Uniform Step Size Research Articles

An improved solution of the probability density evolution equation for analyzing stochastic structural responses with adaptive time steps and initial conditions

The probability density evolution method (PDEM) offers an innovative approach for analyzing the stochastic vibration responses of structures. Given the complexities inherent in real-world challenges, the probability density evolution equations (PDEE) often require numerical methods for effective resolution. Consequently, improving computational efficiency and solution accuracy is of paramount importance. Currently, the impulse function discretization method (IFDM) faces several challenges, including limited differentiability in the spatial direction for initial value conditions. Furthermore, the probability density function produced by this method frequently deviates significantly from the true value, undermining the accuracy of the PDEE. Additionally, when addressing vibration responses characterized by rapid evolutionary velocities, the use of a globally fixed uniform time step size (FUTSS) results in a marked decrease in computational efficiency. To tackle these challenges, this study introduces an improved probability density evolution method (IPDEM), which integrates the Gaussian kernel density estimation discretization method (GKDEDM) and an adaptive non-uniform time step size (ANUTSS). The advantages of the IPDEM are illustrated through two numerical simulation examples. The findings indicate that the GKDEDM not only enhances spatial differentiability but also provides a more accurate estimation of the probability density at the initial moment, thereby improving the accuracy of the PDEE. Furthermore, the ANUTSS significantly reduces the number of iterations required while preserving precision, leading to a substantial optimization of computation time.

A uniform step size, low-voltage piezoelectric motor with dual-channel force loop.

Recently, a variety of piezoelectric motors with remarkable performance have appeared. However, due to the hysteresis effect of piezoelectrics and stress return errors within the mechanical structures, the existing piezoelectric motors still face some challenges, such as inconsistent step size, high working voltage, and considerable speed variances during upward vs downward movements even under identical driving voltage signals. Here, we introduce a novel low-voltage piezoelectric motor with a dual-channel force loop based on piezoelectric stacks, in which each slider has two force loops connected with other sliders and the internal elastic preload element is installed, which can effectively address these issues. This new type of piezoelectric motor has low working voltage (starting voltage is only 0.8V, significantly lower than that of conventional piezoelectric motors), large driving force, uniform step size, and excellent linearity.

A Fitted Numerical Approach for Singularly Perturbed Two-Parameter Parabolic Problem with Time Delay

This paper is aimed at constructing and analyzing a fitted approach for singularly perturbed time delay parabolic problems with two small parameters. The proposed computational scheme comprises the implicit Euler and especially finite difference method for the time and space variable discretization, respectively, on uniform step size. The stability and convergence analysis of the method is provided and is first-order parameter uniform convergent. Further, the numerical results depict that the present method is more convergent than some methods available in the literature.

An operator splitting method for multi-asset options with the Feynman-Kac formula

In this paper, we propose an unconditionally stable numerical technique for a multi-dimensional Black-Scholes equation to price an option with high accuracy. The proposed scheme uses the operator splitting method to reduce the multi-dimensional partial differential equation to a set of one-dimensional sub-problems. The computational domain is discretized with a uniform space and time step size to approximate the option values and asset correlation-related terms by piecewise quadratic polynomials. In order to obtain the numerical solutions of the sub-problems, we analytically integrate the polynomials to estimate the expectation of the Feynman-Kac formula. We compare our method with the implicit operator splitting method (OSM), a widely used finite difference method in the industry. Numerical experiments show that the proposed method outperforms OSM in terms of convergence in space and time directions. We also provide analysis to guarantee the unconditional stability of our method by exploiting the Feynman-Kac recursively.

A linear second-order in time unconditionally energy stable finite element scheme for a Cahn–Hilliard phase-field model for two-phase incompressible flow of variable densities

We propose a novel second-order BDF time stepping method of variable time step sizes combined with a classical residual-based stabilized finite element spatial discretization using the Streameline-Upwind Petrov–Galerkin (SUPG)/pressure stabilization Petrov–Galerkin (PSPG)/grad-div stabilization for solving the phase–field model for two-phase incompressible flow of different densities and viscosities in the advection dominated regime. In the case of uniform time step size and without extra stabilization, the scheme is shown to satisfy a discrete energy law. Benchmark test of the Rayleigh–Taylor instability under high Reynolds number and Péclect number demonstrates that the scheme captures details of the instability comparable to results in the literature by schemes based on sharp-interface models.

A hybrid numerical scheme for singularly perturbed parabolic differential‐difference equations arising in the modeling of neuronal variability

This study aims at constructing a robust numerical scheme for solving singularly perturbed parabolic delay differential equations arising in the modeling of neuronal variability. Taylor's series expansion is applied to approximate the shift terms. The obtained result is approximated by using the implicit Euler method in the temporal discretization on a uniform step size with the hybrid numerical scheme consisting of the midpoint upwind method in the outer layer region and the cubic spline in tension method in the inner layer region on a piecewise uniform Shishkin mesh in the spatial discretization. The constructed scheme is shown to be an ε -uniformly convergent accuracy of order O Λ t + N − 2 ln 3 N . Two model examples are given to testify the theoretical findings.

