Convergence Behavior Research Articles

Finite-Size Effects in Periodic EOM-CCSD for Ionization Energies and Electron Affinities: Convergence Rate and Extrapolation to the Thermodynamic Limit.

We investigate the convergence of quasiparticle energies for periodic systems to the thermodynamic limit using increasingly large simulation cells corresponding to increasingly dense integration meshes in reciprocal space. The quasiparticle energies are computed at the level of equation-of-motion coupled-cluster theory for ionization (IP-EOM-CC) and electron attachment processes (EA-EOM-CC). By introducing an electronic correlation structure factor, the expected asymptotic convergence rates for systems with different dimensionality are formally derived. We rigorously test these derivations through numerical simulations for trans-polyacetylene using IP/EA-EOM-CCSD and the G0W0@HF approximation, which confirm the predicted convergence behavior. Our findings provide a solid foundation for efficient schemes to correct finite-size errors in IP/EA-EOM-CCSD calculations.

Gradient projection method for enforcing crack irreversibility as box constraints in a robust monolithic phase-field scheme

A phase-field monolithic scheme based on the gradient projection method is developed to model crack propagation in brittle materials under cyclic loading. As a type of active set method, the gradient projection method is particularly attractive to enforce the irreversibility condition imposed on the phase-field variables as bound constraints, or box constraints. This method has the advantages of allowing the rapid change of active constraints during iterations and computing the projected gradient with a negligible cost. The gradient projection method is further combined with the limited-memory BFGS (L-BFGS) method to overcome the convergence difficulties arising from the non-convex energy functional. A compact representation of the BFGS matrix is adopted as the limited-memory feature to avoid the storage of fully dense matrices, making this method practical for large-scale finite element simulations. By locating the generalized Cauchy point on the piecewise linear path formed by the projected gradient, the active set of box constraints can be determined. The variables in the active set, which are at the boundary of the box constraints, are kept fixed to form a subspace minimization problem. A primal approach and a dual approach are presented to solve this subspace minimization problem for the remaining free variables at the generalized Cauchy point. Several two-dimensional (2D) and three-dimensional (3D) examples are provided to demonstrate the capabilities of the proposed monolithic scheme, particularly in enforcing the phase-field irreversibility during crack propagation under cyclic loading. In these numerical examples, the proposed monolithic scheme is combined with an adaptive mesh refinement technique to alleviate the heavy computational cost incurred by the fine mesh resolution required around the crack region. The proposed method is further compared with two other phase-field solving techniques regarding the convergence behavior. To ensure a fair comparison, the same problem settings and implementation techniques are adopted. The proposed monolithic scheme provides a unified framework to overcome the numerical difficulties associated with the non-convex energy functional, effectively enforce the phase-field irreversibility to ensure the thermodynamic consistency, and alleviate the heavy computational cost through adaptive mesh refinement in 2D and 3D phase-field crack simulations.

Noise amplification and ill-convergence of Richardson-Lucy deconvolution

Richardson-Lucy (RL) deconvolution optimizes the likelihood of the object estimate for an incoherent imaging system. It can offer an increase in contrast, but converges poorly, and shows enhancement of noise as the iteration progresses. We have discovered the underlying reason for this problematic convergence behaviour using a Cramér Rao Lower Bound (CRLB) analysis. An analytical expression for the CRLB diverges for spatial frequency components that approach the diffraction limit from below. The resulting mean noise variance per pixel diverges for large images. These results imply that a regular optimum of the likelihood does not exist, and that RL deconvolution is necessarily ill-convergent.

Finite Differences on Sparse Grids for Continuous-Time Heterogeneous Agent Models

We present a finite difference method working on sparse grids to solve higher dimensional heterogeneous agent models. If one wants to solve the arising Hamilton–Jacobi–Bellman equation on a standard full grid, one faces the problem that the number of grid points grows exponentially with the number of dimensions. Discretizations on sparse grids only involve O(N(logN)d−1) degrees of freedom in comparison to the O(Nd) degrees of freedom of conventional methods, where N denotes the number of grid points in one coordinate direction and d is the dimension of the problem. While one can show convergence for the used finite difference method on full grids by using the theory introduced by Barles and Souganidis, we explain why one cannot simply use their results for sparse grids. Our numerical studies show that our method converges to the full grid solution for a two-dimensional model. We analyze the convergence behavior for higher dimensional models and experiment with different sparse grid adaptivity types.

