Numerical Integration Error Research Articles

Intractability results for integration in tensor product spaces

We prove lower bounds on the worst-case error of numerical integration in tensor product spaces. The information complexity is the minimal number N of function evaluations that is necessary such that the N-th minimal error is less than a factor ε times the initial error, i.e., the error for N=0, where ε belongs to (0,1). We are interested to which extent the information complexity depends on the number d of variables of the integrands. If the information complexity grows exponentially fast in d, then the integration problem is said to suffer from the curse of dimensionality.Under the assumption of the existence of a worst-case function for the uni-variate problem, we present two methods for providing lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function and can be seen as a generalization of the method of decomposable reproducing kernels. The second method, although only applicable for positive quadrature rules, does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. Several applications of both methods are presented.

Space-filling designs on Riemannian manifolds

This paper proposes a new approach to generating space-filling designs over Riemannian manifolds by using a Hilbert curve. Different from ordinary Euclidean spaces, a novel transformation is constructed to link the uniform distribution over a Riemannian manifold and that over its parameter space. Using this transformation, the uniformity of the design points in the sense of Riemannian volume measure can be guaranteed by the intrinsic measure preserving property of the Hilbert curve. It is proved that these generated designs are not only asymptotically optimal under minimax and maximin distance criteria, but also perform well in minimizing the Wasserstein distance from the target distribution and controlling the estimation error in numerical integration. Furthermore, an efficient algorithm is developed for numerical generation of these space-filling designs. The advantages of the new approach are verified through numerical simulations.

Inf-sup neural networks for high-dimensional elliptic PDE problems

Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen the emergence of neural PDE solvers, leveraging deep networks to effectively tackle high dimensional PDE problems. This study introduces Inf-SupNet, a model-based unsupervised learning approach designed to acquire solutions for a specific category of elliptic PDEs. The fundamental concept behind Inf-SupNet involves incorporating the inf-sup formulation of the underlying PDE into the loss function. The analysis reveals that the global solution error can be bounded by the sum of three distinct errors: the numerical integration error, the duality gap of the loss function (training error), and the neural network approximation error for functions within Sobolev spaces. To validate the efficacy of the proposed method, numerical experiments conducted in high dimensions demonstrate its stability and accuracy across various boundary conditions, as well as for both semi-linear and nonlinear PDEs.

Transient analyses of underwater acoustic propagation with the modified meshfree radial point interpolation method and newmark time integration techniques

Underwater acoustic propagation plays a very important role in underwater communication and ocean engineering field. The radial point interpolation method (RPIM) is an effective numerical approach to solve underwater acoustic propagations though considerable numerical errors can always be caused due to the discontinuously differentiable nodal interpolation function in one integration cell. The origin of this numerical error primarily comes from the mismatch between the background integration cell and the nodal interpolation function support. To address this issue, a modified meshfree RPIM is developed in this work for transient analyses of underwater acoustic propagations. In the proposed modified RPIM, the construction of the interpolation function supports are dependent on the integration cells. In consequence, in one background cell all the quadrature points share one mutual support domain and the continuously differentiable interpolation functions can be obtained. Therefore, the numerical integration errors can be evidently decreased and the related numerical dispersion effects in wave analysis can be largely suppressed. Owing to the adequately small spatial dispersion errors from the proposed modified RPIM, it is observed that the modified RPIM with the Newmark time domain discretization scheme can monotonically improve the solution precision by directly reducing the temporal discretization interval. This property makes the modified RPIM clearly outperform the original RPIM and is especially suitable for handling very complex underwater acoustic propagations.

