Numerical Complexity Research Articles

A Discrete Sine‐Cosine Transforms Galerkin Method for the Conductivity of Heterogeneous Materials With Mixed Dirichlet/Neumann Boundary Conditions

ABSTRACTThis work aims at developing a numerical method for conductivity problems in heterogeneous media subjected to mixed Dirichlet/Neumann boundary conditions. The method relies on a fixed‐point iterative solution of an auxiliary problem obtained by a Galerkin discretization using an approximation space spanned by mixed cosine‐sine series. The solution field is written as a known term verifying the boundary conditions and an unknown term described by cosine‐sine series, having no contribution on the boundary. Discrete sine‐cosine transforms, of Type I and III depending on the boundary conditions, are used to approximate the elementary integrals involved in the Galerkin formulation, which makes the method relying on the numerical complexity of fast Fourier transforms. The method is finally assessed in a problem of a composite material.

A discrete sine–cosine based method for the elasticity of heterogeneous materials with arbitrary boundary conditions

The aim of this article is to extend Moulinec and Suquet (1998)’s FFT-based method for heterogeneous elasticity to non-periodic Dirichlet/Neumann boundary conditions. The method is based on a decomposition of the displacement into a known term verifying the boundary conditions and a fluctuation term, with no contribution on the boundary, and described by appropriate sine–cosine series. A modified auxiliary problem involving a polarization tensor is solved within a Galerkin-based method, using an approximation space spanned by sine–cosine series. The elementary integrals emerging from the weak formulation of the equilibrium are approximated by discrete sine–cosine transforms, which makes the method relying on the numerical complexity of Fourier transforms. The method is finally assessed in several problems including kinematic uniform, static uniform and arbitrary Dirichlet/Neumann boundary conditions.

Some remarks on the effect of the Random Batch Method on phase transition

In this article, we focus on two toy models : the Curie–Weiss model and the system of N particles in linear interactions in a double well confining potential. Both models, which have been extensively studied, describe a large system of particles with a mean-field limit that admits a phase transition. We are concerned with the numerical simulation of these particle systems. To deal with the quadratic complexity of the numerical scheme, corresponding to the computation of the O(N2) interactions per time step, the Random Batch Method (RBM) has been suggested. It consists in randomly (and uniformly) dividing the particles into batches of size p>1, and computing the interactions only within each batch, thus reducing the numerical complexity to O(Np) per time step. The convergence of this numerical method has been proved in other works.This work is motivated by the observation that the RBM, via the random constructions of batches, artificially adds noise to the particle system. The goal of this article is to study the effect of this added noise on the phase transition of the nonlinear limit, and more precisely we study the effective dynamics of the two models to show how a phase transition may still be observed with the RBM but at a lower critical temperature.

The Numerically Leveled Performance Assessment Framework (NLPAF): A Literature-Based Evaluation of Acceptability

ABSTRACT The purpose of this study is to present a literature-based, initial “evaluation of acceptability” (American Educational Research Association et al., 2014, p. 7) of the Numerically Leveled Performance Assessment Framework (NLPAF). Its results enable confidence in continuing its use in the authors’ professional development work and in making it available to professional developers and researchers. The authors developed the NLPAF to guide elementary mathematics teachers enrolled in a professional development program in constructing performance assessments they then administer up to three times annually to their students. These assessments measure growth across time in mathematical proficiency relating to the domain of whole number operations most prominent in the curriculum standards for the grade level taught. Numerically leveled refers to the process teachers learn and apply to estimate the size and numerical complexity of computations to be used in designing performance tasks for their students’ unique levels.

Input-to-state stabilization of discrete-time delayed fuzzy systems via aperiodically intermittent control

This paper studies input-to-state stabilization of delayed discrete-time Takagi–Sugeno (T–S) fuzzy systems via aperiodically intermittent control. We first consider aperiodically intermittent time-triggered control, where we present sufficient conditions via the mathematical induction under the hypotheses of the quasiperiodicity condition. Based on the derived sufficient conditions, we apply a Lyapunov–Krasovskii (L–K) method together with the descriptor method to derive the explicit linear matrix inequalities (LMIs) that ensure the exponential stability and input-to-state stability (ISS), and show the existence of the aperiodically intermittent time-triggered controller that leads to efficient results with much less numerical complexity. We next consider aperiodically intermittent dynamic event-triggered control with an additional parameter that is larger than one. This strategy allows that the introduced dynamical variable does not remain constant but increases during the control rest interval. As a result, the proposed dynamic event-triggered strategy leads to a smaller number of sent signals than that for the case of the additional parameter which equals to one. Finally, numerical examples including a practical inverted pendulum on a cart are presented to verify the validity of the proposed method.