Stochastic convergence behaviour in carbon dioxide emissions: Fourier panel unit root approach

ABSTRACT This study examines the convergence of per capita carbon dioxide emissions using two Fourier-based panel unit root tests allowing for smooth transitions. Additional panel unit root tests without structural breaks and with sharp shifts were conducted for comparison. The analysis spans 68 countries from 1965 to 2019. Results reveal non-stationarity in emissions for low- and middle-income countries, while emissions in high-income countries were stationary under constant and trend components. Evidence indicates limited convergence, with constrained forms observed only in high-income nations. These findings align partially with the Environmental Kuznets Curve hypothesis. This study offers nuanced insights into global carbon dioxide emission dynamics and enriches the empirical discourse on environmental sustainability.

RED-E-Function-Based Equilibrium Parameter Finder: Finding the Best Restraint Parameters in Absolute Binding Free Energy Calculations.

Free energy perturbation (FEP)-based absolute binding free energy (ABFE) calculations have emerged as a powerful tool for the accurate prediction of ligand-protein binding affinities in drug discovery. The restraint addition is crucial in FEP-ABFE calculations; however, due to the non-orthogonal couplings between the restrained degrees of freedom, it typically requires numerous λ windows to ensure the phase-space overlap during restraint addition. This study introduces the RED-E-function-based equilibrium parameter finder (REPF), a novel method that relies on harmonic restraints to optimize the equilibrium values in restraints, enhancing phase-space overlap and improving the convergence of the restraint addition. REPF was applied to 44 protein-ligand complexes across 5 targets and compared to restraint schemes reported in the literature. We found that REPF-optimized restraints achieve an accuracy comparable to that of the 12λ approach while using only 2λ simulations, resulting in a significant reduction in computational costs. Extensive tests confirmed the improved convergence behavior and reduced energy fluctuations of REPF-optimized restraints.

A Cascaded Duplex Organic Vertical Memory with Learning Rate Scheduling for Efficient Artificial Neural Network Training

AbstractLearning rate scheduling (LRS) is a critical factor influencing the performance of neural networks by accelerating the convergence of learning algorithms and enhancing the generalization capabilities. The escalating computational demands in artificial intelligence (AI) necessitate advanced hardware solutions capable of supporting neural network training with LRS. This not only requires linear and symmetric analog programming capabilities but also the precise adjustment of channel conductance to achieve tunable slope in weight update behaviors. Here, a cascaded duplex organic vertical memory is proposed with the coupling of ferroelectric polarization effect and Schottky gate control on the same semiconducting channel, exhibiting adjustable‐slope conductance update with high linearity and symmetry. Therefore, in the chest X‐ray image detection, a fast‐to‐slow LRS is used for a bi‐layer ANN training, achieving a rapid, stable convergence behavior within only 15 epochs and a high recognition accuracy. Moreover, the proposed LRS training is also suitable for the Mackey Glass prediction task using long short‐term memory networks. This work integrates LRS into synaptic devices, enabling efficient hardware implementation of neural networks and thus enhancing AI performance in practical applications.

