Small Step Size Research Articles

Finite difference numerical solution of Troesch’s problem on a piecewise uniform Shishkin mesh

Finding numerical solutions to the Troesch's problem is known to be challenging, especially when the sensitivity parameter $$\lambda $$? is large. In this manuscript, we propose a numerical method for solving the Troesch's problem which combines efficiency and accuracy, even for large sensitivity parameter. Our method can be summarized as a finite difference method formulated on a Shishkin mesh, which is piecewise uniform with smaller step size in the boundary layer in the neighborhood of the point $$x=1$$x=1. In fact, we treat the Troesch's problem as a singularly perturbed problem with a special transition point on the mesh. The transition point for the Shishkin mesh is first defined and computed. The application of the finite difference method on the piecewise uniform mesh leads to a nonlinear matrix system, which is solved numerically using the generalized Newton's method. While, existing numerical solvers suffer significantly when sensitivity parameter $$\lambda $$? becomes large and fail to provide accurate numerical solutions for $$\lambda >20$$?>20 (Chang, Appl Math Comput 216(11):3303---3306, 2010; Khuri and Sayfy, Math Comput Model 54(9):1907---1918, 2011; Raja, Inf Sci 279:860---873, 2014; Temimi, Appl Math Comput 219(2):521---529, 2012), our method provides accurate numerical solution for large values of this sensitivity parameter, up to $$\lambda =100$$?=100. Moreover, the implementation of the method is straight forward and the computation cost is low. Numerical experiments show that the method provides accurate solutions for different values of the sensitivity parameter $$\lambda $$?.

Adaptive burnup stepsize selection using control theory for 2-D lattice depletion simulations

A control theory approach is adopted to determine the temporal discretization during two-dimensional lattice physics depletion simulations. Two primary applications of automated and adaptive stepsize control are identified: (i) the presence of strong absorbers such as gadolinium, where the accurate burnout of the isotopes requires a depletion stepsize smaller than typically required, and (ii) high fidelity multiphysics simulations, e.g. loosely coupled physics, where the coupled physics are nonlinear in time and stepsize changes may be necessary to obtain an accurate coupled solution. A conventional predictor–corrector method is used to address the nonlinearity of the nuclide transmutation and neutron flux. An adaptive stepsize method is developed based on monitoring the one-group scalar neutron flux at both the predictor and corrector steps to approximate the convergence residual of the nonlinear solution. A user-specified tolerance on the L2 relative error norm of the scalar neutron flux is utilized by the stepsize controller. Controllers that include integral, proportional, and/or derivative components are investigated and parameterized using Latin hypercube sampling of the controller input parameters. Three distinct fuel loadings of pressurized water reactor 17 × 17 fuel pin assemblies are considered, including no burnable absorbers, Integral Fuel Burnable Absorber, and gadolinium fuel pins. The required depletion stepsizes, as predicted throughout the cycle by the controller, are compared with a very small stepsize (0.01 MW d/kgHM) reference solution and a solution obtained by a typical rule of thumb depletion stepsize sequence.

Convergence of a fluid–structure interaction problem decoupled by a Neumann control over a single time step

Building off of previous analytical results for recasting fluid–structure interaction into an optimal control setting, an a priori error estimate is given for the optimality system by means of BRR theory. The convergence of the steepest descent method is proven in a discrete setting for a sufficiently small time step and mesh size. A numerical study is included supporting the theoretical rate of convergence over a single time step. Additional results demonstrate optimal convergence in space and time over several time steps.

A GPU-accelerated adaptive discontinuous Galerkin method for level set equation

ABSTRACTThis paper presents a GPU-accelerated nodal discontinuous Galerkin method for the solution of two- and three-dimensional level set (LS) equation on unstructured adaptive meshes. Using adaptive mesh refinement, computations are localised mostly near the interface location to reduce the computational cost. Small global time step size resulting from the local adaptivity is avoided by local time-stepping based on a multi-rate Adams–Bashforth scheme. Platform independence of the solver is achieved with an extensible multi-threading programming API that allows runtime selection of different computing devices (GPU and CPU) and different threading interfaces (CUDA, OpenCL and OpenMP). Overall, a highly scalable, accurate and mass conservative numerical scheme that preserves the simplicity of LS formulation is obtained. Efficiency, performance and local high-order accuracy of the method are demonstrated through distinct numerical test cases.

