Higher-order Partial Derivatives Research Articles

A Spectral Element Solution of the Poisson Equation with Shifted Boundary Polynomial Corrections: Influence of the Surrogate to True Boundary Mapping and an Asymptotically Preserving Robin Formulation

Abstract We present a new high-order spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order hp-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series, which traditionally is used within the shifted boundary method. Here, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small cut cells. Dirichlet, Neumann, and Robin boundary conditions are enforced weakly through a generalized: (i) Nitsche’s method and (ii) Aubin’s method. A consistent asymptotic preserving formulation of the embedded Robin formulations is presented. Several experiments and analyses of the numerical properties of the various weak forms are showcased. We include convergence studies under polynomial increase of the basis functions, p, mesh refinement, h, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. With this, we assess the influence of errors across variational forms, polynomial order, mesh size, and mappings between the true and surrogate boundaries.

New Weighted Inequalities for Functions Whose Higher-Order Partial Derivatives Are Co-Ordinated Convex

The purpose of this study is to establish recent inequalities based on double integrals of mappings whose higher-order partial derivatives in absolute value are convex on the co-ordinates on rectangle from the plane. Also, some special cases of results improved in this study are examined.

Sixth-order hybrid finite difference methods for elliptic interface problems with mixed boundary conditions

In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem −∇⋅(a∇u)=f in Ω﹨Γ, where Γ is a smooth interface inside Ω. The variable scalar coefficient a>0 and source f are possibly discontinuous across Γ. The hybrid FDMs utilize a 9-point compact stencil at any interior regular points of the grid and a 13-point stencil at irregular points near Γ. For interior regular points away from Γ, we obtain a sixth-order 9-point compact FDM satisfying the sign and sum conditions for ensuring the M-matrix property. We also derive sixth-order compact (4-point for corners and 6-point for edges) FDMs satisfying the sign and sum conditions for the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. Thus, for the elliptic problem without interface (i.e., Γ is empty), our compact FDM has the M-matrix property for any mesh size h>0 and consequently, satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. For irregular points near Γ, we propose fifth-order 13-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems. Theoretically, the proposed high order FDMs use high order (partial) derivatives of the coefficient a, the source term f, the interface curve Γ, the two jump functions along Γ, and the functions on ∂Ω. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our proposed FDMs are independent of the choice representing Γ and are also applicable if the jump conditions on Γ only depend on the geometry (e.g., curvature) of the curve Γ. Our numerical experiments confirm the sixth-order convergence in the l∞ norm of the proposed hybrid FDMs for the elliptic interface problem.

HYPAD-UQ: A Derivative-Based Uncertainty Quantification Method Using a Hypercomplex Finite Element Method

Abstract A derivative-based uncertainty quantification (UQ) method called HYPAD-UQ that utilizes sensitivities from a computational model was developed to approximate the statistical moments and Sobol' indices of the model output. Hypercomplex automatic differentiation (HYPAD) was used as a means to obtain accurate high-order partial derivatives from computational models such as finite element analyses. These sensitivities are used to construct a surrogate model of the output using a Taylor series expansion and subsequently used to estimate statistical moments (mean, variance, skewness, and kurtosis) and Sobol' indices using algebraic expansions. The uncertainty in a transient linear heat transfer analysis was quantified with HYPAD-UQ using first-order through seventh-order partial derivatives with respect to seven random variables encompassing material properties, geometry, and boundary conditions. Random sampling of the analytical solution and the regression-based stochastic perturbation finite element method were also conducted to compare accuracy and computational cost. The results indicate that HYPAD-UQ has superior accuracy for the same computational effort compared to the regression-based stochastic perturbation finite element method. Sensitivities calculated with HYPAD can allow higher-order Taylor series expansions to be an effective and practical UQ method.

