Integral Approximation Research Articles

Model predictive control of bidirectional heat transfer in prosumer-based solar district heating networks

District heating networks with decentralized heat production are ideally suited to include a high share of renewable energy sources for heat production in urban areas with limited space. A new concept is a prosumer-based district heating network in which some or even all buildings are equipped with decentralized building-level heat storages and heat generation plants. To exploit the full potential of the interconnected heating network, buildings with surplus heat are intended to transfer heat to buildings with heating demands to prevent the activation of the central heating plant. This work presents an initial feasibility study on using a model predictive control strategy to manage bidirectional heat transfer between buildings in a district heating network. We formulate this problem as an optimal control problem, incorporating binary decision variables for each potential heat transfer connection. This results in a mixed-integer nonlinear optimization problem that is hard to solve. The problem is solved with a fast gradient-based optimization algorithm in combination with a combinatorial integral approximation strategy. A case study concerning a residential heating network with building-level solar thermal collectors and storage tanks is carried out. The optimized operation is compared to the real operation obtained from one month of measurements. The results show that the optimized strategy with bidirectional heat transfer can exploit the total heat generated in the whole network. The central heating provider is only required when the heat produced or stored by all the buildings is insufficient to meet the total heat demand within the network. In the case study, this leads to a reduction of almost 75% of the required thermal energy from the central heat provider.

The semiclassical approximation of multiple functional integrals

In this paper, we study the semiclassical approximation of multiple functional integrals. The integrals are defined through the Lagrangian and the action. Of all possible trajectories, the greatest contribution to the integral is given by the classical trajectory x̅cl for which the action S takes an extremal value. The classical trajectory is found as a solution of the multidimensional Euler – Lagrange equation. To calculate the functional integrals, the expansion of the action with respect to the classical trajectory is used, which can be interpreted as an expansion in powers of Planck’s constant. The numerical results for the semiclassical approximation of double functional integrals are given.

Breaking the symmetric spiral spectrum distribution of a Laguerre-Gaussian beam propagating in moderate-to-strong isotropic atmospheric turbulence.

We demonstrate that the spiral spectrum (also known as orbital angular momentum spectrum) of a Laguerre-Gaussian (LG) beam with topological charge (TC) l is asymmetrically broadened propagating through moderate-to-strong atmospheric turbulence, even the statistics of turbulence is isotropic. This phenomenon is quite different from that predicted in weak turbulence where the spiral spectrum of a disturbed LG beam is symmetric with respect to its TC number l. An explicit analytical expression of the spiral spectrum of the LG beam with l = 1 is derived based on the extend Huygens-Fresnel integral and quadratic approximation, which is used to illustrate the transition scenarios of the spiral spectrum from symmetry to asymmetry in weak-to-strong turbulence. The physical mechanism for the asymmetric spiral spectrum in moderate-to-strong turbulence is thoroughly discussed. Our results are confirmed by the multi-phase screen numerical simulations and are consistent with the experimental results reported in Phys. Rev. A105, 053513 (2022)10.1103/PhysRevA.105.053513 and Opt. Lett.38, 4062 (2013)10.1364/OL.38.004062.

Revisiting Diffusive Representations for Enhanced Numerical Approximation of Fractional Integrals

This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of fractional integrals. The approach seamlessly aligns with established techniques used in addressing problems involving integer-order operators, contributing to a unified framework for numerical solutions.

An extension of the stochastic sewing lemma and applications to fractional stochastic calculus

Abstract We give an extension of Lê’s stochastic sewing lemma. The stochastic sewing lemma proves convergence in $L_m$ of Riemann type sums $\sum _{[s,t] \in \pi } A_{s,t}$ for an adapted two-parameter stochastic process A, under certain conditions on the moments of $A_{s,t}$ and of conditional expectations of $A_{s,t}$ given $\mathcal F_s$ . Our extension replaces the conditional expectation given $\mathcal F_s$ by that given $\mathcal F_v$ for $v<s$ , and it allows to make use of asymptotic decorrelation properties between $A_{s,t}$ and $\mathcal F_v$ by including a singularity in $(s-v)$ . We provide three applications for which Lê’s stochastic sewing lemma seems to be insufficient. The first is to prove the convergence of Itô or Stratonovich approximations of stochastic integrals along fractional Brownian motions under low regularity assumptions. The second is to obtain new representations of local times of fractional Brownian motions via discretization. The third is to improve a regularity assumption on the diffusion coefficient of a stochastic differential equation driven by a fractional Brownian motion for pathwise uniqueness and strong existence.