NIRK-based mixed-type accurate continuous–discrete Gaussian filters with deterministically sampled expectation and covariance for state estimation in continuous-time stochastic process models with discrete measurements

This paper advances further the idea of overall continuous–discrete Gaussian filtering with deterministically sampled expectation and covariance towards efficient mixed-type accurate Kalman-like schemes based on self-regulated Ordinary Differential Equation (ODE) solvers with automatic local and global error controls intended for treating continuous-time nonlinear stochastic process models with discrete measurements. The universal state estimation algorithms of such sort designed recently by Kulikov and Kulikova (2022) possess three potential limitations, which might affect their successful applications in practice. First, these deal with complicated Moment Differential Equations (MDEs) and, hence, can be rather time-consuming in estimating stochastic process models of large size, that compromises their efficiency for online computations. Second, the aforementioned filters do not employ any kind of global error regulation and, hence, can be insufficiently accurate in solving difficult MDEs arisen. Third, these are hardly effective in treating stiff continuous–discrete stochastic systems. Below, all the listed issues are addressed and solved via utilization of the much simpler but more robust MDEs arisen in the usual continuous–discrete Extended Kalman Filter (EKF) within the mixed-type accurate filtering approach initiated by Kulikova and Kulikov (2015), which is rooted in the variable-stepsize Nested Implicit Runge–Kutta (NIRK) methods. For the sake of stability of our novel methods to the numerical integration and round-off errors, we devise square-root versions of the mixed-type Gaussian filters, which are based either on hyperbolic QR factorizations or on hyperbolic Singular Value Decompositions (SVDs). Performances of these new mixed-type square-root Gaussian filters are assessed and compared to their original (non-mixed-type) counterparts as well as to their non-square-root versions in a simulated non-stiff ill-conditioned radar tracking scenario and on a stiff stochastic Oregonator reaction.

ОПТИМАЛЬНЕ ОБЧИСЛЕННЯ ІНТЕГРАЛІВ ВІД ШВИДКООCЦИЛЮВАЛЬНИХ ФУНКЦІЙ ДЛЯ ДЕЯКИХ КЛАСІВ ДИФЕРЕНЦІЙОВНИХ ФУНКЦІЙ

The author considers the problem of calculating integrals of rapidly oscillating functions from some classes of differential functions, in particular, in the case of the interpolation class of functions, where the information operator is specified by a fixed table of its values. Quadrature formulas for calculating integrals of rapidly oscillating functions have been constructed that are optimal in terms of accuracy and optimal in order of accuracy. The optimal estimates for the error of the method are obtained. Keywords: integrals of rapidly oscillating functions, interpolation classes of functions, quadrature formulas optimal in terms of accuracy, method of boundary functions, lower estimate of numerical integration error.

Implementation, Realization and an Effective Solver of Two-Equation Turbulence Models

Currently, when the Reynolds-Averaged Navier–Stokes (RANS) equations are solved using turbulence modelling, most often the one-equation model of Spalart and Allmaras is used. Then, it is only necessary to solve the RANS equations in conjunction with a single transport equation for modeling turbulence. For this model, considerable assessment and analysis has been performed, allowing the possibility of a reliable solution method for an eddy viscosity required to compute the Reynolds stresses in the RANS equations. Such evaluation along with analysis has not been performed for similar performance with two-equation models of the k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document}-ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} type. The primary objective of this paper is to present and discuss the components of an effective numerical algorithm for solving the RANS equations and the two transport equations of k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document}-ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} type turbulence models. All the important details of the turbulence model as actually implemented are given, which is sometimes not done in various papers considering such modeling. The viability and effectiveness of this solution algorithm are demonstrated by solving both two-dimensional and three-dimensional aerodynamic flows. In all applications, a linear rate of convergence without oscillations or other evidence of unstable behavior is observed. This behavior is also particularly true when the proposed algorithm is applied to systematically refined mesh sequences, which is generally not observed with algorithms solving more than one transport equation. Thus numerical integration errors are systematically reduced, allowing for a significantly more reliable assessment of the effectiveness of the model itself. Additionally, in this paper analysis of the solution algorithm, including linear stability, is also performed for a particular flow problem.