Gradient free stochastic training of ANNs, with local approximation in partitions

We present a numerical scheme for computation of Artificial Neural Networks (ANN) weights, which stems from the Universal Approximation Theorem, avoiding costly iterations. The proposed algorithm adheres to the underlying theory, is highly fast, and results in remarkably low errors when applied to regression and classification problems of complex data sets with {textbf{x}} in {mathbb {R}}^{n} (e.g. Griewank, Gomez-Levy, Shekel, and Polynomial functions) with random noise addition (i.e. Uniform, Normal, Generalized Pareto, Log-Normal, and a mixture of Log-Normal, Exponential, and Frechet), as well as the database for handwritten digits recognition MNIST (Modified National Institute of Standards and Technology) with 7times 10^4 images. The same mathematical formulation was found capable of approximating highly nonlinear functions in multiple dimensions, with low errors (e.g. 10^{-10}) for the test set of the unknown functions, their higher-order partial derivatives, as well as numerically solving Partial Differential Equations, such as those appearing in Physics, Engineering, Environmental Sciences, etc. The method is based on the calculation of the weights of each neuron in small neighbourhoods of the data. Accordingly, optimization of hyperparameters is not necessary, as the number of neurons stems directly from the dimensionality of the data, further improving the algorithmic speed. Under this setting, overfitting is inherently avoided, and the results are interpretable and reproducible. The complexity of the proposed algorithm is of class P with {mathcal {O}}(mNni_{cl} + Nmn^2+Nn^3 + mN^2+N^3) computing time, with respect to the observations m, features n, and Neurons N, contrary to the NP-Complete class of standard algorithms for ANN training. The performance of the method is high, irrespective of the size of the data set, and the test set errors are similar or smaller than the training errors, indicating the generalization efficiency of the algorithm. A supplementary computer code in Julia and Python Languages is provided, which can be used to reproduce the validation examples, and/or apply the algorithm to other data sets.

Least‐squares reverse time migration based on an efficient pure qP‐wave equation

AbstractSeismic waves propagating in anisotropic earth media suffer from waveform distortion, which leads to degraded resolution in seismic data imaging if this adverse effect is not corrected. Additionally, the wavefields simulated by the traditional coupled pseudo‐acoustic wave equations contain undesired shear wave artefacts and are unstable when Thomsen's anisotropy parameter ε is less than δ. To address these issues, many pure qP‐wave equations are proposed to describe wave propagation in anisotropic media. However, these equations are either low precision or high accuracy but computationally expensive. Considering the trade‐off between the computation efficiency and simulation accuracy, we derive a pure qP‐wave equation from the exact dispersion formula in tilted transverse isotropic media. The newly derived equation has only two higher order partial derivatives, which can better balance the accuracy and efficiency than the previous pure qP‐wave equations. The least‐squares reverse time migration method can balance amplitudes, suppress migration artefacts and improve imaging resolution. Then, based on the newly derived pure qP‐wave equation, we derive the Born modelling operator and adjoint migration operator and develop an anisotropic least‐squares reverse time migration method. To achieve the accelerated convergence, we introduce a generalized minimum residual algorithm to implement the anisotropic least‐squares reverse time migration method. Numerical simulation results show that the proposed pure qP‐wave equation outperforms the previous equations in balancing the trade‐off between accuracy and efficiency. In addition, synthetic examples demonstrate the effectiveness and robustness of our proposed anisotropic least‐squares reverse time migration method in correcting the deviations due to anisotropic effects.

BEM-Based Magnetic Field Reconstruction by Ensemble Kálmán Filtering

Abstract Magnetic fields generated by normal or superconducting electromagnets are used to guide and focus particle beams in storage rings, synchrotron light sources, mass spectrometers, and beamlines for radiotherapy. The accurate determination of the magnetic field by measurement is critical for the prediction of the particle beam trajectory and hence the design of the accelerator complex. In this context, state-of-the-art numerical field computation makes use of boundary-element methods (BEM) to express the magnetic field. This enables the accurate computation of higher-order partial derivatives and local expansions of magnetic potentials used in efficient numerical codes for particle tracking. In this paper, we present an approach to infer the boundary data of an indirect BEM formulation from magnetic field measurements by ensemble Kálmán filtering. In this way, measurement uncertainties can be propagated to the boundary data, magnetic field and potentials, and to the beam related quantities derived from particle tracking. We provide results obtained from real measurement data of a curved dipole magnet using a Hall probe mapper system.