Expansion mapping in controlled metric space and extended B-metric space

This paper delves into the intricate study of expansion mappings within the frameworks of controlled metric spaces and extended B-metric spaces. Expansion mappings, known for their crucial role in fixed-point theory and iterative processes, are examined under the lens of these generalized metric spaces to uncover their distinct properties and extended applicability. Controlled metric spaces, which incorporate a dynamic control function to modulate the distance measurements, offer a refined approach to traditional metric space concepts. This flexibility allows for a more nuanced understanding of spatial relationships and convergence behaviors. Extended B-metric spaces, by relaxing the stringent requirements of the triangle inequality, open new avenues for theoretical exploration and practical application, accommodating broader classes of functions and sequences. In this study, we aim to provide a comprehensive analysis of the behavior of expansion mappings in these sophisticated metric frameworks. We will present new theoretical results that extend and generalize existing principles, highlighting the interplay between the control functions in controlled metric spaces and the relaxed conditions in extended B-metric spaces. Additionally, we explore practical applications of these findings in various fields, including optimization, computational mathematics, and the analysis of iterative methods. By bridging the gap between classical metric spaces and their generalized counterparts, this paper contributes to a deeper understanding of the mathematical foundations and potential applications of expansion mappings. The results presented herein pave the way for further research and development in this vibrant area of mathematical analysis.

Symmetric Properties of λ-Szász Operators Coupled with Generalized Beta Functions and Approximation Theory

This research work focuses on λ-Szász–Mirakjan operators coupling generalized beta function. The kernel functions used in λ-Szász operators often possess even or odd symmetry. This symmetry influences the behavior of the operator in terms of approximation and convergence properties. The convergence properties, such as uniform convergence and pointwise convergence, are studied in view of the Korovkin theorem, the modulus of continuity, and Peetre’s K-functional of these sequences of positive linear operators in depth. Further, we extend our research work for the bivariate case of these sequences of operators. Their uniform rate of approximation and order of approximation are investigated in Lebesgue measurable spaces of function. The graphical representation and numerical error analysis in terms of the convergence behavior of these operators are studied.

Truncated amplitude flow with coded diffraction patterns

Abstract The phase retrieval problem, which involves Fourier transforms with random masks, has a wide range of applications in signal and image processing, such as diffraction imaging. In recent years, various first-order methods have been explored, including truncated amplitude flow (TAF). These methods are considered to be simple and effective for solving the phase retrieval problem. However, existing convergence results for TAF mainly focus on random Gaussian measurements, which is very far from practical scenarios. In this paper, we analyze the convergence of a revised TAF (RTAF) for the phase retrieval problem. We focus on coded diffraction patterns composed of Fourier transforms and random masks. Then we justify the linear convergence behavior of such algorithm when near the true solution. By utilizing the local initializer found by a non-truncated spectral method, we establish the exact convergence rates in both the noiseless case and the bounded noise case. Further, we have shown that a general signal x ∈ C^n can be estimated
 on an optimal sampling order (specifically, O(nlogn)). Moreover, our demonstrations indicate that sharp stability can be maintained even when the observations are contaminated by bounded noise. Our experiments have revealed that the RTAF algorithm can reliably estimate the original signal using fewer measurements, aligning with our theoretical analysis. Additionally, numerical experiments have highlighted that the RTAF algorithm outperforms the truncated Wirtinger flow (TWF) and truncated Riemannian gradient descent (TRGrad) algorithms in terms of efficiency and stability, showcasing its superior performance over state-of-the-art algorithms.

Stochastic analysis of frequency-domain adaptive filters

This study investigates the convergence behaviors of a family of frequency-domain adaptive filters (FDAFs) under both exact- and under-modeling situations. The stochastic analysis is conducted by transforming the frequency-domain equations into their time-domain counterparts. We discuss the transient and steady-state convergence behaviors of four FDAF versions, i.e., the constrained FDAFs with and without step-normalization, the unconstrained FDAFs with and without step-normalization, and we also present the upper bounds of step size for mean stability and mean-square stability. Starting from the expression for the steady-state mean weight vector, this study investigates whether the FDAFs can converge to unknown system impulse responses and optimum Wiener solutions. Moreover, we provide the closed-form minimum mean-square error (MMSE) that each FDAF can achieve. The difference between the current work and our previous one is threefold. First, the presented time-domain analysis is much easier to handle and has a more explicit physical meaning than that in the frequency domain. Second, we here consider an arbitrary overlap factor between consecutive blocks, while our previous analysis only focuses on 50% overlap. Third, the presented MMSE expressions and excess mean-square error (EMSE) approximations have not been given before. Simulations reveal high consistency between the experimental and theoretical results.