Analytic solution of chemical evolution models with Type Ia supernovae

Context. In recent years, a significant number of works have been focussed on finding analytic solutions for the chemical enrichment models of galactic systems, including the Milky Way. Some of these solutions, however, are not able to account for the enrichment produced by Type Ia supernovae (SNe) due to the presence of the delay time distributions (DTDs) in the models. Aims. We present a new analytic solution for the chemical evolution model of the Galaxy. This solution can be used with different prescriptions of the DTD, including the single- and double-degenerate scenarios, and allows for the inclusion of an arbitrary number of pristine gas infalls. Methods. We integrated the chemical evolution model by extending the instantaneous recycling approximation with the contribution of Type Ia SNe. This implies an extra term in the modelling that depends on the DTD. For DTDs that lead to non-analytic integrals, we describe them as a superposition of Gaussian, exponential, and 1/t functions using a restricted least-squares fitting method. Results. We obtained the exact solution for a chemical model with Type Ia SNe widely used in previous works, while managing to avoid numerical integration errors. This solution is able to reproduce the expected chemical evolution of the α and iron-peak elements in less computing time than numerical integration methods. We compare the pattern in the [Si/Fe] versus [Fe/H] plane observed by APOGEE DR17 with that predicted by the model. We find the low α sequence can be explained by a delayed gas infall. We exploit the applicability of our solution by modelling the chemical evolution of a simulated Milky Way-like galaxy from its star formation history. The implementation of our solution has been released as a PYTHON package. Conclusions. Our solution constitutes a promising tool for Galactic archaeology studies and it is able to model the observed trends in α element abundances versus [Fe/H] in the solar neighbourhood. We infer the chemical information of a simulated galaxy modelled without chemistry.

Fast and accurate computation of polar harmonic Fourier moments for image description.

Continuous orthogonal moments are widely used in various image techniques due to their simplicity and good rotational invariance and stability. In recent years, numerous excellent continuous orthogonal moments have been developed, among which polar harmonic Fourier moments (PHFMs) exhibit strong image description capabilities. However, the numerical integration error is large in the calculation, which seriously affects the calculation accuracy, especially in higher-order calculation. In this paper, a continuous orthogonal moments-fast and accurate PHFM (FAPHFM) is proposed. It utilizes the polar pixel tiling technique to reduce numerical errors in the computation; this method particularly improves the accuracy of higher-order moments of traditional PHFMs. However, as accuracy increases, calculation complexity also increases. To address this issue, an eight-way symmetric/anti-symmetric calculation of the angular and radial functions was performed using the symmetry and anti-symmetry of traditional PHFMs, and clustering of pixels was performed as a way to improve the computational speed. The experimental results show that FAPHFMs perform better in image reconstruction (including noise), with higher computational accuracy, lower time complexity, and better image description ability.

Deep Ritz method with adaptive quadrature for linear elasticity

In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the H1/2(ΓD) norm is introduced and analyzed for linear elasticity equation in order to deal with the (essential) Dirichlet boundary condition. We show that the resulting deep Ritz method provides the best approximation among the set of deep neural network (DNN) functions with respect to the “energy” norm. Furthermore, we demonstrate that the total error of the deep Ritz simulation is bounded by the sum of the network approximation error and the numerical integration error, disregarding the algebraic error. To effectively control the numerical integration error, we propose an adaptive quadrature-based numerical integration technique with a residual-based local error indicator. This approach enables efficient approximation of the modified energy functional. Through numerical experiments involving smooth and singular problems, as well as problems with stress concentration, we validate the effectiveness and efficiency of the proposed deep Ritz method with adaptive quadrature.