Review and Comparison of Variable Step-Size LMS Algorithms

The inherent feature of the Least Mean Squares (LMS) algorithm is the step size, and it requires careful adjustment. Small step size, required for small excess mean square error, results in slow convergence. Large step size, needed for fast adaptation, may result in loss of stability. Therefore, many modifications of the LMS algorithm, where the step size changes during the adaptation process depending on some particular characteristics, were and are still being developed. The paper reviews seventeen of the best known variable step-size LMS (VS-LMS) algorithms to the degree of detail that allows to implement them. The performance of the algorithms is compared in three typical applications: parametric identification, line enhancement, and adaptive noise cancellation. The paper suggests also one general modification that can simplify the choice of the upper bound for the step size, which is a crucial parameter for many VS-LMS algorithms.

Convergence Study on the Symmetric Version of ADMM with Larger Step Sizes

The alternating direction method of multipliers (ADMM), also well known as a special split Bregman algorithm in imaging, is being popularly used in many areas including the image processing field. One useful modification is the symmetric version of the original ADMM, which updates the Lagrange multiplier twice at each iteration and thus the variables are treated in a symmetric manner. The symmetric version of ADMM, however, is not necessarily convergent. It was recently found that the convergence of symmetric ADMM can be sufficiently ensured if both the step sizes for updating the Lagrange multiplier are shrunk conservatively. Despite the theoretical significance in ensuring convergence, however, smaller step sizes should be strongly avoided in practice. On the contrary, we actually have the desire of seeking larger step sizes whenever possible in order to accelerate the numerical performance. Another technique leading to numerical acceleration of ADMM is enlarging its step size by a constant originally p...

Analogue‐to‐digital converter phase insensitive dispersion search method with large search step size for low computational cost

Accumulated chromatic dispersion (CD) estimation is crucial for frequency-domain equalisation in an optical coherent digital receiver. To solve the problems of small search step size and analogue-to-digital converter (ADC) sampling phase sensitivity from conventional CD searching algorithm, a CD searching method with modified shift parameters and waveforms is proposed in this study. The modified CD searching method allows a much larger search step size and can achieve faster CD estimation with low computational complexity. The performance of the modified CD searching method is also insensitive to ADC sampling phase. On the basis of simulation results, a very low search error probability is confirmed with a search step size of 50 km. About 1 K symbols are found to be enough for dispersion estimation. After the frequency-domain equaliser (FDE) with the estimated dispersion value, the residual dispersion length <30 km can be well compensated with a 9-tap adaptive time-domain equaliser (TDE) with fast convergence. The overall error rate performance of the tunable electrical dispersion compensation is also studied and compared with lumped and distributed optical dispersion compensation for dynamically switched optical networks.

MEAN-SQUARE EXPONENTIAL DICHOTOMY OF NUMERICAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS

We present the ability of numerical simulations to reproduce the mean-square exponential dichotomy of stochastic differential equations. Under some conditions, we show that the mean-square exponential dichotomy of stochastic differential equations is equivalent to that of the numerical method for sufficient small step sizes.

Collapse simulation of reinforced concrete frame structures

Summary Study of collapse-resisting properties of structures has attracted widespread attention because of frequently occurring earthquakes and extreme events (e.g. blast) around the world. The developments in computational methods have enabled researchers to numerically simulate the collapse of structures under different kinds of loadings and provide reliable assessments of the collapse performance of structures. The dynamic nature of structural collapse requires a direct integration algorithm to solve the equations of motion of the numerical simulation model. A major concern in such simulations is the computational efficiency, which stems from the need to use a small time step size in both implicit algorithm and explicit algorithm. In this paper, modeling techniques to simulate typical failure mechanisms in reinforced concrete frame structures combined with the application of the recently developed explicit, unconditionally stable, parametrically dissipative KR-α integration method to investigate collapse simulation are presented. A fiber beam-column element is used to model the frame members, where the material nonlinearities, especially material softening, are simulated by a plastic damage model combined with a failure criterion. Numerical examples are presented to illustrate the proposed collapse simulation technique. The results indicate that the proposed technique provides an accurate result and has exceptional computational efficiency. Copyright © 2015 John Wiley & Sons, Ltd.

