Conventional Numerical Algorithms Research Articles

A Dynamic Gain Fixed-Time Robust ZNN Model for Time-Variant Equality Constrained Quaternion Least Squares Problem With Applications to Multiagent Systems.

A dynamic gain fixed-time (FXT) robust zeroing neural network (DFTRZNN) model is proposed to effectively solve time-variant equality constrained quaternion least squares problem (TV-EQLS). The proposed approach surmounts the shortcomings of conventional numerical algorithms which fail to address time-variant problems. The DFTRZNN model is constructed with a novel dynamic gain parameter and a novel activation function (NAF), which differs from previous zeroing neural network (ZNN) models. Moreover, the comprehensive theoretical derivation of the FXT stability and robustness of the DFTRZNN model is presented in detail. Simulation results further confirm the availability and superiority of the DFTRZNN model for solving TV-EQLS. Finally, the consensus protocols of multiagent systems are presented by utilizing the design scheme of the DFTRZNN model, which further demonstrates its practical application value.

Multi-receptive-field physics-informed neural network for complex electromagnetic media

Acquiring the electromagnetic response of intricate media at the nanoscale constitutes a pivotal phase in the design intricacies of nanophotonic apparatuses. Conventional numerical algorithms often necessitate intricate and specialized treatments to accommodate the unique properties of the medium, coupled with substantial computational time and resource demands. In recent years, the advent of deep learning technology has heralded numerous advancements in the domain of computational electromagnetics, albeit with a scarcity of solvers tailored for versatile complex media. Consequently, this study introduces an innovative multi-receptive-field physics-informed neural network (MRF-PINN) designed to tackle nano optical scattering predicaments inherent in media exhibiting dispersion, inhomogeneity, anisotropy, nonlinearity, and chirality. This framework adeptly captures electromagnetic perturbations surrounding scatterers via variable-scale receptive fields, thereby enhancing prediction precision. Within the training regimen, a scale balancing algorithm is proposed to expedite network convergence. Empirical findings demonstrate that a fully trained MRF-PINN proficiently reconstructs electromagnetic field distributions within complex nanomaterials within a mere tens of milliseconds of inference time. Such quasi real-time capabilities herald a novel approach to supplant the arduous forward calculation processes inherent in nanomaterial design workflows.

Coupled multiphysics solver for irregular regions based on graph neural network

Multiphysics analysis, involving the coupled phenomena of various physics fields, such as electro-mechanical, thermofluidic, thermoelectric, and thermo-hygric phenomena, plays a critical role in advancing technology and engineering disciplines. Conventional numerical algorithms are generally time-consuming and resource-demanding. Thanks to the rapid development in deep learning (DL), data-driven, physics-informed, and operator learning methods have been proposed to accelerate the calculation process. However, existing DL methods, encompassing convolutional neural network (CNN) and graph neural network (GNN) based approaches, exhibit limitations in effectively handling irregular regions and addressing the challenges posed by multiphysics tasks. To tackle these problems, we propose a DL framework based on GNN with a coupled solver. In the solving process of the proposed framework, a graph is generated according to the given region, and then a graph coupled-field solving network (GCFSN) is adopted. We have conducted three experiments using GCFSN to solve the coupled-field and the mean relative error is about 5%. The experimental results have verified the superior performance and generalization ability of the GCFSN method in solving the thermal-coupled physics problems on an irregular region.

Solving time fractional partial differential equations with variable coefficients using a spatio-temporal meshless method

<p>This paper presents a novel spatio-temporal meshless method (STMM) for solving the time fractional partial differential equations (TFPDEs) with variable coefficients based on the space-time metric. The main idea of the STMM is to directly approximate the solutions of fractional PDEs by using a multiquadric function with the space-time distance within a space-time scale framework. Compared with the existing methods, the present meshless STMM entirely avoids the difference approximation of fractional temporal derivatives and can be easily applied to complicated irregular geometries. Furthermore, both regular and irregular nodal distribution can be used without loss of accuracy. For these reasons, this new space-time meshless method could be regarded as a competitive alternative to the conventional numerical algorithms based on difference decomposition for solving the TFPDEs with variable coefficients. Numerical experiments confirm the ability and accuracy of the proposed methodology.</p>