The curse of dimensionality for the Lp-discrepancy with finite p

The Lp-discrepancy is a quantitative measure for the irregularity of distribution of an N-element point set in the d-dimensional unit-cube, which is closely related to the worst-case error of quasi-Monte Carlo algorithms for numerical integration. It's inverse for dimension d and error threshold ε∈(0,1) is the minimal number of points in [0,1)d such that the minimal normalized Lp-discrepancy is less or equal ε. It is well known, that the inverse of L2-discrepancy grows exponentially fast with the dimension d, i.e., we have the curse of dimensionality, whereas the inverse of L∞-discrepancy depends exactly linearly on d. The behavior of inverse of Lp-discrepancy for general p∉{2,∞} has been an open problem for many years. In this paper we show that the Lp-discrepancy suffers from the curse of dimensionality for all p in (1,2] which are of the form p=2ℓ/(2ℓ−1) with ℓ∈N.This result follows from a more general result that we show for the worst-case error of numerical integration in an anchored Sobolev space with anchor 0 of once differentiable functions in each variable whose first derivative has finite Lq-norm, where q is an even positive integer satisfying 1/p+1/q=1.

IPHFMs: Fast and accurate Polar Harmonic Fourier Moments

Polar Harmonic Fourier moments (PHFMs) are widely used in image processing due to their excellent reconstruction ability. However, PHFMs suffer from geometric errors and numerical integration errors. Also, direct computation of PHFMs from their definition is usually slow. This paper proposes a fast and accurate calculation method for PHFMs, named improved polar Harmonic Fourier moments (IPHFMs). The variable equidistant discrete approach and fast Fourier transform (FFT) reduce the geometric errors and time complexity. Extensive experimental results on a real-world application demonstrate the efficacy and superiority of the proposed IPHFMs, concerning speed and accuracy.

Integration and calibration of UBCSAND model for drained monotonic and cyclic triaxial compression of aggregates

We report on results from triaxial drained monotonic and cyclic compression tests for four different aggregates (gabbro, limestone, demolished concrete, steel slag), and discuss the applicability of UBCSAND model for simulating the relevant non-linear stress-strain curves. First, UBCSAND is integrated analytically for the elastic and plastic volumetric and deviatoric strains, leading to a pair of novel closed-form solutions expressed in terms of hypergeometric functions. These solutions are inherently free of numerical errors and can be used for the evaluation of results from numerical integration. Second, three calibration techniques for the model parameters are presented and compared: (1) a graphical “chart” solution based on results from forward numerical simulations; (2) a numerical approach based on minimization of residuals; and (3) an informed trial-and-error approach based on the analytical solution. Errors in numerical integration of the model are explored by comparing the analytical solution against results from an Euler integration. Variations in the calibrated parameters are discussed for drained monotonic and cyclic triaxial loading considering different types of aggregates.

Reducing roundoff errors in numerical integration of planetary ephemeris

Reducing roundoff errors in numerical integration of planetary ephemeris

The Hermite-Hadamard type inequalities for quasi $ p $-convex functions

<abstract><p>In this paper, the Hermite-Hadamard inequality and its generalization for quasi $ p $-convex functions are provided. Also several new inequalities are established for the functions whose first derivative in absolute value is quasi $ p $-convex, which states some bounds for sides of the Hermite-Hadamard inequalities. In the context of the applications of results, we presented some relations involving special means and some inequalities for special functions including digamma function and Fresnel integral for sinus. In addiditon, an upper bound for error in numerical integration of quasi p-convex functions via composite trapezoid rule is given.</p></abstract>

An Analytical Solution for Saturable Absorption in Pharmacokinetics Models

ObjectiveThe first-order absorption is a common model used in Pharmacokinetics. The absorption of some drugs follows carrier mediated transport. It has been proposed that the amount of drug available may saturate the transport mechanism resulting in an absorption slower than the one predicted by the first-order model. Saturable absorption has been modeled at the differential equation level by substituting the constant rate absorption by a Hill kinetics absorption. However, its exact solution is so far unknown. The goal of this is to know the exact solution of different Hill kinetic absorption models.MethodsWe start defining different absorption models and increasing then their complexity. The simplest case is the first-order absorption model and the most complex will be a generalized Hill kinetic absorption model. The differential equation of each model is integrated.ResultsThe complexity of the models their solutions may be not expressed in a close-form, or in term of elementary functions. We obtain and discuss the exact solutions of the different Hill kinetics absorption models. To do that, the solutions are studied according to the possible values of the free parameters of the models. We show the differences between models through simulations.ConclusionsThe knowledge of closed-form solutions allows to illustrate the differences between the different absorption models and minimizes the errors of numerical integration.