Analysis of Scattering characteristics of PEMC sphere by vortex Bessel beams

Generalized Lorenz-Mie theory (GLMT) is well known to analyze the scattering characteristics of spherical surfaces and metamaterials such as perfect electromagnetic conductor (PEMC) sphere. Also, metamaterials are explored very well when illuminated by electromagnetic waves/beams and their characteristics depend upon propagation and scattering response. So, in this manuscript, vortex Bessel beam (VBB) interaction from PEMC sphere has been investigated. The VBB interaction with material particles is of quite interesting due to association of orbital angular momentum (OAM) with them. By employing the Integral localized approximation (ILA) method, the beam‐shape coefficients (BSCs) regarding VBB are obtained. Computations for the scattering efficiency QSca and extinction efficiency QExt for PEMC sphere by considering the co‐polarized and cross‐polarized scattering components are presented and analyzed. Moreover, the influences of OAM mode number, electromagnetic admittance parameter, and diffraction angle are discussed. Numerical results have applications regarding transfer of OAM of VBB to PEMC sphere and associated applications in light manipulation and scattering, optical tweezers, metamaterial fields, and remote sensing.

Approximation of the Hilbert transform in the Lebesgue spaces

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. The Hilbert transform is the main part of the singular integral equations on the real line. Therefore, approximations of the Hilbert transform are of great interest. Many papers have dealt with the numerical approximation of the singular integrals in the case of bounded intervals. On the other hand, the literature concerning the numerical integration on unbounded intervals is by far poorer than the one on bounded intervals. The case of the Hilbert Transform has been considered very little. This article is devoted to the approximation of the Hilbert transform in Lebesgue spaces by operators which introduced by V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions in a strip. In this paper, we prove that the approximating operators are bounded maps in Lebesgue spaces and strongly converges to the Hilbert transform in these spaces.

Exploring Locality in Molecular Dirac-Coulomb-Breit Calculations: A Perspective.

The Dirac-Coulomb-Breit (DCB) operator is widely recognized for its ability to accurately capture relativistic effects and spin-physics in molecular calculations. However, due to its high computational cost, there is a need to develop low-scaling approximations without compromising accuracy. To tackle this challenge, it becomes essential to gain a deeper understanding of the DCB operator's behavior. This work aims to explore local integral approximations, shedding light on the locality of the parts of the charge-current distribution due to the small component. In particular, we propose an atomic Breit approximation that leverages an analysis of the behavior observed in a series of gold chains. Through benchmark studies of metal complexes, we evaluated the accuracy and performance of the proposed atomic Breit approximation. This work provides a comprehensive understanding of the behavior of the charge-current distribution in terms of its contributions from its AO basis constituents, facilitating the development of low-scaling methods that strike a balance between computational efficiency and accuracy.

Discrete approximation of complete p-elliptic integral of the second kind and its application

Discrete approximation of complete p-elliptic integral of the second kind and its application

Expected integration approximation under general equal measure partition

In this paper, we first use an L2−discrepancy bound to give the expected uniform integration approximation for functions in the Sobolev space H1(K) equipped with a reproducing kernel. The concept of stratified sampling under general equal measure partition is introduced into the research. For different sampling modes, we obtain a better convergence order O(N−1−1d) for the stratified sampling set than for the Monte Carlo sampling method and the Latin hypercube sampling method. Second, we give several expected uniform integration approximation bounds for functions equipped with boundary conditions in the general Sobolev space Fd,q∗, where 1p+1q=1. Probabilistic Lp−discrepancy bound under general equal measure partition, including the case of Hilbert space-filling curve-based sampling are employed. All of these give better general results than simple random sampling, and in particular, Hilbert space-filling curve-based sampling gives better results than simple random sampling for the appropriate sample size.

