Neumann Series Research Articles

A novel multistage damage detection method for trusses using time-history data based on model order reduction and deep neural network

This article first proposes a multistage damage identification approach for trusses using time-series data relied upon model order reduction (MOR) and deep neural network (DNN). In the first step, an acceleration–displacement-based strain energy indicator (ADSEI), which is computed from the acceleration and displacement data incompletely measured at limited sensors and unmeasured ones inferred from a second-order Neumann series expansion (SNSE)-relied MOR technique, is utilized to eliminate low-risk damage candidates, aiming to only keep high-potential ones. This can dramatically reduce the number of output neurons of the DNN model in the second step. The input data employed to build such a DNN are the finite element method (FEM)-simulated acceleration and displacement signals corresponding to measured degrees of freedom (DOFs). Low-risk flawed candidates predicted by this DNN are then excluded via a suggested damage threshold. By repeating such a manner, the accuracy of the DNN models constructed in the subsequent stages is therefore enhanced continuously, although these upgraded DNNs only require a moderate dataset, simple architectures, and much less computational cost for their training and testing processes. Accordingly, both the location and severity of damaged members can be reliably and precisely diagnosed by time-series data measured in a fairly short interval at a few sensors, even with high noises. Several numerical examples of 3D trusses are tested to affirm the efficiency and feasibility of the current approach.

Performance analysis and comparison of approximate detectors with linear detectors in massive MIMO systems under imperfect channel state information

SummaryMassive MIMO systems are a key technology for enhancing the capacity and reliability of wireless communication systems. However, to unlock the full potential of massive MIMO systems, accurate channel state information is crucial at the receivers. In practice, channel estimation errors are inevitable, leading to significant performance degradation in traditional massive MIMO equalizers that assume perfect channel state information. In this work, we investigate the performance of approximate massive MIMO detectors, specifically the Gauss‐Seidel and Neumann series, over a time‐invariant channel with imperfect channel state information. We derive the expressions of signal‐to‐interference‐plus‐noise ratio and average bit error rate for approximate massive MIMO detectors in a unicellular environment using M‐ary quadrature amplitude modulation. Additionally, we compare the performance of linear detectors, including zero‐forcing, minimum‐mean‐squared‐error, and maximal‐ratio combining, with that of the approximate detectors. Through extensive Monte Carlo numerical simulations and various system configurations, we demonstrate that the Gauss‐Seidel detector, under imperfect channel state information, achieves comparable bit error rate performance to the zero‐forcing and minimum‐mean‐squared‐error detectors while requiring less computational complexity. Our study suggests that the Gauss‐Seidel detector has the potential to be used in massive MIMO systems in the presence of imperfect channel state information, highlighting its practical relevance in real‐world scenarios.

Mitigating quantum errors via truncated Neumann series

Mitigating quantum errors via truncated Neumann series

Iterative algorithms and architectures in massive MIMO detection

Massive multiple-input multiple-output (MIMO) wireless systems play an important role in the 5G networks. The complexity of signal detection in massive MIMO is increasing rapidly due to the growth of the number of transmitting antennas. In this paper, we introduced different iterative algorithms to decrease the computational cost of the approximate minimum mean-square error (MMSE) algorithm, including the Neumann Series algorithm, Jacobi method, Gauss-Seidel method, Successive Over Relaxation method and the Conjugate Gradient method. In addition, the VLSI architecture implementations of algorithms mentioned above are also discussed in the article.

A Dual Integral Equation Approach for Evaluating the Shielding of Thick Circular Disks against a Coaxial Loop

The electromagnetic interaction between a circular disk with finite conductivity and finite thickness and a coaxial circular loop of constant current is addressed here. The finite conductivity and thickness of the material disk lead to the adoption of suitable generalized boundary conditions, and the problem is thereby reduced to the solution of two sets of dual integral equations in the Hankel transform domain. Such equations are then solved by expanding the spectral unknowns in Neumann series of Bessel functions. An alternative formulation that is valid for purely conductive screens with no magnetic properties, which is computationally much faster, is proposed as well. The magnetic shielding effectiveness of the structure is studied in detail, pointing out its dependencies and possible critical situations.