Stochastic Gradient Descent for Kernel-Based Maximum Correntropy Criterion.

Maximum correntropy criterion (MCC) has been an important method in machine learning and signal processing communities since it was successfully applied in various non-Gaussian noise scenarios. In comparison with the classical least squares method (LS), which takes only the second-order moment of models into consideration and belongs to the convex optimization problem, MCC captures the high-order information of models that play crucial roles in robust learning, which is usually accompanied by solving the non-convexity optimization problems. As we know, the theoretical research on convex optimizations has made significant achievements, while theoretical understandings of non-convex optimization are still far from mature. Motivated by the popularity of the stochastic gradient descent (SGD) for solving nonconvex problems, this paper considers SGD applied to the kernel version of MCC, which has been shown to be robust to outliers and non-Gaussian data in nonlinear structure models. As the existing theoretical results for the SGD algorithm applied to the kernel MCC are not well established, we present the rigorous analysis for the convergence behaviors and provide explicit convergence rates under some standard conditions. Our work can fill the gap between optimization process and convergence during the iterations: the iterates need to converge to the global minimizer while the obtained estimator cannot ensure the global optimality in the learning process.

Abstract and Acoustic Targets in Phonetic Imitation

ABSTRACTPhonetic imitation (also called convergence or accommodation) occurs when talkers alter their pronunciation towards speech they hear. This can happen spontaneously with only a few minutes of exposure in a laboratory experiment even without instruction to imitate. While there is considerable evidence for spontaneous imitation, in many cases it is not clear exactly which aspects of the model talker are being imitated. It is possible to imitate the raw acoustics of the model's voice or more abstract targets like normalised acoustics of the model's voice, phonological patterns the model exhibits, or speech style. Although there is substantial literature demonstrating convergent speech behaviour, existing work typically does not distinguish between these different types of targets. This has theoretical implications for accounts of imitation and normalisation, methodological significance for analysis of imitation studies, and potential applied significance for how imitation is used in language teaching and clinical speech therapy. This paper will review these issues, discuss the statistical challenges associated with measuring convergence to competing targets, and make methodological recommendations for future studies.

Computational Analysis of Parallel Techniques for Nonlinear Biomedical Engineering Problems

In this study, we develop new efficient parallel techniques for solving both distinct and multiple roots of nonlinear problems at the same time. The parallel techniques represent an innovative contribution to the discipline, with local convergence of the ninth order. Theoretical research shows the rapid convergence and effectiveness of the proposed parallel schemes. To assess the suggested scheme’s stability and consistency, we look at certain biomedical engineering applications, such as osteoporosis in Chinese women, blood rheology, and differential equations. Overall, detailed analyses of convergence behavior, memory utilization, computational time, and percentage computational efficiency show that the novel parallel techniques outperform the traditional methods. The proposed methods would be more suitable for large-scale computational problems in biomedical applications due to their advantages in memory efficiency, CPU time, and error reduction.

Swarm-based gradient descent method for non-convex optimization

We introduce a new swarm-based gradient descent (SBGD) method for non-convex optimization. The swarm consists of agents, each is identified with a position, x \mathbf {x} , and mass, m m . The key to their dynamics is communication: masses are being transferred from agents at high ground to low(-est) ground. At the same time, agents change positions with step size adjusted to their relative mass. Accordingly, the crowd of agents is dynamically divided between heavier ‘leaders’, which proceed with small time steps and are expected to approach local minima, and lighter ‘explorers’ which proceed with larger-step protocol and are expected to encounter improved position for the swarm. If they do, then they assume the role of heavy swarm leaders and so on. We prove local convergence and we present convincing numerical simulations which demonstrate that the added layer of SBGD explorers performs well in improving its global convergence behavior in one-, two-, and 20-dimensional benchmarks.