Improved explicit quartic B-spline time integration scheme for dynamic response analysis of viscoelastic systems

A quartic B-spline based explicit scheme is extended for dynamic analysis of viscoelastic systems. In the calculation formulation, different integration approximation schemes for exponential damping models and other damping models are presented. The mathematical derivation of the convolution solving strategy is formulated for viscoelastic systems. For the damping models composed of exponential function, this scheme only requires once convolution calculation in each time step, even though the dynamic equations are used twice. This significantly reduces the calculation time for the damping models composed of exponential function. Numerical examples show that the proposed quartic B-spline based explicit scheme achieves higher accuracy and efficiency compared to other competitive explicit schemes, not only for damping models composed of exponential functions, but also for other viscoelastic damping models.

A Class of Quadrature Rules for Complex Cauchy Principal Value Integrals

This article is fully devoted to the numerical approximation of Cauchy-type integrals in the complex plane. A class of degree eight quadrature rules is formulated from a family of Gauss-type two-point rules based on the method of extrapolation. The basic rules are developed and then composite rules are constructed from the basic rules. The Error bounds for each rule are determined. The validity of the method has been demonstrated by the provision of numerical experiments and their results.

A fast, rapidly convergent Nyström solver for electromagnetic scattering from orthotropic media

This article presents a simple, efficient, and high-order numerical scheme for the approximation of the convolution integral of the form(1)∫R2∇x′Φ(x,x′)⋅U(x′)dx′, where Φ(x,x′)=i4H0(1)(κ|x−x′|) and U is smooth and compactly supported vector function on R2. Moreover, we employ this integration scheme to obtain a Nyström solver for the numerical solution of the electromagnetic wave scattering problem by inhomogeneous orthotropic media, that is, media with refractive index of the form[n11(x)n12(x)0n21(x)n22(x)000n33(x)], where nij(x1,x2,x3) are independent of x3 coordinate. On a uniform grid of size n×n, the suggested approach has computational complexity O(n2log⁡n) and displays high-order convergence when the material properties are smooth. When the material attributes are real analytic, the approach actually converges exponentially. We demonstrate the performance of our integration scheme and the Nyström solver through several numerical experiments.

The fast reduced QMC matrix–vector product

We study the approximation of integrals of the form ∫Df(x⊤A)dμ(x), where the measure μ is of product form and A is a matrix, by quasi-Monte Carlo (QMC) rules N−1∑k=0N−1f(xk⊤A). We are interested in cases where the main computational cost in the approximation arises from calculating the products xk⊤A. We design QMC rules for which the computation of xk⊤A, k=0,1,…,N−1, can be done in a fast way, and for which the approximation error of the QMC rule is similar to the standard QMC error. We do not require that the matrix A has any particular structure.Problems of this form arise in some important applications in statistics and uncertainty quantification. For instance, this approach can be used when approximating the expected value of some function with a multivariate normal random variable with some given covariance matrix, or when approximating the expected value of the solution of a PDE with random coefficients.The speed-up of the computation time of our approach is sometimes better and sometimes worse than the fast QMC matrix–vector product from [Josef Dick, Frances Y. Kuo, Quoc T. Le Gia, and Christoph Schwab, Fast QMC Matrix-Vector Multiplication, SIAM J. Sci. Comput. 37 (2015), no. 3, A1436–A1450]. As in that paper, our approach applies to lattice point sets and polynomial lattice point sets, but also applies to digital nets (we are currently not aware of any approach which allows one to apply the fast QMC matrix–vector paper from the aforementioned paper of Dick, Kuo, Le Gia, and Schwab to digital nets).The method in this paper does not make use of the fast Fourier transform, instead we use repeated values in the quadrature points to derive a significant reduction in the computation time. Such a situation naturally arises from the reduced CBC construction of lattice rules and polynomial lattice rules. The reduced CBC construction has been shown to reduce the computation time for the CBC construction. Here we show that it can additionally be used to also reduce the computation time of the underlying QMC rule. One advantage of the present approach is that it can be combined with random (digital) shifts, whereas this does not apply to the fast QMC matrix–vector product from the earlier paper of Dick, Kuo, Le Gia, and Schwab.