High resolution acoustic identification of clusters of small blockages in fluid-filled pipe using maximum likelihood estimation

This paper presents a method for identifying a cluster of small blockages (i.e., blockages with length on the order of centimeters and radial extent on the order of millimeters and separated by a distance on the order of few centimeters) in pressurized fluid-filled pipes using sound waves. This focus on defects with small scale, and, thus, small scattering strength is exploited to develop a Neumann series solution for the scattered acoustic wave field. The probing waves are such that the Helmholtz number (ratio of blockage longitudinal length scale and probing acoustic wavelength) is of order 1 or larger. A high resolution inverse technique for identifying a cluster of small blockages based on the maximum likelihood estimation principle is developed. The proposed technique uses two-dimensional search space to resolve each blockage in the cluster and requires a single measurement point only. The method is successfully tested through both numerical and laboratory experiments. The proposed methodology allows an early identification of a cluster of small defects and leads to reliable condition assessment of pipelines, which is necessary to inform decisions as to when remedial actions are required.

A new preconditioned AOR-type method for [formula omitted]-tensor equation

By using a truncated Neumann series approximation to the inverse of a given matrix, we propose a new preconditioner for improving the AOR-type method, then we develop preconditioned AOR-type methods to solve M-tensor equation and prove that the methods are convergent. Some comparison theorems are provided to show that the preconditioned AOR-type methods perform better than the AOR-type method. Finally, numerical examples illustrate that our methods are feasible and effective.

English

In next-generation high throughput satellite (NGHTS), precoding (a preprocessing technique at the gateway (GW) can mitigate the interbeam interference with no additional resource at the user terminal. However, the calculation of precoding requires matrix inversion, which involves huge complexity in terms of hardware resources with real-time calculation for near error-free results. This paper proposes a novel approach for precoding matrix calculation using the Neumann series. In this way, the algorithm does not directly compute the matrix inversion. Hence the computational complexity is highly reduced. The proposed method uses the 4th-order iterative series with the initial guess as an inverse of the diagonal matrix of the input to construct the Neumann series. This leads to fast convergence of the matrix inversion process with fewer resources. The algorithm is claimed to be generic as it is seamlessly applied to any linear precoding scheme. The paper examines the resource complexity, convergence probability and Bit Error Rate (BER) performance for the proposed method with Zero Forcing (ZF) and Regularized Zero Forcing (RZF) based linear precoding scheme. Experiment results demonstrate that the proposed algorithm accomplishes superior performance with fewer resources compared to the Neumann series and Joint iterative Newton/Chebyshev-based Neumann series methods.

Asymptotic Expansion of the Angular Flux Applied to Discrete-Ordinates Source Iterations in Lattice Depletion Calculations

This work presents an asymptotic method based on angular flux expansion in a Neumann series. The technique is aimed at effective reduction of the memory imprint of numerical methods based on collision probabilities (CPs). The asymptotic method has been implemented in the heterogeneous Cartesian cells of the integro-differential transport solver (IDT). The IDT solves the neutral-particle transport equation by discrete ordinates combined with angular-dependent CP matrices. In lattice depletion calculations, because of the change of isotopic concentration along the burnup, methods based on CP discretization, such as current-coupling CP or the one presented in this paper, would require construction and storage of a set of CP coefficients for any depleted pin cell. When the number of media grows, the performances of the solver are bounded by the memory pressure caused by the growth of coefficients. Application of the asymptotic technique, presented in this paper, transforms by two user’s parameters the memory-bound solver in a compute-bound application, where the principal workload is transferred from coefficients to source iterations. In this work, a theoretical study of the method is presented together with two applications to two-dimensional assembly simulations. The effects on self-shielded and depleted materials are highlighted. Preliminary results show an encouraging reduction of memory occupation by a factor 10 without any significant loss of accuracy.

A novel univariate dimension‐reduction based interval finite element method for static response prediction of uncertain structures

AbstractTo eliminate the errors caused by the conventional interval perturbation finite element method due to classic interval arithmetic and neglect of higher‐order terms, we propose a novel univariate dimension‐reduction based interval finite element method to predict the static response bounds of structures with uncertain but bounded parameters. First, a univariate dimension‐reduction algorithm is derived using the generalized Taylor expansion. The global stiffness matrix is expressed as the sum of the median and the univariate disturbance radius. Compared with Taylor expansion approximation, the univariate dimension‐reduction approximation has higher accuracy and does not increase the amount of calculation. Then the inverse of the interval global stiffness matrix is approximated as an improved Neumann series. Higher‐ order terms are included by summing up the geometric terms in the Neumann series. Finally, the improved interval algorithm is used to solve the upper and lower bounds of the structural displacement response and the element stress response. The dependence between the interval parameters is accounted in comparison with the classic interval algorithm. The accuracy and effectiveness of the new method are validated by numerical cases on 2D truss, 3D frame and truck frame with multiple interval parameters.