Buongiorno Model for MHD Nanofluid Flow Between Rotating Parallel Plates in the Presence of Variable Liquid Properties

In this article, the effects of Brownian motion and thermophoresis on the flow, heat, and mass transfer from a rotating parallel plate is investigated. The Buongiorno mathematical model is considered and variable liquid properties are taken into account. MHD nanofluid terms are fused in the governing equations. Keller Box method is employed to solve the set of ordinary differential equations which are treated by suitable similarity transformations. A uniform step size of �� = 0.01 is observed to be satisfactory with an error tolerance of 10−6. The graphical analysis of velocity, temperature, and concentration is presented. It is recorded that; upsurge in variable viscosity parameter diminishes the axial velocity however it has a twofold effect on transverse velocity. Furthermore, nanoparticle concentration and temperature are inversely related with reference to thermophoresis and Brownian motion parameters

BLOCK PROCEDURE WITH IMPLICIT SIXTH ORDER LINEAR MULTISTEP METHOD USING LEGENDRE POLYNOMIALS FOR SOLVING STIFF INITIAL VALUE PROBLEMS

In this paper, a discrete implicit linear multistep method in block form of uniform step size for the solution of first-order ordinary differential equations is presented using the power series as a basis function. To improve the accuracy of the method, a perturbation term is added to the approximated solution. The method is based on collocation of the differential equation and interpolation of the approximate solution using power series at the grid points. The procedure yields four linear multistep schemes which are combined as simultaneous numerical integrators to form block method. The method is found to be consistent and zero-stable, and hence convergent. The accuracy of the method is tested with some standard stiff first order initial value problems. The results are compared with the two point implicit block backward difference formula and two point implicit block extended backward difference formula. All numerical examples show that our proposed method has a better accuracy than the existing literature.

Finite difference scheme for the time-fractional Fokker–Planck equation with time- and space-dependent forcing

This paper presents the finite difference scheme for the time-fractional Fokker–Planck equation with the external force and source term which depends on space and time. We prove the stability and convergence of the numerical scheme by the energy method. The spatial error is in the discrete -norm, and for the time discretization with uniform stepsize τ when . The numerical examples verify these theoretical results.

Adaptive order of block backward differentiation formulas for stiff ODEs

In this paper, Adapative Order of Block Backward Differentiation Formulas (ABBDFs) are formulated using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs). These ABBDF methods are of order four, five and six. The benefit of the ABBDF methods is the computation time in the computation of solutions. Numerical results are presented to demonstrate the advantage of implementing adaptive order selection in a single code.

A Note on Maxima in Random Walks

We give a combinatorial proof that a random walk attains a unique maximum with probability at least $1/2$. For closed random walks with uniform step size, we recover Dwass's count of the number of length $\ell$ walks attaining the maximum exactly $k$ times. We also show that the probability that there is both a unique maximum and a unique minimum is asymptotically equal to $\frac14$ and that the probability that a Dyck word has a unique minimum is asymptotically $\frac12$.

Comparison of Split-Step Fourier Schemes for Simulating Fiber Optic Communication Systems

This paper mainly focuses on efficient schemes for simulating propagation in optical fibers. Various schemes based on split-step Fourier techniques to solve the nonlinear Schrodinger equation (NLSE), which describes the propagation in optical fibers, are compared. In general, the schemes in which the loss operator is combined with nonlinearity operator are found to be more computationally efficient than the schemes in which the loss is combined with dispersion. When the global error is large, the schemes with variable step size outperform the ones with uniform step size. The schemes based on local error and/or minimum area mismatch (MAM) further improve the computational efficiency. In this scheme, by minimizing the area mismatch between the exponential profile and its stepwise approximation, an optimal step size distribution is found. The optimization problem is solved by the steepest descent algorithm. The number of steps to get the desired accuracy is determined by the local error method. The proposed scheme is found to have higher computational efficiency than the other schemes studied in this paper. For QPSK systems, when the global error is 10 -8 , the number of fast Fourier transforms (FFTs) needed for the conventional scheme (loss combined with dispersion and uniform step size) is 5.8 times that of the proposed scheme. When the global error is 10 -6 , the number of FFTs needed for the conventional scheme is 3.7 times that of the proposed scheme.

Hybrid Monte Carlo with non-uniform step size.

The Hybrid Monte Carlo method offers a rigorous and potentially efficient approach to the simulation of dense systems, by combining numerical integration of Newton's equations of motion with a Metropolis accept-or-reject step. The Metropolis step corrects for sampling errors caused by the discretization of the equations of motion. The integration is usually performed using a uniform step size. Here, we present simulations of the Lennard-Jones system showing that the use of smaller time steps in the tails of each integration trajectory can reduce errors in energy. The acceptance rate is 10-15 percentage points higher in these runs, compared to simulations with the same trajectory length and the same number of integration steps but a uniform step size. We observe similar effects for the harmonic oscillator and a coarse-grained peptide model, indicating generality of the approach.