The versatility of the Cholesky decomposition in electronic structure theory

AbstractThe resolution‐of‐the‐identity (RI) or density fitting (DF) approximation for the electron repulsion integrals (ERIs) has become a standard component of accelerated and reduced‐scaling implementations of first‐principles Gaussian‐type orbital electronic‐structure methods. The Cholesky decomposition (CD) of the ERIs has also become increasingly deployed across quantum chemistry packages in the last decade, even though its early applications were mostly limited to high‐accuracy methods such as coupled‐cluster theory and multiconfigurational approaches. Starting with a summary of the basic theory underpinning both the CD and RI/DF approximations, thus underlining the extremely close relation of the CD and RI/DF techniques, we provide a brief and largely chronological review of the evolution of the CD approach from its birth in 1977 to its current state. In addition to being a purely numerical procedure for handling ERIs, thus providing robust and computationally efficient approximations to the exact ERIs that have been found increasingly useful on modern computer platforms, CD also offers highly accurate approaches for generating auxiliary basis sets for the RI/DF approximation on the fly due to the deep mathematical connection between the two approaches. In this review, we aim to provide a concise reference of the main techniques employed in various CD approaches in electronic structure theory, to exemplify the connection between the CD and RI/DF approaches, and to clarify the state of the art to guide new implementations of CD approaches across electronic structure programs.This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Software > Quantum Chemistry Quantum Computing > Theory Development

On the Influence of Scattered Errors over Full-Field Receptances in the Rayleigh Integral Approximation of Sound Radiation from a Vibrating Plate

Spatially dense operative deflection shapes and receptances, acquired in broad frequency bands, increase the detail in the spatial and frequency domains of the responses of parts in actual dynamic loading, manufacturing and mounting conditions. This work remarks the potential benefits of greater spatial resolution in the Rayleigh integral approximation of sound pressure—here reformulated to exploit the increased quality output from experiment-based optical full-field technologies in contactless structural dynamics—radiated by a vibrating surface in a broad frequency band. But in some cases the noise that is scattered over the estimated receptance maps might be heavier, or with different patterns, than expected, with potential repercussions on the sound pressure simulations that come thereof. This work covers this specific latter issue with insight over examples from experiment-based receptances of a lightweight vibrating plate. The effects of error spreading are analysed in the space and frequency domains, with special attention to the contribution of the experiment-based full-field receptance maps to the accuracy of the vibro-acoustic frequency response function maps.

Evaluation of Electromagnetic Fields of Extremely Low-Frequency Horizontal Electric Dipoles at Sea–Air Boundaries

The technologies of undersea detection and communication, seabed sensor networks, and geophysical detection using electromagnetic waves have emerged as research focal points within the field of marine science and engineering. However, most studies have focused on the propagation of electromagnetic fields over long distances within the shallow “sea-seabed” environment. This paper introduces a quasi-static approximation method to address the Sommerfeld numerical integration challenge within the near-field region, employing the horizontal electric dipole (HED) as a model. It derives the Sommerfeld numerical integral expressions under conditions where the wave-number ratio at the “seawater-air” boundary does not adhere to the requirement of |k0/k1| << 1 (where subscripts 0 and 1 denote seawater and air media, respectively). Building upon this, the paper simplifies the Bessel-Fourier infinite integral term within the integral expression to obtain Sommerfeld numerical integral approximations for the propagation of electromagnetic fields in the near region of extremely low frequency (ELF) within seawater. The study further conducts simulations and calculations to determine amplitude variations in electromagnetic field intensity generated by an ELF HED at different frequencies, dipole heights, and observation point depths. It concludes with an analysis of electromagnetic field propagation characteristics at the seawater-air boundary. Experimental findings highlight the lateral wave as the primary mode of electromagnetic wave propagation at this interface.