A simulation method on target strength and circular SAS imaging of X-rudder UUV including multiple acoustic scattering

Target strength (TS) and circular synthetic aperture sonar (CSAS) images provide essential information for active acoustic detection and recognition of non-cooperative unmanned undersea vehicles (UUVs), which pose a significant threat to underwater preset facilities. To access them, we propose an iterative physical acoustics (IPA)-based method to simulate the multiple acoustic scattered fields on rigid surfaces in high-frequency cases. It uses the Helmholtz integral equation with an appropriate Green's function in terms of the Neumann series, and then incorporates the ideas of triangulation and iteration into a numerical implementation. Then two approximate analytic formulae with precise physical meanings are derived to predict the TS and CSAS images of concave targets, respectively. There are no restrictions on the surface's curvature and the order of multiple scattering. The method is validated against the finite element method (FEM) for acoustic scattering from a sphere segment and against an experiment involving an X-rudder UUV's stern. On this basis, we simulate and analyze the TS and CSAS images of an X-rudder UUV. In addition, the influence of the angle of adjacent rudders on the multiple scattering characteristics is discussed. Results show that this method can potentially predict accurate UUV features, especially the multiple scattered features.

Parallel implementations of randomized vector algorithm for solving large systems of linear equations

The results of a parallel implementation of a randomized vector algorithm for solving systems of linear equations are presented in the paper. The solution is represented in the form of a Neumann series. The stochastic method computes this series by sampling only random columns, avoiding multiplication of matrix by matrix and matrix by vector. We consider the case when the matrix is too large to fit in random-access memory (RAM). We use two approaches to solve this problem. In the first approach, the matrix is divided into parts that are distributed among MPI processes and stored in the available RAM of the cluster nodes. In the second approach, the entire matrix is stored on each node’s hard drive, loaded into RAM, and processed in parts. Independent Monte Carlo experiments for random column indices are distributed among MPI processes or OpenMP threads for both approaches to matrix storage. The efficiency of parallel implementations is analyzed. Results are given for a system governed by dense matrices of size $$10^4$$ and $$10^5$$ .

A Comparison Between Different Solutions for the Marchenko Multiple Eliminations Scheme

Seismic reflection data can be used to generate high-resolution in-depth images capable of facilitating, with high precision, the correct positioning of wells in hydrocarbon exploration and production. However, images produced by migrating seismic data are often contaminated by artifacts due to multiple internal reflections. Different schemes can be used to avoid problems caused by these noises and to attenuate them, such as the Marchenko multiple elimination scheme (MME). Various solutions based on the MME method have been proposed in the literature. Therefore, in this work, we explore the MME based on the least-squares schemes (LSMME), the MME as Neumann series approximation solution (NMME), and the MME based on beyond Neumann method (BNMME), and compare them in terms of effectiveness and efficiency in different numerical examples.

Raytracing in Galerkin Boundary Integral form

Raytracing is an established method for computing the late-time part of room impulse responses. But it has the drawback that only very crude Monte Carlo models of boundary scattering and diffraction are possible to include without losing its attractive computational cost scaling. This happens because higher-resolution models of these processes output multiple child rays for every parent ray received, making the number of rays grow with reflection order. An emerging solution is so-called 'Surface-Based' Geometrical Acoustics. Here the distribution of rays arriving at a boundary is mapped onto a predefined set of spatial elements and angular interpolation functions, producing a vector of coefficients. Re-radiation of subsequent reflections is then a matrix multiplication, with the steady state solution being solvable via a Neumann series. Since rays only every propagate one reflection order before being collected, diffraction and scattering process that cause them to multiply can be included without issue. Here we present a new formulation based on a Galerkin Boundary Element Method (BEM). A unique feature is its ability to readily change interpolation functions, so their effect on accuracy and convergence can be assessed. In this preliminary work, it is verified against an Image Source Model for a rectangular room.