Digital Back Propagation With Optimal Step Size for Polarization Multiplexed Transmission

A digital back propagation (DBP) scheme with optimal step size for polarization division multiplexed transmission system is proposed. For a fixed number of steps in DBP, the optimum step size is calculated by minimizing the mismatch between the area under the exponentially increasing nonlinearity profile and its stepwise approximation. In simulations, the vector nonlinear Schodinger equation or Manakov equations are used for forward propagation and Manakov equations are used for backward propagation. The simulation results show that the proposed scheme using the optimum step size outperforms that using the uniform step size at the same computational cost.

Gradient descent and fast artificial time integration

The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated minimization prob- lem. Greedy algorithms such as steepest descent for determining the step size are as slow to reach steady state as is forward Euler integration with the best uniform step size. But other, much faster methods using bolder step size selection exist. Various alternatives are investigated from both theo- retical and practical points of view. The steepest descent method is also known for the regularizing or smoothing effect that the first few steps have for certain inverse problems, amounting to a finite time regularization. We further investigate the retention of this property using the faster gradient descent variants in the context of two applications. When the combination of regularization and accuracy demands more than a dozen or so steepest descent steps, the alternatives offer an advantage, even though (indeed because) the absolute stability limit of forward Euler is carefully yet severely violated.

Numerical techniques for solving the quantum constraint equation of generic lattice-refined models in loop quantum cosmology

To avoid instabilities in the continuum semiclassical limit of loop quantum cosmology models, refinement of the underlying lattice is necessary. This lattice refinement leads to new dynamical difference equations which, in general, do not have a uniform step size, implying complications in their analysis and solutions. We propose a numerical method based on Taylor expansions, which can give us the necessary information to calculate the wave function at any given lattice point. The method we proposed can be applied in any lattice-refined model, while in addition the accuracy of the method can be estimated. Moreover, we confirm numerically the stability criterion which was earlier found following a von Neumann analysis. Finally, the ''motion'' of the wave function due to the underlying discreteness of the space-time is investigated, for both a constant lattice, as well as lattice refinement models.

Unified Convergence Results on a Minimax Algorithm for Finding Multiple Critical Points in Banach Spaces

A minimax method for finding multiple critical points in Banach spaces is successfully developed in [X. Yao and J. Zhou, SIAM J. Sci. Comput., 26 (2005), pp. 1796-1809] by using a projected pseudogradient as a search direction. Since several different techniques can be used to compute a projected pseudogradient, the uniform stepsize and the continuity of a search direction, two key properties for the convergence results in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24 (2002), pp. 865-885], get lost. In this paper, instead of proving convergence results of the algorithm for each technique, unified convergence results are obtained with a weaker stepsize assumption. An abstract existence-convergence result is also established. It is independent of the algorithm and explains why function values always converge faster than their gradients do. The weaker stepsize assumption is then verified for several different cases. As an illustration to the new results, the Banach space $W_0^{1,p}(\Omega)$ is considered and the conditions posed in the new results are verified for a quasi-linear elliptic PDE.

Global Stability of a General Class of Discrete-Time Recurrent Neural Networks

A general class of discrete-time recurrent neural networks (DTRNNs) is formulated and studied in this paper. Several sufficient conditions are obtained to ensure the global stability of DTRNNs with delays based on induction principle (not based on the well-known Liapunov methods). The obtained results have neither assumed the symmetry of the connection matrix, nor boundedness, monotonicity or the differentiability of the activation functions. In addition, discrete-time analogues of a general class of continuous-time recurrent neural networks (CTRNNs) are derived and studied. The convergence characteristics of CTRNNs are preserved by the discrete-time analogues without any restriction imposed on the uniform discretization step size. Finally, the simulating results demonstrate the validity and feasibility of our proposed approach.

Exponential stability of continuous-time and discrete-time bidirectional associative memory networks with delays

First, convergence of continuous-time Bidirectional Associative Memory (BAM) neural networks are studied. By using Lyapunov functionals and some analysis technique, the delay-independent sufficient conditions are obtained for the networks to converge exponentially toward the equilibrium associated with the constant input sources. Second, discrete-time analogues of the continuous-time BAM networks are formulated and studied. It is shown that the convergence characteristics of the continuous-time systems are preserved by the discrete-time analogues without any restriction imposed on the uniform discretionary step size. An illustrative example is given to demonstrate the effectiveness of the obtained results.

DOA estimation based on the database retrieval technique with nonuniform quantization and clustering

A direction-of-arrival (DOA) estimation method using an array antenna has been developed based on a database retrieval technique. This method uses a database consisting of a set of correlation matrices of the array output vectors for various combinations of the quantized angles and signal powers. When a correlation matrix is estimated from an observed output vector, several correlation matrices close to the estimated one are searched out from the database, and the DOA is estimated based on the retrieved data. This method gives an accurate estimation, but the use of uniform quantization step size requires a large amount of storage space. In this paper, the relation between the quantization step size and the estimation accuracy is analyzed, and a nonuniform quantization scheme is developed to reduce the database size without sacrificing the estimation accuracy. A clustering technique is also introduced to alleviate the performance degradation caused by the retrieval of data which have similar correlation matrices but have much different angles. We show by simulations that the nonuniform quantization reduces the database size and the clustering improves the estimation accuracy, and that the proposed method is applicable to the array of three elements at the present.