Eccentric catastrophes & what to do with them

Analytic modeling of gravitational waves (GWs) from inspiraling eccentric binaries poses an interesting mathematical challenge. When constructing analytic waveforms in the frequency domain, one has to contend with the fact that the phase of the Fourier integral in non-monotonic, resulting in a breakdown of the standard stationary phase approximation (SPA). In this work, we study this breakdown within the context of catastrophe theory. We find that the SPA holds in the context of eccentric Keplerian orbits when the Fourier frequency satisfies , where are integer multiples of the apocenter/pericenter frequencies, respectively. For values outside of this interval, the phase undergoes a fold catastrophe, giving rise to an Airy function approximation of the Fourier integral. Using these two different approximations, we generate a matched asymptotic expansion that approximates generic Fourier integrals of Keplerian motion for bound orbits across all frequency values. This asymptotic expansion is purely analytic and closed-form. We discuss several applications of this investigation and the resulting approximation, specifically: (1) the development and improvement of effective fly-by waveforms for binary black holes, (2) the transition from burst emission in the high eccentricity limit to wave-like emission in the quasi-circular limit, which results in an analogy between eccentric GW bursts and Bose–Einstein condensates, and (3) the calculation of f-mode amplitudes in eccentric binary neutron stars and black hole-neutron star binaries in terms of complex Hansen coefficients. The techniques and approximations developed herein are generic, and will be useful for future studies of GWs from eccentric binaries within the context of post-Newtonian theory.

Uneven internal time-step adjustment for fast power system dynamic simulations based on Trapezoidal integration of elementary transfer function blocks

With the development of renewable energy sources, the scale and complexity of power systems are continuously increasing, resulting in a high requirement for the efficiency of dynamic simulation. Increasing the simulation time-step is a possible measure for the fixed time-step (FTS) method to accelerate the simulation, yet also a potential cause for deteriorating the simulation accuracy. To improve the overall power system dynamic simulation speed without jeopardizing the expected accuracy, this paper proposes a novel Uneven Internal Time-Step Adjustment method based on Trapezoidal integration of elementary transfer function blocks (ETFBs). For ETFBs with different time constants, three different integration modes are defined, including single-step integration, multiple-step integration, and static approximation. Each ETFB can select an appropriate integration mode and adjust its internal time-step, automatically expanding the size of the simulation time-step. Implementation of static approximation for ETFBs with small time constants is given. Discussion on the determination of threshold values of internal time-step is provided, which mainly affect the accuracy of the system. The proposed method is applied to the open-source Simulation Toolkit for Electrical Power Systems (STEPS). Different cases are tested to obtain the maximum simulation time-step that can maintain numerical stability and validate the efficiency and accuracy of the proposed method.

CFD-DEM coupled simulation of fluidized beds with improved lumped formulation for heat transfer

PurposeThis paper aims to present a novel approach for computing particle temperatures in simulations coupling computational fluid dynamics (CFD) and discrete element method (DEM) to predict flow and heat transfer in fluidized beds of thermally thick spherical particles.Design/methodology/approachAn improved lumped formulation based on Hermite-type approximations for integrals to relate surface temperature to average temperature and surface heat flux is used to overcome the limitations of classical lumped models. The model is validated through comparisons with analytical solutions for a convectively cooled sphere and experimental data for a fixed particle bed. The coupled CFD-DEM model is then applied to simulate a Geldart D bubbling fluidized bed, comparing the results to those obtained using the classical lumped model.FindingsThe validation cases demonstrate that ignoring internal thermal resistance can significantly impact the temperature in cases where the Biot number is greater than 0.1. The results for the fixed bed case clearly demonstrate that the proposed method yields significantly improved outcomes compared to the classical model. The fluidized bed results show that surface temperature can deviate considerably from the average temperature, underscoring the importance of accurately accounting for surface temperature in convective heat transfer predictions and surface processes.Originality/valueThe proposed approach offers a physically more consistent simulation without imposing a significant increase in computational cost. The improved lumped formulation can be easily and inexpensively integrated into a typical DEM solver workflow to predict heat transfer for spherical particles, with important implications for various industrial applications.