Alternative fan-beam backprojection and adjoint operators

Abstract We present in this work alternative analytic formulations for the fan-beam tomographic backprojection operation and its associated adjoint transform in standard (equiangular) and linear (equidistant) detector geometries. The proposed formulations are obtained from a recent backprojection theorem in parallel tomography. Such formulations are written as a Bessel–Neumann series in the frequency domain that can be implemented as an O ⁢ ( N 2.3729 ) {O(N^{2.3729})} matrix multiplication. Proofs are provided together with numerical simulations compared with conventional fan-beam O ⁢ ( N 3 ) {O(N^{3})} backprojection representations showing more robustness when dealing with highly noisy data.

Solving integral equations in free space with inverse-designed ultrathin optical metagratings.

As standard microelectronic technology approaches fundamental limitations in speed and power consumption, novel computing strategies are strongly needed. Analogue optical computing enables the processing of large amounts of data at a negligible energy cost and high speeds. Based on these principles, ultrathin optical metasurfaces have been recently explored to process large images in real time, in particular for edge detection. By incorporating feedback, it has also recently been shown that metamaterials can be tailored to solve complex mathematical problems in the analogue domain, although these efforts have so far been limited to guided-wave systems and bulky set-ups. Here, we present an ultrathin Si metasurface-based platform for analogue computing that is able to solve Fredholm integral equations of the second kind using free-space visible radiation. A Si-based metagrating was inverse-designed to implement the scattering matrix synthesizing a prescribed kernel corresponding to the mathematical problem of interest. Next, a semitransparent mirror was incorporated into the sample to provide adequate feedback and thus perform the required Neumann series, solving the corresponding equation in the analogue domain at the speed of light. Visible wavelength operation enables a highly compact, ultrathin device that can be interrogated from free space, implying high processing speeds and the possibility of on-chip integration.

Method for solving inverse spectral problems on quantum star graphs

Abstract A new method for solving inverse spectral problems on quantum star graphs is proposed. The method is based on Neumann series of Bessel function representations for solutions of Sturm–Liouville equations. The representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse spectral problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the representation is sufficient for the recovery of the potential. The method for solving the inverse spectral problem on the graph consists in reducing the problem to a two-spectra inverse Sturm–Liouville problem on each edge. Then a system of linear algebraic equations is derived for computing the first coefficient of the series representation for the solution on each edge and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.

Unique Solvability of the Initial-Value Problem for Fractional Functional Differential Equations—Pantograph-Type Model

Contrary to the initial-value problem for ordinary differential equations, where the classical theory of establishing the exact unique solvability conditions exists, the situation with the initial-value problem for linear functional differential equations of the fractional order is usually non-trivial. Here we establish the unique solvability conditions for the initial-value problem for linear functional differential equations of the fractional order. The advantage is the lack of the calculation of fractional derivatives, which is a complicated task. The unique solution is represented by the Neumann series. In addition, as examples, the model with a discrete memory effect and a pantograph-type model from electrodynamics are studied.

On Neumann series of Macdonald functions

The main aim of this article is to establish a closed-form expression for the Neumann series built with Macdonald functions in which order contains the summation index and the coefficients include the summation index exponentially and factorially. Motivated by this problem which occurs in several problems of physics or electrochemistry such as diffusion phenomena, we derive two integral representations of a general Neumann series of this type containing Macdonald functions as building blocks.

A novel model order reduction-based two-stage damage detection paradigm for trusses using time–history acceleration

This paper presents a novel model order reduction (MOR)-based two-stage damage detection method for trusses employing time-series acceleration measured by limited sensors. In the first step, an acceleration-based strain energy indicator (ASEI) is newly proposed to evaluate the most doubtfully damaged candidates. Accordingly, the number of design variables defined in an inverse optimization problem of the second phase is reduced significantly. The location and severity of damaged members are then determined by minimizing an objective function with a newly added dynamic penalty parameter to achieve a faster convergence speed and better optimal solutions. Since acceleration signals are incompletely measured at limited sensors, the second-order Neumann series expansion (SNSE) is firstly employed to condense the proportionally damped trusses for inferring the unmeasured time–history information. An adaptive hybrid evolutionary firefly algorithm (AHEFA) is utilized to resolve the above inverse optimization problem. Four numerical examples of 2D and 3D trusses with various damage scenarios including noises are examined to demonstrate the reliability of the proposed methodology. Attained outcomes indicate that the suggested paradigm can reliably diagnose both multidamage sites and extents of trusses with only relatively short time histories and a few measurement sensors.