UWB RCS calculations of smooth targets via the multi-band beam summation method

We present an ultra-wide-band beam-summation (UWB-BS) algorithm for calculating the radar cross-section (RCS) of large smooth targets whose asymptotic numerical complexity is of O ( 1 ) . In this approach, the incident field is expanded into a sparse set of iso-diffracting Gaussian beams (ID-GB), tiling the phase space (PS) of beam origins and directions. The field of each ID-GB is then tracked through reflections from the scatterer until the beam emerges from the target zone and propagates to the far zone using a closed-form near-to-far (N2F) zone transformation, and finally, the beam contributions are aggregated to obtain the scattered field. The algorithm can accommodate a multi-octave frequency band by dividing it into octave-wide sub-bands and introducing a UWB-BS scheme such that the PS beam lattices in the higher sub-bands are obtained by spatial-spectral upsampling those of the lower bands. The PS summation of ID-GBs is a priori localized and involves a roughly constant relatively small number of contributing beams in all the sub-bands, leading to an O ( 1 ) (frequency-independent) asymptotic computational complexity. Furthermore, with the ID parametrization, the beam axes and their propagation and scattering parameters are frequency-independent and, therefore, need to be calculated only once and then reused for all frequencies, thus greatly reducing the numerical cost of computing the RCS at multiple frequencies. The paper describes all aspects of the algorithm, including the choice of the beam parameters for asymptotic error convergence and the utilization of localization for algorithmic efficacy.

Numerical Analysis of the Discrete, Fractional Order PID Controller Using FOBD Approximation

This paper proposes a methodology of the numerical testing of the discrete, approximated Fractional Order PID Controller (FOPID). The fractional parts of the controller are approximated using the Fractional Order Backward Difference (FOBD) operator. The goal of the analysis is to find the memory length optimum from point of view both accuracy and duration of computations. To do it new cost functions describing both accuracy and numerical complexity were proposed and applied. Results of tests indicate that the optimum memory length lies between 200 and 400. The proposed approach can be also useful to examine of another discrete implementations of a fractional order operator using FOBD.

Marginal models with individual-specific effects for the analysis of longitudinal bipartite networks

AbstractA new modeling framework for bipartite social networks arising from a sequence of partially time-ordered relational events is proposed. We directly model the joint distribution of the binary variables indicating if each single actor is involved or not in an event. The adopted parametrization is based on first- and second-order effects, formulated as in marginal models for categorical data and free higher order effects. In particular, second-order effects are log-odds ratios with meaningful interpretation from the social perspective in terms of tendency to cooperate, in contrast to first-order effects interpreted in terms of tendency of each single actor to participate in an event. These effects are parametrized on the basis of the event times, so that suitable latent trajectories of individual behaviors may be represented. Inference is based on a composite likelihood function, maximized by an algorithm with numerical complexity proportional to the square of the number of units in the network. A classification composite likelihood is used to cluster the actors, simplifying the interpretation of the data structure. The proposed approach is illustrated on simulated data and on a dataset of scientific articles published in four top statistical journals from 2003 to 2012.

A combinatorial model reduction method for the finite element analysis of wind instruments

AbstractA high‐fidelity finite element model is proposed for the complete simulation of the time‐harmonic acoustic propagation in wind instruments. The challenge is to meet the extremely high accuracy required by professional musicians, in a complex domain, for all fingerings and over a wide frequency range, within an affordable computational time. Several modelling assumptions are made to limit the numerical complexity of the problem while preserving all relevant physics. A dedicated high‐performance solution strategy is also proposed, based on partitioning, condensation and model order reduction, exploiting the combinatorial nature of wind instrument fingerings. Finally, the proposed approach is applied to the simulation of an alto saxophone. An order of magnitude reduction in memory and computational cost is achieved.

