Large Step Size Research Articles

Finite Rate Distributed Weight-Balancing and Average Consensus Over Digraphs

This article proposes the first distributed algorithm that solves the weight-balancing problem using only finite rate and simplex communications among nodes, compliant with the directed nature of the graph edges. It is proved that the algorithm converges to a weight-balanced solution at sublinear rate. The analysis builds upon a new metric inspired by positional system representations, which characterizes the dynamics of information exchange over the network, and on a novel step-size rule. Building on this result, a novel distributed algorithm is proposed that solves the average consensus problem over digraphs, using, at each timeslot, finite rate simplex communications between adjacent nodes—some bits for the weight-balancing problem and others for the average consensus. Convergence of the proposed quantized consensus algorithm to the average of the node's unquantized initial values is established, both almost surely and in the moment generating function of the error; and a sublinear convergence rate is proved for sufficiently large step-sizes. Numerical results validate our theoretical findings.

Dynamical behaviour of HIV Infection with the influence of variable source term through Galerkin method

In this paper, we formulate the intricate process of Human Immunodeficiency Virus (HIV) infection in a mathematical model. In the formulation of the model, the density of CD4+ T-cells is categorized into healthy and infected classes while the density of free HIV viruses in the human host’s blood is considered a separate class. It is reported that when HIV enters a human’s body, it infects a large amount of CD4+ T-cells and unbalance the supply of new cells from the thymus. Therefore, we incorporate variable source term relying on the viral load for the supply of new cells instead of using a fixed value for the supply of these cells. We implement a continuous Galerkin Petrov time discretization scheme, particularly cGP(2)-scheme to visualize the dynamical behavior of the mentioned model and a detailed description of the effects of different physical parameters of interest are depicts graphically and discuss that how the level of healthy, infected CD4+ T-cells and free HIV particles varies concerning the emerging parameters in the model. In the proposed cGP(2)-method, two unknowns terms can be found by solving a 2×2 block system. This method is accurate of order 3 in the whole time interval and shows even superconvergence of order 4 in the discrete-time points. Furthermore, determine the solution of the model utilizing the fourth-order Runge-Kutta (RK4)-method and find out the absolute errors between the solutions obtained from both approaches. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with the RK-4 method. We examine the accuracy of the cGP(2)-sheme and observe that it produces more accurate results at relatively larger step sizes as in comparison to RK-4 scheme. Correspondence between the findings of cGP(2)-method and RK4-method reveal that the new technique is a promising tool for obtaining approximate solutions to the nonlinear systems of real-world problems.

Energy correction and analytic energy gradients due to triples in CCSD(T) with spin–orbit coupling on graphic processing units using single-precision data

ABSTRACT Calculating the contribution of triples ((T)) to the correlation energy, the density matrices and the constant terms in the Λ equation is the most expensive steps in obtaining analytic energy gradients for the CCSD(T) approach. In this work, we report the implementation of these steps for the CCSD(T) method with spin–orbit coupling (SOC) included in post-self-consistent-field calculations (SOC-CCSD(T)) using single-precision data on a consumer GPU card to accelerate calculations. The developed program can be used for calculations on GPU with single-precision data or on CPU with either single- or double-precision data. According to our results, calculating the (T) correlation energy in SOC-CCSD(T) on GPU with single-precision data is about 7–10 times faster for the investigated molecules than that on CPU with double-precision data, and it is 5–9 times faster in calculating the (T) part of analytic energy gradients for the SOC-CCSD(T) method. Our results indicate that loss of accuracy for energy gradients and equilibrium structures using single-precision data is negligible, while a mixed-precision calculation is needed for the (T) correlation energy. In calculating harmonic frequencies based on finite difference of analytic energy gradients, a larger step size is required to achieve accurate results for certain vibrational modes with single-precision data.