Applications of intrinsic coordinates for simulation of thin shocks in relaxing media

The augmented Burgers equation that describes nonlinear propagation in a relaxing fluid can be expressed using intrinsic coordinates [Hammerton and Crighton, JFM 252, 585 (1993)]. A multivalued pressure waveform in physical coordinates can be represented as single-valued with intrinsic coordinates. Shocks can then be inserted into the multivalued pressure waveform using the equal area rule from weak shock theory to obtain a single-valued waveform solution, thus avoiding the high computational cost associated with discretizing thin shocks with conventional algorithms. By first solving the evolution equation in intrinsic coordinates, and then using the solution as the input to a conventional algorithm based on the augmented Burgers equation, a two-stage approach can be developed that achieves both high accuracy and significantly reduced computational cost when compared to solving the Burgers equation with conventional numerical algorithms alone. Approaches based on intrinsic coordinates are applied to realistic problems in relaxing media, which include air and seawater, to allow for comparisons of computational efficiency and accuracy. [WAW is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Modeling variable-density flow in saturated-unsaturated porous media: An advanced numerical model

Modeling variable-density flow in unconfined aquifers is a challenging task because of the nonlinear coupling between variably saturated flow and contaminant transport. This results in a highly nonlinear system since the strongly nonlinear Richards flow equation is, in addition, coupled to the advection-dispersion transport equation by viscosity and density variation. The solution of such a nonlinear system is often subject to convergence issues and can be very expansive in terms of computational time, especially for large-scale problems. Conventional numerical algorithms based on the sequential approach and the classical finite difference or finite element methods with the first-order backward Euler time integration scheme are generally inefficient and/or do not provide satisfactory results. In this work, we develop a new efficient and accurate 2D numerical model for the transport of dense contaminants in unsaturated porous media that allows for the simulation of large-scale problems. This research describes a new model that combines advanced spatial discretization methods (mixed hybrid finite element method, discontinuous Galerkin finite element method, and multipoint flux approximation method) with higher-order time integration techniques via the method of lines (MOL). The latter allows one to adapt the time step's size and the order of the time integration to improve the computational efficiency while maintaining accuracy. The robustness and accuracy of the new model are shown by comparison against a widely used commercial code based on the standard finite element method. The applicability of the developed model to a large-scale problem is then investigated by simulating saltwater intrusion under a climate change projection and long-term pumping regimes for the Akkar coastal aquifer in Lebanon using a simplified 2D conceptual model.

Machine learning based fast forward modelling of ground-based time-domain electromagnetic data

Inversion of large-scale time-domain electromagnetic surveys are computationally expensive and time consuming. Deterministic or probabilistic inversion schemes usually require calculations of forward responses, and often thousands to millions of forward responses are computed. We propose a machine learning based forward modelling approach as a computationally feasible alternative to approximate numerical forward modelling where a neural network is employed to model the relationship between the resistivity models and corresponding forward responses. For training of the neural network, we generated forward responses using conventional numerical algorithm for 93,500 resistivity models derived from different surveys conducted in Denmark representing typical resistivities of sedimentary geological layers. The input resistivity models and the network target outputs, i.e. forward responses, are scaled using a novel normalization strategy to ensure each gate is equally prioritized. The performance of the network is evaluated on two test datasets consisting of 8942 resistivity models by comparing the forward responses generated by the neural network and the conventional algorithm. We also measure the performance for the time derivatives of forward responses, i.e. dB/dt, by incorporating a system response. The results show that the proposed strategy is at least 13 times faster than commonly used accurate modelling methods and achieves an accuracy of 98% within 3% relative error, which is comparable to data uncertainty. Additional experiments on surveys from two other continents show that the results generalize in similar geological settings. Thus, under certain geological constraints, the proposed methodology may be incorporated into the pre-existing inversion structures, allowing for significantly faster inversion of large datasets.