Optimization of tuned mass dampers – minimization of potential energy of elastic deformation

A tuned mass damper (TMD) optimization can be performed under various assumptions and objectives. All the variables of the optimization, such as structural model, performance index and load type affect the optimal parameters of the TMD. This paper presents a new optimization method that implements straightforward performance index and allows taking load spectral characteristics into account. Thanks to the usage of modal coordinates, the method allows fast numerical optimization of TMD attached to large or complicated structures with numerous degrees of freedom. One of the complicated tasks while optimizing TMD is the choice of a performance index. In this paper, the mean value of potential energy stored in the elastic deformation of a structure under periodic load serves as a performance index, which leads to a low numerical complexity task if the optimization is performed in the frequency domain. The new method also allows a simple inclusion of load spectral characteristics and permits TMD optimization for any loading spectral range. When applied to a structure with a single degree of freedom, this method leads to H2 optimization in the case of white noise excitation. However, it is applicable to multiple degrees of freedom structures with single or multiple TMDs and any given load. The paper also presents several examples of numerical optimization of the TMD attached to both single and multiple degrees of freedom structures under various loads, including white noise excitation, pedestrian load, and earthquake strong motion.

Modeling polydispersed droplets in non-equilibrium condensing CO2 flows through turbine cascades using moment-based methods for efficient energy utilization

The efficiency of a power cycle is significantly influenced by the condensation process and the size distribution of the droplets, as these factors directly affect heat transfer rates and overall energy conversion efficiency. Accurate numerical modeling of thermodynamic non-equilibrium condensation is crucial for predicting droplet nucleation and growth in carbon dioxide flows. This study applies moment-based polydispersed droplet models, such as the quadrature method of moments, to account for the droplet size spectrum. Additionally, discrete methods based on a predefined shape of the polydispersed droplet size spectrum are utilized and evaluated to reduce numerical complexity. Our results demonstrate that polydispersed models provide higher accuracy compared to monodispersed models when simulating steam and supercritical carbon dioxide flows in converging–diverging nozzles. In turbine cascades, the choice of model significantly influences the predicted mean droplet size and efficiency. Isentropic efficiency evaluations reveal higher values for carbon dioxide compared to steam, with efficiencies of 94.6–95.8% versus 91.8–92.9% for wet cases and 96.5–97.2% versus 95.8–96.5% for dry cases. Notably, spontaneous droplet nucleation slightly affects the efficiency of the blade cascade in carbon dioxide flows, whereas steam flows experience efficiency reductions between 3.58% and 4.01%. This work advances the understanding of droplet condensation processes in turbine cascades, highlighting the superior performance of polydispersed models and providing quantitative insights into their impact on efficiency.

Energies and spectra of solids from the algorithmic inversion of dynamical Hubbard functionals

Energy functionals of the Green's function can simultaneously provide spectral and thermodynamic properties of interacting electrons' systems. Although powerful in principle, these formulations need to deal with dynamical (frequency-dependent) quantities, increasing the algorithmic and numerical complexity and limiting applications. We first show that, when representing all frequency-dependent propagators as sums over poles—a truncated Lehmann representation—, the typical operations of dynamical formulations become closed (i.e., all quantities are expressed as sums over poles) and analytical. In the framework, the Dyson equation is mapped into a nonlinear eigenvalue problem that can be solved exactly; this is achieved by introducing a fictitious noninteracting system with additional degrees of freedom, which shares, upon projection, the same Green's function of the real system. In addition, we introduce an approximation to the exchange-correlation part of the Klein functional adopting a localized GW approach; this is a generalization of the static Hubbard extension of density-functional theory with a dynamical screened potential U(ω). We showcase the algorithmic efficiency of the method, and the physical accuracy of the functional, by computing the spectral, thermodynamic, and vibrational properties of SrVO3, finding results in close agreement with experiments and state-of-the-art methods, at highly reduced computational costs and with a transparent physical interpretation. Published by the American Physical Society 2024

Learning quantum symmetries with interactive quantum-classical variational algorithms

A symmetry of a state |ψ⟩ is a unitary operator of which |ψ⟩ is an eigenvector. When |ψ⟩ is an unknown state supplied by a black-box oracle, the state’s symmetries provide key physical insight into the quantum system; symmetries also boost many crucial quantum learning techniques. In this paper, we develop a variational hybrid quantum–classical learning scheme to systematically probe for symmetries of |ψ⟩ with no a priori assumptions about the state. This procedure can be used to learn various symmetries at the same time. In order to avoid re-learning already known symmetries, we introduce an interactive protocol with a classical deep neural net. The classical net thereby regularizes against repetitive findings and allows our algorithm to terminate empirically with all possible symmetries found. An iteration of the learning algorithm can be implemented efficiently with non-local SWAP gates; we also give a less efficient algorithm with only local operations, which may be more appropriate for current noisy quantum devices. We simulate our algorithm on representative families of states, including cluster states and ground states of Rydberg and Ising Hamiltonians. We also find that the numerical query complexity scales well for up to moderate system sizes.