Inverse differential quadrature method for structural analysis of composite plates

A novel two-dimensional inverse differential quadrature method is proposed to approximate the solution of high-order system of differential equations. A critical aspect of the proposed scheme is to circumvent the error arising from high sensitivity to noise associated with high-order numerical differentiation operations during direct approximation. A general framework for approximating arbitrary functions from high-order partial derivatives is developed and a comprehensive insight for the implementation of different orders of the analysis within the context of first-order shear deformation theory kinematical assumptions governing laminated plates bending and buckling behaviours is presented. Concerning buckling analysis, a Moore–Penrose pseudo-inverse preconditioning procedure is further proposed to formulate the eigenvalue problem, since the developed algebraic expressions constitute an underdetermined system. According to the numerical tests for bending and buckling analyses of laminated plates under different loading and boundary conditions, the accuracy of current solutions compares satisfactorily with Navier’s solution, differential quadrature method and commercial finite element analysis models. Furthermore, the developed method demonstrates the potential for improved convergence in comparison with differential quadrature solutions highlighting its computational merits.

Stochastic emulation of quantum algorithms

Quantum algorithms profit from the interference of quantum states in an exponentially large Hilbert space and the fact that unitary transformations on that Hilbert space can be broken down to universal gates that act only on one or two qubits at the same time. The former aspect renders the direct classical simulation of quantum algorithms difficult. Here we introduce higher-order partial derivatives of a probability distribution of particle positions as a new object that shares these basic properties of quantum mechanical states needed for a quantum algorithm. Discretization of the positions allows one to represent the quantum mechanical state of n bit qubits by 2(n bit + 1) classical stochastic bits. Based on this, we demonstrate many-particle interference and representation of pure entangled quantum states via derivatives of probability distributions and find the universal set of stochastic maps that correspond to the quantum gates in a universal gate set. We prove that the propagation via the stochastic map built from those universal stochastic maps reproduces up to a prefactor exactly the evolution of the quantum mechanical state with the corresponding quantum algorithm, leading to an automated translation of a quantum algorithm to a stochastic classical algorithm. We implement several well-known quantum algorithms, analyse the scaling of the needed number of realizations with the number of qubits, and highlight the role of destructive interference for the cost of the emulation.

Dynamic texture representation based on oriented magnitudes of Gaussian gradients

Efficiently capturing shape and turbulent motions of dynamic textures (DTs) for video description is a challenge in real applications due to the negative influences of the well-known problems: environmental elements, illumination, scale, and noise. In this paper, we propose an efficient and simple framework for DT representation based on the oriented features of high-order Gaussian gradients. Firstly, 2D/3D Gaussian-based filtering kernels in high-order partial derivatives are taken into account the video analysis as a preprocessing step to obtain corresponding gradient-filtered images/volumes. After that, the oriented features, which are robust against the above issues, are extracted by decomposing the Gaussian derivative magnitudes into oriented components. Finally, a shallow local encoding is utilized for structuring spatio-temporal features from these oriented magnitudes. This allows constructing discriminative descriptors with promising performances compared to those based on the non-oriented ones. Experimental results for DT classification task on benchmark datasets have verified the interest of our proposal.

A deep artificial neural network architecture for mesh free solutions of nonlinear boundary value problems

Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation.