Fractal impedance for passive controllers: a framework for interaction robotics

There is increasing interest in control frameworks capable of moving robots from industrial cages to unstructured environments and coexisting with humans. Despite significant improvement in some specific applications (e.g., medical robotics), there is still the need for a general control framework that improves interaction robustness and motion dynamics. Passive controllers show promising results in this direction; however, they often rely on virtual energy tanks that can guarantee passivity as long as they do not run out of energy. In this paper, a Fractal Attractor is proposed to implement a variable impedance controller that can retain passivity without relying on energy tanks. The controller generates a Fractal Attractor around the desired state using an asymptotic stable potential field, making the controller robust to discretization and numerical integration errors. The results prove that it can accurately track both trajectories and end-effector forces during interaction. Therefore, these properties make the controller ideal for applications requiring robust dynamic interaction at the end-effector.

A Modified Radial Point Interpolation Method (M-RPIM) for Free Vibration Analysis of Two-Dimensional Solids

The classical radial point interpolation method (RPIM) is a powerful meshfree numerical technique for engineering computation. In the original RPIM, the moving support domain for the quadrature point is usually employed for the field function approximation, but the local supports of the nodal shape functions are always not in alignment with the integration cells constructed for numerical integration. This misalignment can result in additional numerical integration error and lead to a loss in computation accuracy. In this work, a modified RPIM (M-RPIM) is proposed to address this issue. In the present M-RPIM, the misalignment between the constructed integration cells and the nodal shape function supports is successfully overcome by using a fixed support domain that can be easily constructed by the geometrical center of the integration cell. Several numerical examples of free vibration analysis are conducted to evaluate the abilities of the present M-RPIM and it is found that the computation accuracy of the original RPIM can be markedly improved by the present M-RPIM.

How Reliable Are Modern Density Functional Approximations to Simulate Vibrational Spectroscopies?

We show that properties of molecules with low-frequency modes calculated with density functional approximations (DFAs) suffer from spurious oscillations along the nuclear displacement coordinate due to numerical integration errors. Occasionally, the problem can be alleviated using extensive integration grids that compromise the favorable cost-accuracy ratio of DFAs. Since spurious oscillations are difficult to predict or identify, DFAs are exposed to severe performance errors in IR and Raman intensities and frequencies or vibrational contributions to any molecular property. Using Fourier spectral analysis and digital signal processing techniques, we identify and quantify the error due to these oscillations for 45 widely used DFAs. LC-BLYP and BH&H are revealed as the only functionals showing robustness against the spurious oscillations of various energy, dipole moment, and polarizability derivatives with respect to a nuclear displacement coordinate. Given the ubiquitous nature of molecules with low-frequency modes, we warrant caution in using modern DFAs to simulate vibrational spectroscopies.

Improved Cubature Kalman Filtering on Matrix Lie Groups Based on Intrinsic Numerical Integration Error Calibration with Application to Attitude Estimation

This paper investigates the numerical integration error calibration problem in Lie group sigma point filters to obtain more accurate estimation results. On the basis of the theoretical framework of the Bayes–Sard quadrature transformation, we first established a Bayesian estimator on matrix Lie groups for system measurements in Euclidean spaces or Lie groups. The estimator was then employed to develop a generalized Bayes–Sard cubature Kalman filter on matrix Lie groups that considers additional uncertainties brought by integration errors and contains two variants. We also built on the maximum likelihood principle, and an adaptive version of the proposed filter was derived for better algorithm flexibility and more precise filtering results. The proposed filters were applied to the quaternion attitude estimation problem. Monte Carlo numerical simulations supported that the proposed filters achieved better estimation quality than that of other Lie group filters in the mentioned studies.