An improved phase field method by using statistical learning theory-based optimization algorithm for simulation of martensitic transformation in NiTi alloy

An improved phase field method by using statistical learning theory based optimization algorithm is developed for solving the phase field equations through building simple relationships between the key phase field variables and the phase evolution driving force, and using statistical analysis of mass computed data during phase field simulation. Phase field simulation results of growth of R phase and the B2–R phase transformation in a Ni-rich Ni50.5Ti49.5 alloy by using the proposed statistical strategy algorithm are compared with that using the conventional numerical algorithm, which demonstrates that with coupling the statistical learning theory, i.e., by means of the optimization algorithm, the credible simulated microstructure is obtained while maintaining high accuracy, and meanwhile the computational time has been significantly reduced.

Position and Velocity Estimation Using TOA and FOA Based on Lagrange Programming Neural Network

This paper addresses the problem of estimating the position and velocity of a moving source utilizing the time-of-arrival (TOA) and frequency-of-arrival (FOA) measurements. Since the concerned estimation problem is highly non-linear and non-convex, we propose to utilize a novel neural circuit, namely the Lagrange programming neural network (LPNN) framework, to solve this problem. LPNN equips the abilities of fast convergence and the robustness of resisting high noise level, and thus these two advantages have drawn much attention for it recently. Since LPNN is able to solve the optimization problem with constraint, we first reformulate the original non-linear and non-convex maximum likelihood (ML) problem by introducing additional variables and constraints, and thus a neural network is built up based on the LPNN framework. Subsequently, the convergence and stability of the proposed neural network is mathematically proved and then verified by the results of numerical experiments. Different from the conventional numerical algorithms, the analog neural network can be utilized to fulfil the task of real-time calculation, especially when there are limited computation resources in some applications. The simulation results demonstrate that the proposed LPNN model equips the basic properties of convergence and stability, and also show the superior localization accuracy of the proposed method than other numerical algorithms.

Numerical recipes for successfully modeling the phase transitions in thermal energy storage adsorption systems

AbstractStrong convergence problems may arise during the dynamic simulation of the phase transitions of adsorption systems subjected to larger and/or random fluctuations of the boundary conditions, which do not follow the conventional cycle sequence (e.g., adsorptive thermal storage systems). This sequence could even be reversed at some instants and its stages not completed. Such convergence problems preclude solving those phase transitions, and thus the successful dynamic simulation of the adsorption systems. A set of numerical procedures for solving those problems is proposed, consisting of a small group of new conditions to be added to the conventional numerical algorithms. Its successful use is demonstrated in the dynamic simulation of a thermal energy storage system provided with an adsorption module, modeled in two distinct but coupled platforms (TRNSYS and MATLAB), for which the convergence difficulties are identified and described. Results indicate successful convergence by applying the proposed methodology, in contrast with the original algorithm, attesting for its capability to overcome the convergence problems that occur during the system's dynamic simulation. This set of numerical recipes can be adapted for other systems' algorithms, or to serve as a basis for its successful dynamic simulation, thus showing a promising and enlarged range of use.

Calculating for surface electric field of converter valve shield system with fast multipole curved boundary element method

The accurate and fast calculation of surface electric field of DC converter valve shield system have guiding significance in the designing process. However, the electrostatic field analysis is a large-scale problem with low efficiency using the conventional numerical algorithm. In order to solve this problem, fast multipole curved boundary element method (FMCBEM) was proposed. The FMCBEM is based on the indirect boundary integral equation of electrostatic field. Galerkin curved boundary element method based on coordinate transformation is used to improve the accuracy, then fast multipole method is used to improve the calculation speed and reduce the memory usage. Through simple examples, the accuracy and efficiency of the algorithm were verified. Compared with finite element method, it was found that the accuracy can meet the requirements of engineering. The efficiency was obviously better than the conventional boundary element method. Applying this algorithm, the surface electric fields of different shield systems of ±160 kV converter valve, which the degree of freedoms is more than 220 thousand, were analysed using desktop computer. This algorithm can calculate the large-scale electric field problem fast and accurately, which is an effective tool for the optimisation design.