Data-driven fluid simulations using regression forests

Traditional fluid simulations require large computational resources even for an average sized scene with the main bottleneck being a very small time step size, required to guarantee the stability of the solution. Despite a large progress in parallel computing and efficient algorithms for pressure computation in the recent years, realtime fluid simulations have been possible only under very restricted conditions. In this paper we propose a novel machine learning based approach, that formulates physics-based fluid simulation as a regression problem, estimating the acceleration of every particle for each frame. We designed a feature vector, directly modelling individual forces and constraints from the Navier-Stokes equations, giving the method strong generalization properties to reliably predict positions and velocities of particles in a large time step setting on yet unseen test videos. We used a regression forest to approximate the behaviour of particles observed in the large training set of simulations obtained using a traditional solver. Our GPU implementation led to a speed-up of one to three orders of magnitude compared to the state-of-the-art position-based fluid solver and runs in real-time for systems with up to 2 million particles.

A nonnegativity preserved efficient algorithm for atmospheric chemical kinetic equations

Air pollution models plays a critical role in atmospheric environment research. Chemical kinetic equations is an important component of air pollution models. The chemical equations is numerically sticky because of its stiffness, nonlinearity, coupling and nonnegativity of the exact solutions. Over the past decades, numerous papers about chemical equation solvers have been published. However, these solvers cannot preserve the nonnegativity of the exact solutions. Therefore, in the calculation, the negative numerical concentration values are usually set to zero artificially, which may cause simulation errors. To obtain real nonnegative numerical concentration values, very small step-size has to be adopted. Then enormous amount of CPU time is consumed to solve the chemical equations. In this paper, we revisit this topic and derive a new algorithm. Our algorithm Modified-Backward-Euler (MBE) Method can unconditionally preserve the nonnegativity of the exact solutions. MBE is a simple, robust and efficient solver. It is much faster and more precise than the traditional solvers such as LSODE and QSSA. The numerical results and parameter suggestions are shown at the end of the paper. MBE is based on the P-L structure of the chemical equations and a deeper view into the nature of Euler Methods. It cannot only be used to solve chemical equations, but can also be applied to conquer ordinary differential equations (ODEs) with similar P-L structure.

Jump Phenomena and Bifurcations in Stochastic Vehicle‐Road Dynamics

AbstractWhen vehicles ride on uneven roads, they are excited to vertical random vibrations whose stationary rms‐values (root‐mean‐square) strongly depend on the velocity of the vehicle. To investigate this vibration behavior, it is appropriate to introduce road models in way domain which are based on the theory of stochastic differential equations and transformed from way to time by means of velocity‐dependent way and noise increments.The random base excitations by roads are applied to nonlinear quarter car models. They lead to stationary rms‐values of the vertical vehicle vibrations which become resonant for critical velocities and show jump phenomena similar to those of the Duffing oscillator under harmonic excitations. In the stochastic case, jump phenomena are only observable for narrow‐banded road excitations. They vanish for increasing car damping and excitation bandwidth.For efficient simulations of the road‐vehicle model, the n state equations are utilized to derive n(n + 1)/2 stochastic covariance equations. For small step sizes, their numerical mean square solutions coincide with the nonlinear results of fix‐point iterations obtained when the noise terms of the covariance equations are omitted. It can easily be shown, that this deterministic approach leads to the correct stationary covariances in the linear case. (© 2015 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Design and implementation of 15‐level cascaded multi‐level voltage source inverter with harmonics elimination pulse‐width modulation using differential evolution method

This study describes the application of a soft computing method - known as differential evolution (DE) to implement the harmonics elimination pulse-width modulation (HEPWM) for a 15-level, multi-level voltage source inverter (MVSI). The main feature of this technique is the wide range of fundamental output voltage that can be achieved, while maintaining a very low total harmonics distortion (below 6%). The fundamental voltage can be controlled very precisely (very small step size) and the trajectories of the HEPWM angles are continuous over the whole range of modulation index - making it very suitable for utility and variable speed drives application. The study details the procedures to obtain the HEPWM angles trajectories, which include the formulation of objective functions, appropriate selection of the DE evolutionary parameters and stopping criterions. The viability of the proposed method is simulated using MATLAB/Simulink and verified by a three-phase, cascaded-type MVSI test rig. Experimentally, The HEPWM waveforms are implemented using the Altera field programmable gate array, whereas the 15-level MVSI is built using metal-oxide semiconductor field effect transistor H-bridge. Selected results are presented; the simulation and hardware results are found to be in very close agreement with the theoretical predictions.