Analysis of Equilibrium Points and Convergent Behaviors for Constrained Signed Networks

Analysis of Equilibrium Points and Convergent Behaviors for Constrained Signed Networks

EVALUATION OF INTERPOLATION METHODS FOR GNSS VELOCITIES IN CRUSTAL DEFORMATION: A CASE STUDY ON STRAIN RATE CALCULATION IN THE SOUTHERN SUMATRA

This study evaluates three interpolation methods—Inverse Distance Weighting (IDW), Kriging, and Sandwell—for analyzing crustal deformation in southern Sumatra, focusing on the on-site velocity consistency test. This test is crucial for assessing interpolation accuracy by comparing interpolated velocities to actual GNSS velocities at specific sites. Given the region's sparse GNSS network, accurate interpolation is vital for reliable deformation analysis. GNSS data from continuous and campaign sites, collected between 2017 and 2022 with a 30-second sampling interval, were processed to generate velocity. The Sandwell method, particularly with a Poisson’s ratio of 0, demonstrated superior performance, achieving the lowest mean residuals in the on-site velocity consistency test. This method consistently provided accurate interpolated velocities that closely matched original site data. IDW and Kriging methods also showed effectiveness but had different convergence behaviors: IDW required higher interpolation degrees for accuracy, while Kriging excelled in the east-west residuals. The Sandwell method’s velocities were used to calculate strain rates, revealing significant spatial variability. The findings underscore the importance of detailed on-site velocity consistency testing and highlight the need for expanding GNSS networks to improve accuracy and better assess seismic hazards in southern Sumatra.

Transparent boundary condition and its high frequency approximation for the Schrödinger equation on a rectangular computational domain

This paper addresses the numerical implementation of the transparent boundary condition (TBC) and its various approximations for the free Schrödinger equation on a rectangular computational domain. In particular, we consider the exact TBC and its spatially local approximation under high frequency assumption along with an appropriate corner condition. For the spatial discretization, we use a Legendre–Galerkin spectral method where Lobatto polynomials serve as the basis. Within variational formalism, we first arrive at the time-continuous dynamical system using spatially discrete form of the initial boundary-value problem incorporating the boundary conditions. This dynamical system is then discretized using various time-stepping methods, namely, the backward-differentiation formula of order 1 and 2 (i.e., BDF1 and BDF2, respectively) and the trapezoidal rule (TR) to obtain a fully discrete system. Next, we extend this approach to the novel Padé based implementation of the TBC presented by Yadav and Vaibhav (2024). Finally, several numerical tests are presented to demonstrate the effectiveness of the boundary maps (incorporating the corner conditions) where we study the stability and convergence behaviour empirically.

Error estimates for a bubble-enriched [formula omitted] interior penalty method to a reduced state constrained elliptic optimal control problem

In this article, we study a bubble-enriched C0 interior penalty method for the elliptic optimal control problem with pointwise control constraints. We have used the strategy of converting the control constrained optimization problem into a state constrained optimization problem by removing the control variable. The convergence behavior is obtained in H2-like energy norm. We present numerical tests to validate our theoretical results.

Stochastic convergence of renewable energy intensity across US States: evidence from a panel unit root test with asymmetries and common factors

ABSTRACT This study examines the stochastic convergence of renewable energy intensity (REI) across US states. We test for the stationarity of relative REI (stochastic convergence) using a recently developed quantile panel unit root test that accommodates both cross-sectional dependence and asymmetric behaviour. Our overall finding is that most states exhibit convergence, which implies that economic growth is sustainable with a higher share of renewable energy in the energy mix. This result provides support for those advocating that state governments should increase investment in renewable energy in advancing the clean energy transition. However, over half of the states display asymmetric stochastic convergence, which implies that how effective policies will be in promoting renewable energy efficiency will depend on the size and type of shocks impacting REI. Our results imply that in states that encounter relatively large positive shocks to REI, it may be necessary to invest more in the diffusion of renewable energy technologies and set long-term targets. Asymmetric convergence behavior also makes it difficult to coordinate policy responses across states, providing support for a decentralized approach in which each state invests in renewable energy technologies specific to their unique economic circumstances, rather than adopting a mandated overall national policy approach.