Joint Multiple Sources Localization Using TOA Measurements Based on Lagrange Programming Neural Network

Multiple sources’ localization using the time-of-arrival (TOA) measurements in the presence of sensor position uncertainty is studied in this paper. The non-cooperative scenario where the clock between the source and sensors is not synchronized is also considered. First, we theoretically prove that the Cramer–Rao lower bound of multi-source joint localization is lower than that of the single source case when the TOA measurements from different sources possess the same sensor position errors. Moreover, different from the conventional numerical algorithms, we propose to employ a neural circuit named Lagrange programming neural network (LPNN) to fulfill this non-trivial task of jointly locating multiple sources. The maximum likelihood problem of multi-source localization is reformulated by utilizing the LPNN framework, and then we build up a neural model, which is proved to be asymptotically stable through both mathematical analysis and numerical experiments. The simulation results show that the proposed method is superior to other positioning algorithms, and it has excellent localization performance and robustness even in the case of large measurement noise, synchronization error, and sensor position displacements.

Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms

Quantum lattice algorithms originated with the Feynman checkerboard model for the one-dimensional Dirac equation. They offer discrete models of quantum mechanics in which the complex numbers representing wavefunction values on a discrete spatial lattice evolve through discrete unitary operations. This paper draws together some of the identical, or at least unitarily equivalent, algorithms that have appeared in three largely disconnected strands of research. Treated as conventional numerical algorithms, they are all only first order accurate under refinement of the discrete space/time grid, but may be raised to second order by a unitary change of variables. Much more efficient implementations arise from replacing the evolution through a sequence of unitary intermediate steps with a short path integral formulation that expresses the wavefunction at each spatial point on the most recent time level as a linear combination of values at immediately preceding time levels and neighbouring spatial points. In one dimension, a particularly elegant reformulation replaces two variables at two time levels with a single variable over three time levels. The resulting algorithm is a variational integrator arising from a discrete action principle, and coincides with the Ablowitz–Kruskal–Ladik finite difference scheme for the Klein–Gordon equation.

Semi-analytic texturing algorithm for polygon computer-generated holograms.

A texturing method for the semi-analytic polygon computer-generated hologram synthesis algorithm is studied. Through this, the full-potential and development direction of the semi-analytic polygon computer-generated holograms are discussed and compared to that of the conventional numerical algorithm of polygon computer-generated hologram generation based on the fast Fourier transform and bilinear interpolation. The theoretical hurdle of the semi-analytic texturing algorithm is manifested and an approach to resolve this problen. A key mathematical approximation in the angular spectrum computer-generated hologram computation, as well as the trade-offs between texturing effects and computational efficiencies are analyzed through numerical simulation. In this fundamental study, theoretical potential of the semi-analytic polygon computer-generated hologram algorithm is revealed and the ultimate goal of research into the algorithm clarified.

Minimum risk facility location-allocation problem with type-2 fuzzy variables.

Facility location decision is basically viewed as a long-term strategy, so the inherited uncertainty of main parameters ought to be taken into account in order to make models applicable. In this paper, we examine the impact of uncertain transportation costs and customers' demands on the choice of optimal location decisions and allocation plans. This leads to the formulation of the facility location-allocation (FLA) problem as a fuzzy minimum risk programming, in which the uncertain parameters are assumed to be characterized by type-2 fuzzy variables with known type-2 possibility distributions. Since the inherent complexity of type-2 fuzzy FLA may be troublesome, existing methods are no longer effective in handling the proposed problems directly. We first derive the critical value formula for possibility value-at-risk reduced fuzzy variable of type-2 triangular fuzzy variable. On the basis of formula obtained, we can convert original fuzzy FLA model into its equivalent parametric mixed integer programming form, which can be solved by conventional numerical algorithms or general-purpose software. Taking use of structural characteristics of the equivalent optimization, we design a parameter decomposition method. Finally, a numerical example is presented to highlight the significance of the fuzzy FLA model. The computational results show the credibility and superiority of the proposed parametric optimization method.