θ-Maruyama methods for nonlinear stochastic differential delay equations

In this paper, mean-square convergence and mean-square stability of θ-Maruyama methods are studied for nonlinear stochastic differential delay equations (SDDEs) with variable lag. Under global Lipschitz conditions, the methods are proved to be mean-square convergent with order 12, and exponential mean-square stability of SDDEs implies that of the methods for sufficiently small step size h>0. Further, the exponential mean-square stability properties of SDDEs and those of numerical methods are investigated under some non-global Lipschitz conditions on the drift term. It is shown in this setting that the θ-Maruyama method with θ=1 can preserve the exponential mean-square stability for any step size. Additionally, the θ-Maruyama method with 12≤θ≤1 is asymptotically mean-square stable for any step size, provided that the underlying system with constant lag is exponentially mean-square stable. Applications of this work to some special problem classes show that the results are deeper or sharper than those in the literature. Finally, numerical experiments are included to demonstrate the obtained theoretical results.

Elastic and inelastic transmission of electrons through insulating polyethylene terephthalate nanocapillaries

The transmission of electrons with incident energies of 300, 500, and 800 eV through insulating polyethylene terephthalate (PET) nanocapillaries for small tilt angles near 0\ifmmode^\circ\else\textdegree\fi{} was studied. The exiting electrons showed elastic and inelastic behaviors with the inelasticity increasing with the tilt angle. This paper is an extension of our earlier work for electrons on PET reported by Das et al. [Phys. Rev. A 76, 042716 (2007)], but the present study goes further by investigating the transmission for very small tilt angles near 0\ifmmode^\circ\else\textdegree\fi{} with the small step sizes revealing distinct regions of behavior. The work performed here was conducted for two samples with different capillary diameters and pore densities: sample 1 with 100-nm diameter capillaries in a foil of 12-\textmu{}m thickness and a pore density of $5\ifmmode\times\else\texttimes\fi{}{10}^{8}/{\mathrm{cm}}^{2}$, and sample 2 with 200-nm diameter capillaries with the same foil thickness and a pore density of $5\ifmmode\times\else\texttimes\fi{}{10}^{7}/{\mathrm{cm}}^{2}$. Two distinct regions of transmission depending on the sample tilt angle for both samples corresponding to direct and guided electrons as well as the transition region between them were observed. The angular centers and widths of the transmission for each angle investigated were found and are discussed in terms of the tilt angle variation in the foil. For the widths, comparison with the inherent angular width of the electron spectrometer is made. Energy loss of electrons was studied from the centroids of the energy spectra with these energies corresponding to the peak centers of the angular profiles. The broadness of the energy distributions due to inelasticity was investigated from the full width at half maxima (in eV) of the spectra. The results for both samples agreed well despite the differences in capillary diameters and pore densities. The results are in partial agreement with the previous work performed for PET nanocapillaries by Das et al. [Phys. Rev. A 76, 042716 (2007)], but differences are also found.

An analysis of a three-factor model proposed by the Danish Society of Actuaries for forecasting and risk analysis

This paper provides the explicit solution to the three-factor diffusion model recently proposed by the Danish Society of Actuaries to the Danish industry of life insurance and pensions. The solution is obtained by use of the known general solution to multidimensional linear stochastic differential equation systems. With offset in the explicit solution, we establish the conditional distribution of the future state variables which allows for exact simulation. Using exact simulation, we illustrate how simulation of the system can be improved compared to a standard Euler scheme. In order to analyze the effect of choosing the exact simulation scheme over the traditional Euler approximation scheme frequently applied by practitioners, we carry out a simulation study. We show that due to its recursive nature, the Euler scheme becomes computationally expensive as it requires a small step size in order to minimize discretization errors. Using our exact simulation scheme, one is able to cut these computational costs significantly and obtain even better forecasts. As probability density tail behavior is key to expected investment portfolio performance, we further conduct a risk analysis in which we compare well-known risk measures under both schemes. Finally, we conduct a sensitivity analysis and find that the relative performance of the two schemes depends on the chosen model parameter estimates.