Nambu-covariant many-body theory II: Self-consistent approximations

The theory of Self-Consistent Green’s Function (SCGF) is reformulated in an explicit Nambu-covariant fashion for applications to many-body systems at non-zero temperature in symmetry-broken phases. This is achieved by extending the Nambu-covariant formulation of perturbation theory, presented in the first part of this work, to non-perturbative schemes based on self-consistently dressed propagators and vertices. We work out in detail the self-consistent ladder approximation, motivated by a trade-off between numerical complexity and many-body phenomenology. Taking a complex general Hartree–Fock–Bogoliubov (HFB) propagator as a starting point, we also formulate and prove a sufficient condition on the stability of the HFB self-energy to ensure the convergence of the initial series of ladders at any energy. The self-consistent ladder approximation is written purely in terms of spectral functions and the resulting set of equations, when expressed in terms of Nambu tensors, are remarkably similar to those in the symmetry-conserving case. This puts the application of the self-consistent ladder approximation to symmetry-broken phases of infinite nuclear matter within reach.

On design of preprocessing and postprocessing in cascaded-resonator-based harmonic analyzers and filter banks

The cascaded-resonator (CR)-based filter structure provides high attenuation in the stopbands thanks to the serially coupled resonators, which is an advantage in many applications of harmonic analyses and/or filtering. In addition to that, it has good properties provided by its parallel structure. On the other side, there are applications which besides the high stopband attenuation need wider (or even flat-top) passbands, lower group delays, etc. These performances can be modified/improved by preprocessing and postprocessing through reshaping of the primary CR-based filter bank frequency responses. The numerical complexity is also an important aspect where infinite impulse response (IIR) filters provide implementation with a smaller number of numerical operations, but involving nonlinearity of the phase responses and stability control issue. Although most of these issues have been discussed in the previous papers dealing with the design of the CR-based harmonic filtering and/or filter banks, the aim of this paper is to highlight them all together. In addition, some improvements of the previously described design techniques are suggested too.

The Heston–Queue-Hawkes process: A new self-exciting jump–diffusion model for options pricing, and an extension of the COS method for discrete distributions

We propose a new self-exciting jump–diffusion process, the Heston–Queue-Hawkes (HQH) model, which integrates the well-known Heston model with the recently introduced Queue-Hawkes (Q-Hawkes) jump process. Similar to the Heston–Hawkes process (HH), the HQH model effectively captures both the slow and continuous evolution of prices, and the sudden and impactful market movements due to self-excitation and contagion. But a significant advantage of the HQH model is that its characteristic function is available in closed form, allowing for the efficient application of Fourier-based fast pricing algorithms, such as the COS method (which we extend to deal with discrete distributions). We also demonstrate that, by leveraging partial integrals of the characteristic function, which are explicitly known for the HQH process, we can reduce the dimensionality of the COS method, thereby decreasing its numerical complexity. Our numerical results for pricing European and Bermudan options indicate that the HQH model provides a broader range of volatility smiles compared to the Bates model, while maintaining a substantially lower computational burden than the HH process.

Resilient and robust management policy for multi‐stage supply chains with perishable goods and inaccurate forecast information: A distributed model predictive control approach

AbstractAn efficient supply chain (SC) management requires that decisions are taken to minimize the effects of parametric uncertainties and unpredictable external disturbances. In this article, we consider this problem with reference to a multi‐stage SC (MSSC) whose dynamics is characterized by the following elements of complexity: perishable goods with uncertain perishability rate, an uncertain future customer demand that is only known to fluctuate inside a given compact set. The problem we face is to define a resilient and robust Replenishment Policy (RP) such that at any stage the following requirements are satisfied: the fulfilled demand is maximized, overstocking is avoided, the bullwhip effect (BE) is mitigated. These objectives should be pursued despite the mentioned uncertainties and unexpected customer demand behaviors violating the bounds of the compact set. Robustness is here intended with respect to uncertainty on the perishability rate, and resiliency as the ability to quickly react to the mentioned unforeseen customer demands. We propose a method based on a distributed resilient robust model predictive control (DRRMPC) approach. Each local robust MPC (RMPC) involves solving a Min‐Max constrained optimization problem (MMCOP). To drastically reduce the numerical complexity of each MMCOP, we parametrize its solution by means of B‐spline functions.