Bivariate Infinite Series Solution of Kepler’s Equations

A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure for the analytical computation of all the higher-order partial derivatives of the eccentric anomaly with respect to the eccentricity e and mean anomaly M in a given base point (ec,Mc) of the (e,M) plane. Explicit examples of such bivariate infinite series are provided, corresponding to different choices of (ec,Mc), and their convergence is studied numerically. In particular, the polynomials that are obtained by truncating the infinite series up to the fifth degree reach high levels of accuracy in significantly large regions of the parameter space (e,M). Besides their theoretical interest, these series can be used for designing 2-D spline numerical algorithms for efficiently solving Kepler’s equations for all values of the eccentricity and mean anomaly.

Approximation by multivariate generalized Poisson–Cauchy type singular integral operators

This research and survey work deals exclusively with the study of the approximation of generalized multivariate Poisson–Cauchy type singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. These operators are not in general positive linear operators. In particular we study the rate of convergence of these integral operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these integral operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of approximation. The above properties are studied with respect to $$L_{p}$$ norm, $$1\le p\le \infty $$ .

Complete Approximations by Multivariate Generalized Gauss-Weierstrass Singular Integrals

Abstract This research and survey article deals exclusively with the study of the approximation of generalized multivariate Gauss-Weierstrass singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. The multivariate generalized Gauss-Weierstrass operators are not in general positive linear operators. In particular we study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of multivariate approximation. The above properties are studied with respect to Lp norm, 1 ≤ p ≤ ∞.

Some Weighted Inequalities for Higher-Order Partial Derivatives in Two Dimensions and Its Applications

We establish some Ostrowski type inequalities involving higher-order partial derivatives for two-dimensional integrals on Lebesgue spaces (L_{∞}, L_{p} and L₁). Some applications in Numerical Analysis in connection with cubature formula are given. Finally, with the help of obtained inequality, we establish applications for the kth moment of random variables.

Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables

Abstract In this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-valued functions analytic in the unit ball B 2 = { z ∈ C 2 : | z | = | z 1 | 2 + | z 2 | 2 < 1 } , $\begin{array}{} \mathbb{B}^2\! = \!\{z\!\in\!\mathbb{C}^2: |z|\! = \!\small\sqrt{|z_1|^2+|z_2|^2}\! \lt \! 1\}, \end{array} $ where L = (l 1, l 2): 𝔹2 → R + 2 $\begin{array}{} \mathbb{R}^2_+ \end{array} $ is a positive continuous vector-valued function. Particularly, we deduce analog of Hayman’s theorem for this class of functions. The theorem shows that in the definition of boundedness of L-index in joint variables for vector-valued functions we can replace estimate of norms of all partial derivatives by the estimate of norm of (p + 1)-th order partial derivative. This form of criteria could be convenient to investigate analytic vector-valued solutions of system of partial differential equations because it allow to estimate higher-order partial derivatives by partial derivatives of lesser order. Also, we obtain sufficient conditions for index boundedness in terms of estimate of modulus of logarithmic derivative in each variable for every component of vector-valued function outside some exceptional set by the vector-valued function L(z).

Performance Analysis of Parameter Estimator Based on Closed-Form Newton Method for Ultrawideband Positioning