Parallel computing subgradient method for nonsmooth convex optimization over the intersection of fixed point sets of nonexpansive mappings

Nonsmooth convex optimization problems are solved over fixed point sets of nonexpansive mappings by using a distributed optimization technique. This is done for a networked system with an operator, who manages the system, and a finite number of users, by solving the problem of minimizing the sum of the operator’s and users’ nondifferentiable, convex objective functions over the intersection of the operator’s and users’ convex constraint sets in a real Hilbert space. We assume that each of their constraint sets can be expressed as the fixed point set of an implementable nonexpansive mapping. This setting allows us to discuss nonsmooth convex optimization problems in which the metric projection onto the constraint set cannot be calculated explicitly. We propose a parallel subgradient algorithm for solving the problem by using the operator’s attribution such that it can communicate with all users. The proposed algorithm does not use any proximity operators, in contrast to conventional parallel algorithms for nonsmooth convex optimization. We first study its convergence property for a constant step-size rule. The analysis indicates that the proposed algorithm with a small constant step size approximates a solution to the problem. We next consider the case of a diminishing step-size sequence and prove that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem. We also give numerical examples to support the convergence analyses.

Optical coherent burst‐mode receivers with delayed channel equalisation feedback from parallel and pipelined design

A digital optical coherent burst-mode receiver (BMR) with 112-Gb/s polarisation-division multiplexing-quaternary phase-shift keying (QPSK) modulation is proposed with parallel digital signal processing and pipelined design. The convergence rate property of the digital channel equaliser with non-negligible computational delay is studied extensively using simulations. Owing to the interaction between the channel equalisation and frequency offset estimation (FOE), a novel equaliser adaptation method using the latest available equaliser output and the latest available FOE result is proposed for a faster convergence than the conventional method. Simulation results reveal that the computational delay have a great impact on the convergence rate. A smaller step size for tap coefficient updating with slower convergence rate should be used to avoid divergence. Compared with the FOE module, the computational delay from the channel equaliser has the dominant effect on the convergence. The design of the equaliser with short computational delay is the key for a fast convergence rate. The parallel and pipelined digital optical coherent BMR after successful initialisation has negligible error rate performance loss when compared with the ideal serial signal processing within a wide frequency offset range.

Implementation of a new maximum power point tracking control strategy for small wind energy conversion systems without mechanical sensors

This paper proposes a modified perturbation and observation maximum power point tracking algorithm for small wind energy conversion systems to overcome the problems of the conventional perturbation and observation technique, namely rapidity/efficiency trade-off and the divergence from peak power under a fast variation of the wind speed. Two modes of operation are used by this algorithm, the normal perturbation and observation mode and the predictive mode. The normal perturbation and observation mode with small step-size is switched under a slow wind speed variation to track the true maximum power point with fewer fluctuations in steady state. When a rapid change of wind speed is detected, the algorithm tracks the new maximum power point in two phases: in the first stage, the algorithm switches to the predictive mode in which the step-size is auto-adjusted according to the distance between the operating point and the estimated optimum point to move the operating point near to the maximum power point rapidly, and then the normal perturbation and observation mode is used to track the true peak power in the second stage. The dc-link voltage variation is used to detect rapid wind changes. The proposed algorithm does not require either knowledge of system parameters or of mechanical sensors.The experimental results confirm that the proposed algorithm has a better performance in terms of dynamic response and efficiency compared with the conventional perturbation and observation algorithm.

Exponential mean square stability of the theta approximations for neutral stochastic differential delay equations

In this paper, a split-step theta (SST) method is introduced and analyzed for neutral stochastic differential delay equations (NSDDEs). It is proved that the SST method with θ∈[0,1/2] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize and the drift coefficient, but for θ∈(1/2,1], the SST can reproduce the exponential mean square stability unconditionally. Then, based on the stability results of SST scheme, we examine the exponential mean square stability of the stochastic linear theta (SLT) approximation for NSDDEs and obtain the similar stability results to that of the SST method. Moreover, for sufficiently small stepsize, we show that the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately. These results show that different values of theta will drastically affect the exponential mean square stability of the two classes of theta approximations for NSDDEs.