A Surrogate Model of Heat Transfer Mechanism in a Domestic Gas Oven: A Numerical Simulation Approach for Premixed Flames

This paper introduces an innovative analytical model to compute flame velocities and temperatures within a premix burner in a domestic gas oven. This model significantly streamlines the heat transfer simulation process by simplifying the modeling of the thermo-chemical energy release during combustion, effectively reducing complexity and computation time. Accelerated solutions are essential at the initial design stages when comparing the effect of the oven parameter variation on the overall performance. The validation of the proposed analytical model involved experimental assessments of the temperature of the false bottom plate in a natural gas oven. The resulting data were then compared against CFD simulations performed utilizing the proposed model. The results revealed a marginal discrepancy of 4% between the experimental measurements and the outcomes generated by the model. Simulations were executed under differing conditions, encompassing scenarios with and without radiation effects. This exploration identified natural convection as the predominant heat transfer mechanism, with heat radiation contributing only modestly to the heating of the false bottom plate. Among its advantages, the proposed model offers a notable reduction in the numerical complexity of the modeling of the combustion process. Furthermore, its straightforward integration into numerical simulations involving premixed flames underscores its practical utility and versatility in evaluating design performance at the early stages of the design. Highly accurate models can be left for the final oven configuration validation.

Data-driven identification and comparison of full multivariable models for propofol–remifentanil induced general anesthesia

In this paper, we present results with clinical data to enable a 2x2 input–output multivariable patient model for hypnosis and analgesia. Nonlinear multi-drug interaction models are identified from data recorded from 70 patients undergoing surgery during total intravenous anesthesia (TIVA) with several medical monitors for variables such as Bispectral Index, Nociception level (Medasense), skin conductance (Medstorm) and advanced spectral analysis conductance (AnspecPro). Bispectral index measures the depth of hypnosis (lack of consciousness), while nociception related indices from Medasense, Medstorm, and AnspecPro devices measure levels related to analgesia (lack of reaction to noxious stimuli). A comparison is given among three response surface model (RSM) structures – Minto, Greco, and Reduced Greco – for hypnotic and analgesic states during Propofol–Remifentanil interaction. The identified models capture the pharmacodynamic properties of dose–effect concentrations of Propofol/Remifentanil while the pharmacokinetic part of the patient model is calculated from patient’s biometric values using Schnider/Minto (SM), and Eleveld/Eleveld (EE) models. In presence of strict clinical protocols delivering data under poor identifiability conditions, we propose two methods of identification: (i) based on steady-state gains, and (ii) using all available data which includes part of the dynamic transient. The model parameters are optimized with Genetic Algorithm based on a goodness of fit performance measure complemented with root mean square error. The results suggest that the EE model combination is advantageous for Bispectral index pharmacokinetic modeling at the cost of numerical complexity, therefore reducing the uncertainty left to be identified in the pharmacodynamic part of the patient model. By contrast, the SM model combination is less computationally demanding and provides some improvement in the RSM accuracy for nociception level indicators. The comparison of three devices for nociception levels evaluation suggests that clinical data captured with the Medasense monitor provides best fitted RSMs with the Reduced Greco RSM structure, despite having fewer parameters.

Learning Lyapunov terminal costs from data for complexity reduction in nonlinear model predictive control

AbstractA classic way to design a nonlinear model predictive control (NMPC) scheme with guaranteed stability is to incorporate a terminal cost and a terminal constraint into the problem formulation. While a long prediction horizon is often desirable to obtain a large domain of attraction and good closed‐loop performance, the related computational burden can hinder its real‐time deployment. In this article, we propose an NMPC scheme with prediction horizon and no terminal constraint to drastically decrease the numerical complexity without significantly impacting closed‐loop stability and performance. This is attained by constructing a suitable terminal cost from data that estimates the cost‐to‐go of a given NMPC scheme with long prediction horizon. We demonstrate the advantages of the proposed control scheme in two benchmark control problems.