Task scheduling in budget-constrained cloud computing systems to maximise throughput

We consider the problem of running a large amount of independent equal-sized tasks in the cloud with a budget constraint from the perspective of the cloud users. We model the cloud infrastructure by a node-weighted edge-weighted star-shaped graph and focus on the maximisation of the steady-state throughput. We show that the specific budget-constrained steady-state throughput maximisation problem can be formulated and solved as a linear programming (LP) problem. We incorporate the compute nodes of different communication capacity into our problem formulation in a unified way. We identify two modes of the system where closed-form solutions exist, i.e., the budget-bound mode and the communication-bound mode. The best scheduling scheme is benefit-first (resp. communication-first) when the system is budget-bound (resp. communication-bound), where tasks should be scheduled onto the compute nodes in the descending order of the effective benefit (resp. bandwidth). When the system is under conditions other than these two modes, we propose a simple heuristic to solve it instead of resolving to the conventional numerical algorithms for the LP. Simulation results show that the simple heuristic outperforms other intuitive heuristics under varied sampled system setups.

Wiener Chaos Expansion and Simulation of Electromagnetic Wave Propagation Excited by a Spatially Incoherent Source

First, we propose a new stochastic model for a spatially incoherent source in optical phenomena. The model naturally incorporates the incoherent property into the electromagnetic wave equation through a random source term. Then we propose a new numerical method based on Wiener chaos expansion (WCE) and apply it to solve the resulting stochastic wave equation. The main advantage of the WCE method is that it separates random and deterministic effects and allows the random effects to be factored out of the primary partial differential equation (PDE) very effectively. Therefore, the stochastic PDE is reduced to a set of deterministic PDEs for the coefficients of the WCE method which can be solved by conventional numerical algorithms. We solve these secondary deterministic PDEs by a finite-difference time domain (FDTD) method and demonstrate that the numerical computations based on the WCE method are considerably more efficient than the brute-force simulations. Moreover, the WCE approach does not require generation of random numbers and results in less computational errors compared to Monte Carlo simulations.

Reducing the computational requirements of time-reversal device optimizations

Time-reversal simulations using conventional numerical algorithms provide the basis for a simple component optimization procedure. However, the computational requirements of the approach can become excessive, requiring the recording of the complete field time histories on a surface surrounding the problem space. In this work temporal, spatial and modal filtering methods are employed to significantly reduce the computational resources demanded by the time-reversal process. The design of a number of WR90 (X-band) waveguide components is used to quantify the balance between the computational efficiency gains and deterioration in the final design quality. Copyright © 2009 John Wiley & Sons, Ltd.

A Cooperative Recurrent Neural Network for Solving L1 Estimation Problems with General Linear Constraints

The constrained L(1) estimation is an attractive alternative to both the unconstrained L(1) estimation and the least square estimation. In this letter, we propose a cooperative recurrent neural network (CRNN) for solving L(1) estimation problems with general linear constraints. The proposed CRNN model combines four individual neural network models automatically and is suitable for parallel implementation. As a special case, the proposed CRNN includes two existing neural networks for solving unconstrained and constrained L(1) estimation problems, respectively. Unlike existing neural networks, with penalty parameters, for solving the constrained L(1) estimation problem, the proposed CRNN is guaranteed to converge globally to the exact optimal solution without any additional condition. Compared with conventional numerical algorithms, the proposed CRNN has a low computational complexity and can deal with the L(1) estimation problem with degeneracy. Several applied examples show that the proposed CRNN can obtain more accurate estimates than several existing algorithms.

05/02107 Thermal effect on electric power network components under short-circuit currents

In this paper a novel approach for power system contingency analysis is proposed. The developed expert system (HESCA), incorporates operator knowledge into the contingency selection process towards reducing the number of contingencies to be computed. The proposed hybrid approach combines the conventional numerical algorithms and knowledge-based methods to enhance the accuracy of final results.The proposed method was applied to modified IEEE 30 bus and an actual national grid power system to show its feasibility and capabilities. This paper demonstrates the improved results in comparison to conventional methods and shows that the proposed approach is reliable and efficient to be used in power network control centers.