained optimization problems, based on orthogonal decomposition of the Hessian approximation matrix and division of the entire space into the sum of two orthogonal subspaces. On one subspace (the kernel of the Hessian approximation matrix), a method is applied where derivatives in the direction of the 4th order are computed, while on the orthogonal complement to it, a quasi-Newtonian method is applied. A separate one-dimensional search is applied on each of these subspaces to determine the step multiplier in the respective direction. The effectiveness of the presented combined method is confirmed by numerical experiments conducted on widely accepted test functions for unconstrained optimization problems. The proposed method allows obtaining fairly accurate solutions to test tasks in case of degeneracy of the minimum point with significantly lower computational costs for gradient calculations compared to optimization procedures of well-known mathematical packages. Conclusions. The idea of dividing the entire space into the sum of two (or possibly more) orthogonal subspaces when solving complex optimization problems is quite promising in terms of applying a combination of different numerical methods on separate subspaces. In the future, it is planned to conduct theoretical research on the convergence rate of the presented combined method in solving degenerate unconstrained optimization problems. Keywords: unconstrained optimization, quasi-Newton methods, degenerate minimum point, spectral matrix decomposition, Machine Learning.

Predicting ELNES/XANES spectra by machine learning with an atomic coordinate-independent descriptor and its application to ground-state electronic structures

ELNES/XANES spectra can be observed using TEM or synchrotron radiation and can elucidate the unoccupied state electronic structures of an excited states. The computation of their features is usually demanding substantial computational resources due to the requisite structure optimization and electronic structure calculations. Herein, we leverage a machine learning technique alongside an atomic-coordinate-independent descriptor, SMILES, to yield the ELNES/XANES spectra, directly, with heightened precision. Moreover, our approach extends to obtain ground state electronic structure, namely PDOS at both occupied and unoccupied ground states, underscoring its viability for a ground-state spectroscopy. Our study revealed that incorporation of long-SMILES molecules into the training dataset enhances prediction accuracy for such molecular structures. This study's direct derivation of spectroscopy from SMILES strings holds promise for expediting spectroscopic inquiries.

Adaptive ray tracing in freeform gradient-index media using an index directional derivative.

Freeform gradient index (F-GRIN) media are increasingly used in optical systems. The existing ray tracing methods for F-GRIN have low calculation efficiency and require manual adjustment of step size, making the design process complex. To address this, we propose an adaptive ray tracing (ART) method for F-GRIN. ART obtains an initial step size based on an index directional derivative and the element's length. During the subsequent transmission process, the step size is adaptively adjusted. The accuracy and speed of ART were verified through multiple comparison tests. ART can reduce manual participation and significantly improve efficiency in the optical design of F-GRIN.

Gravity and magnetism of southern Africa in relation to Craton structures and belts

The general tectonic structural architecture of southern Africa is an ensemble comprised of Kalahari Craton, a vast range of mosaics of the best-preserved geological Belts, and exposed crustal blocks. In the region, demarcation of geological boundaries, tracing of magnetic and gravitational bodies, and detrending lineaments are essential to understanding the structural limits. The study presents the prevalent gravitational and magnetic investigation of major geological Belts, enhancing edges of Cratons, and evaluation of surface invariants to improve the geological understanding and evaluate the correlation of structures with the general structural tectonic framework. The utilized filter methods in the gravity maps are the Directional derivatives along x, y, and z-directions, the Modulus tensor, the Total horizontal derivative, and the Tilt angle techniques.The phase-based filters used in the magnetic section include the Total horizontal derivative, Analytical signal, Tilt angle method, Tilt of total horizontal derivative, Theta method, Horizontal tilt angle method, Enhanced tilt filter, and Enhanced total derivative of the tilt angle. The Directional derivatives of the gravity field and the Modulus of the gravity gradient tensor demarcate the boundaries of geological structures. The Total horizontal derivative method gives an immediate and easy-to-read image of the linear structures and fault systems. The signal cluster on the Analytical signal and Tilt angle maps delineate the boundaries of causative geological sources. The Theta map is comparable with the enhanced total derivative of the tilt angle. The Tilt of the total horizontal derivative, Theta map, and Enhanced tilt filter accentuate the traces of magnetic bodies, including the Cratons. The study finds regional lineaments surrounding the Cratons and concentrated along the geological Belts. The approach of using joint elegant filters is effective and increases the certainty of the interpretation.

Improving the Sensitivity of Task-Based Multi-Echo Functional Magnetic Resonance Imaging via T2* Mapping Using Synthetic Data-Driven Deep Learning.

(1) Background: Functional magnetic resonance imaging (fMRI) utilizing multi-echo gradient echo-planar imaging (ME-GE-EPI) has demonstrated higher sensitivity and stability compared to utilizing single-echo gradient echo-planar imaging (SE-GE-EPI). The direct derivation of T2* maps from fitting multi-echo data enables accurate recording of dynamic functional changes in the brain, exhibiting higher sensitivity than echo combination maps. However, the widely employed voxel-wise log-linear fitting is susceptible to inevitable noise accumulation during image acquisition. (2) Methods: This work introduced a synthetic data-driven deep learning (SD-DL) method to obtain T2* maps for multi-echo (ME) fMRI analysis. (3) Results: The experimental results showed the efficient enhancement of the temporal signal-to-noise ratio (tSNR), improved task-based blood oxygen level-dependent (BOLD) percentage signal change, and enhanced performance in multi-echo independent component analysis (MEICA) using the proposed method. (4) Conclusion: T2* maps derived from ME-fMRI data using the proposed SD-DL method exhibit enhanced BOLD sensitivity in comparison to T2* maps derived from the LLF method.

Direct In‐Situ Estimates of Energy and Force Balance Associated With Magnetopause Reconnection

AbstractFundamental processes in plasmas act to convert energies into different forms, for example, electromagnetic, kinetic and thermal. Direct derivation from the Vlasov‐Maxwell equation yields sets of equations that describe the temporal evolution of magnetic, kinetic and internal energies in either the monofluid or multifluid frameworks. In this work, we focus on the main terms affecting the changes in kinetic energy. These are pressure‐gradient‐related terms and electromagnetic terms. The former account for plasma acceleration/deceleration from a pressure gradient, while the latter from an electric field. Although limited spatial and temporal deviations are expected, a statistical balance between these terms is fundamental to ensure the overall conservation of energy and momentum. We use in‐situ observations from the Magnetospheric MultiScale (MMS) mission to study the relationship between these terms. We perform a statistical analysis of those parameters in the context of magnetic reconnection by focusing on small‐scale Electron Diffusion Regions and large‐scale Flux Transfer Events. The analysis reveals a correlation between the two terms in the monofluid force balance, and in the ion force and energy balance. However, the expected relationship cannot be verified from electron measurements. Generally, the pressure‐gradient‐related terms are smaller than their electromagnetic counterparts. We perform an error analysis to quantify the expected underestimation of gradient values as a function of the spacecraft separation compared to the gradient scale. Our findings highlight that MMS is capable of capturing energy and force balance for the ion fluid, but that care should be taken for energy conversion terms based on electron pressure gradients.

A new regularity criterion for the 3D nematic liquid crystal flows

The aim of this study was to establish a new regularity criterion for local-in-time smooth solutions for the 3D nematic liquid crystal flows in terms of the horizontal gradient of two horizontal velocity components in the framework of the homogeneous Besov [Formula: see text] space and one directional derivative of molecular orientations in the framework of the homogeneous Morrey–Campanato [Formula: see text] space. More precisely, we show that the smooth solution [Formula: see text] can be extended beyond [Formula: see text], provided that [Formula: see text]