Ultrawideband (UWB) is considered as a promising technology in indoor positioning due to its accurate time resolution and good penetration through objects. Since the functional model of UWB positioning is nonlinear, iterative algorithms are often considered for solving the nonlinear problem. With a rough initial value, we can obtain the optimal solution by the way of continuous iteration. As an iterative descent method of high efficiency, the Gauss–Newton method is widely used to estimate the position. However, in the UWB indoor positioning, since the location information of the user is rather limited, it is not easy to get a good initial value for iteration. Besides, the positioning system is prone to become ill-posed. These factors make the Gauss–Newton method not easy to converge to a global optimal solution or even diverge, especially under ill-conditioned positioning configuration. Furthermore, the linearization of the positioning functional model results in biased least-squares estimation. The Gauss–Newton method only includes the first-order Taylor expansion of distance equations. The bias comes from neglected higher order terms. In this study, the closed-form Newton method which considers the high order partial derivatives is introduced and proposed. The simulation and measurement experiments are implemented to analyze the performance of the closed-form Newton method in UWB positioning. Both the initial value factor and geometric configuration factor are discussed. Experimental results are given to demonstrate that the closed-form Newton method can converge to the global optimal solution with better convergence and higher efficiency regardless of whether the initial iteration value is reasonable or not. Meanwhile, the closed-form Newton method can improve the accuracy of positioning results, particularly when the anchors are not uniformly and symmetrically distributed, or the ranging error is relatively large. The study shows that closed-form of the Newton method has better convergence and positioning performance than the Gauss–Newton method, especially under ill-conditioned positioning configuration or relatively low measurement precision, even though it adds a little computational cost.

Prominent Local Representation for Dynamic Textures Based on High-Order Gaussian-Gradients

Understanding dynamic textures (DTs) is a challenge in various computer vision applications due to the negative impacts of noise, changes of environment, illumination, and scales on capturing turbulent characteristics. In this work, we propose an efficient shallow framework for DT representation by addressing the following novel concepts. First, it is the first time in DT analysis that 2D/3D Gaussian-gradient filterings are taken into account as a pre-processing step to point out robust components against those influences in effect. Second, high-order partial derivatives of the Gaussian kernels and their informative magnitudes are exploited to forcefully capture multi-order Gaussian-gradient features. Third, these gradient kernels are investigated in multi-scale analysis of different orders and standard deviations in order to enrich more useful scale-gradient information. Finally, the obtained complementary components are shallowly encoded using a simple local operator to construct robust descriptors of High-order 2D/3D Gaussian-gradient-based Features ( $\mathrm{HoGF}^{2D/3D}$ ) against the well-known issues of DT description. Experiments for DT classification on various benchmarks have validated the interest of our approach since its performance is comparable to state-of-the-art results, including that of deep-learning methods, while it only has a small dimension.

Seidel aberration coefficients: an alternative computational method.

An alternative method to compute Seidel aberrations is presented that utilizes real pseudoparaxial skew rays traveling through a rotationally-symmetric lens system as a Taylor series expansion in terms of object height and ray-direction spherical coordinates. Expressions for defocus, lateral magnification, and Seidel primary ray aberration coefficients are obtained in terms of numerically-determined higher-order partial derivatives of rays which are proximate to the optical axis. In contrast to commonly used methods, the new form of the aberration coefficients is related to unit entrance pupil radius and unit object height. The proposed methodology can be extended to derive higher-order ray aberration coefficients.

Omission Error Analysis in Gravity Gradient Measurement

Gravity gradient plays an important role in many fields of science, and many methods are used to achieve the measurement of it. To improve measurement accuracy, various error analyses have been conducted in previous studies about positioning and orientation errors and system noise, among others. However, knowledge on the influence of omission errors from the theoretical models of gravity gradient measurement is limited. In this paper, we investigated omission errors in gravity gradient measurement, which was accomplished with the principle of differential acceleration. First, we determined the source of the omission errors to be the omission of high-order terms. Second, we calculated these terms on the basis of the Earth Gravitational Model 2008. Specifically, the expression of the partial derivative of the high order for the gravity potential in the spherical coordinates and the recursive equations for the high-order partial derivatives of the Legendre function were derived. Moreover, we transformed these high-order terms from the spherical coordinate system to the local north-oriented frame. The analysis led to three findings. First, a positive correlation was found between the omission errors and the distance between two measuring points. Second, the influences of the omission errors varied across different regions. Third, $\Gamma _{zz} $ was the least affected by the omission errors among the components $\Gamma _{xz} $ , $\Gamma _{yz} $ , and $\Gamma _{zz} $ . In conclusion, our study demonstrates that omission errors affect gravity gradient measurement.