A non-standard finite difference scheme for a class of predator–prey systems with non-monotonic functional response

We design and analyse a non-standard finite difference (NSFD) scheme for a class of predator–prey models with non-monotonic functional response. Such functional response can be used, for instance, to model the prey's group defence ability. The numerical scheme is unconditionally dynamically consistent with respect to the positivity and boundedness of the solutions. We establish analytically the convergence of the scheme and provide numerical simulations for a Holling type-IV functional response that support the theoretical findings. The simulations further show that the NSFD scheme is more efficient than some standard numerical methods. In particular, the standard methods show dynamic inconsistencies when large step-sizes were used. On the other hand, the NSFD scheme replicated different behaviours, including the oscillatory dynamics common to predator–prey systems.

A dissipation-preserving scheme for damped oscillatory Hamiltonian systems based on splitting

In this paper, a new dissipation-preserving scheme is established for weakly dissipative perturbations of oscillatory Hamiltonian systems. The system exhibits a nonlinear oscillatory structure. The main oscillation is governed by a matrix M and the damping is governed by a matrix Γ. The new scheme preserves the oscillatory structure of the systems by incorporating the matrix M in the scheme based on the idea of ERKN methods. Meanwhile, the discrete gradient and splitting are used to construct the scheme such that the numerical solution possesses a nearly correct damping rate of the system. A main feature of the new scheme is that a relatively large stepsize can be chosen since the convergence of the implicit iterations in the scheme is shown to be independent of the matrices M and Γ. Three numerical experiments of perturbed Hamiltonian systems are conducted to show the effectiveness and the efficiency of the new scheme in comparison with the traditional discrete gradient methods.

A Study on the Variable Inductor Design by Switching the Main Paths and the Coupling Coils

In a radio frequency (RF) system, it is possible to use variable inductors for providing tunable or selective frequency range. Variable inductors can be implemented by the microelectromechanical system (MEMS) process or by using transistors as switches to change the routing of coils or coupling quantities. In this paper, we investigated the design method of a variable inductor by using MOS transistors to switch the main coil paths and the secondary coupled coils. We observed the effects of different metal layers, turn numbers, and layout arrangements for secondary-coupled coils and compared their characteristics on the inductances and quality factors. We implemented two chips in the 0.18 μm CMOS process technology for each kind of arrangement for verification. One inductor can achieve inductance values from about 300 pH to 550 pH, and the other is between 300 pH and 575 pH, corresponding to 59.3% and 62.5%, respectively, inductance variation range at 4 GHz frequency. Additionally, their fine step sizes of the switched inductances are from 0.5% to 6% for one design, and 1% to 12.5% for the other. We found that both designs achieved a large inductance tuning range and moderate inductance step sizes with a slight difference behavior on the inductance variation versus frequency.

A Projected Extrapolated Gradient Method with Larger Step Size for Monotone Variational Inequalities

A Projected Extrapolated Gradient Method with Larger Step Size for Monotone Variational Inequalities

Numerical solutions for point-source high frequency Helmholtz equation through efficient time propagators for Schrödinger equation

For high frequency Helmholtz equations with point-source conditions, geometrical optics provides an efficient way to compute the solutions without ‘pollution effect’ that is usually unavoidable in finite difference and finite element methods. However, geometrical optics generally can only provide locally valid approximations for the wavefield near the primary source. In order to obtain globally valid wavefield, we will propagate the wavefield with efficient time propagators for an appropriate time-dependent Schrödinger equation. The geometrical optics is first applied to compute locally valid approximations for the wavefield, then the wavefield is propagated through the time-dependent Schrödinger equation until its change is small enough in the domain of interest, which results in the globally valid approximation for the solution to the Helmholtz equation in the domain of interest. The adapted time propagator for the time-dependent Schrödinger equation is a Strang operator splitting based pseudospectral approach, which assures the overall efficiency of the method since the pseudospectral approach is unconditionally stable such that large time step sizes are allowed to propagate the wavefield efficiently, and the number of points per wavelength is independent of the frequency. Both two-dimensional and three-dimensional numerical experiments are presented to demonstrate the method.

Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations

In this paper, we study the classical Allen-Cahn equations and investigate the maximum-principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen-Cahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its flexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stiff source given in the equation, we use the conservative modified exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

A semi-analytic method for FM-EDFA intensity model

Based on the intensity model of few-mode erbium-doped fiber amplifiers (FM-EDFAs), we put forward a semi-analytic method for efficiently calculating the signal mode gains by solving the analytical expression for the average population inversion over each fiber segment. Our studies show that, the semi-analytic method can offer a desirable calculation accuracy, very close to that of the full approach method. According to the length dependency of the overlap integral between the normalized transverse distribution of population inversion and the signal modal field, a large iterative step size is verified to be feasible for the semi-analytic method, which is helpful for greatly reducing the iterative times. We also take the example of designing FM-EDFA’s overall gain using the semi-analytic method. The semi-analytic method is not only suitable for the FM-EDFAs with complex erbium-ion doping profile, the cladding pumping scheme and the cascaded FM-EDFAs, but also helpful for greatly improving the optimization efficiency of FM-EDFAs in combination with relevant optimization algorithms.

Undulation of a moving fluid membrane pushed by filament growth

Biomembranes experience out-of-equilibrium conditions in living cells. Their undulation spectra are different from those in thermal equilibrium. Here, we report on the undulation of a fluid membrane pushed by the stepwise growth of filaments as in the leading edge of migrating cells, using three-dimensional Monte Carlo simulations. The undulations are largely modified from equilibrium behavior. When the tension is constrained, the low-wave-number modes are suppressed or enhanced at small or large growth step sizes, respectively, for high membrane surface tensions. In contrast, they are always suppressed for the tensionless membrane, wherein the wave-number range of the suppression depends on the step size. When the membrane area is constrained, in addition to these features, a specific mode is excited for zero and low surface tensions. The reduction of the undulation first induces membrane buckling at the lowest wave-number, and subsequently, other modes are excited, leading to a steady state.

Algorithmic formulation of clay and sand pipe–soil interaction models for on-bottom stability analysis

This paper presents a new algorithmic formulation of the clay and sand pipe–soil interaction models recommended by the DNV-RP-F109 code for dynamic on-bottom stability analysis of submarine pipelines. The pipe–soil force update algorithm is formulated within the framework of computational elasto-plasticity and applies Backward-Euler integration to ensure stability and robustness for large time step sizes. Algorithmic optimization techniques are developed by utilizing a closed-form solution and subincrementation. A numerical verification study covering full cyclic displacement ranges of a 12 inch pipeline is presented. The new formulation is shown to increase the time step size by a factor of up to 50 compared to commercial software tools for on-bottom stability analysis. This achievement will be particularly beneficial for long-duration 3D nonlinear time domain on-bottom stability analysis.

Femtosecond laser fabrication of square pillars integrated Siberian-Cocklebur-like microstructures surface for anti-icing

In a low-temperature environment, water vapor in the ambient air is easy to condense and form frost on superhydrophobic surfaces, causing a reduction of their anti-icing performance. In this paper, a femtosecond laser was used to fabricate two new structures on polytetrafluoroethylene (PTFE) substrates to compare anti-icing properties with conventional structures. The two new structures are Siberian-Cocklebur-like microstructures and square pillars integrated Siberian-Cocklebur-like microstructures; whereas, the conventional structures include microgrooves with small step size, microgrooves with large step size, and square pillars. We illustrate the mechanism of forming the new structures in the interaction between the femtosecond laser and PTFE. The two new structures have greater water contact angles in a low-temperature environment than the conventional structures. We set up corresponding models for various microstructures contacted with water droplets and discuss hydrophobicity diversity. The two new structures on PTFE can effectively inhibit the vapor condensation in the microstructures and improve the hydrophobic performance and anti-icing property of PTFE surfaces in a low-temperature environment.

Exact and nonstandard finite difference schemes for the Burgers equationB(2,2)

In this paper, we consider the Burgers equation B(2,2). Exact and nonstandard finite difference schemes (NSFD) for the Burgers equation B(2,2) are designed. First, two exact finite difference schemes for the Burgers equation B(2,2) are proposed using traveling wave solution. Then, two NSFD schemes are represented for this equation. These two NSFD schemes are compared with a standard finite difference (SFD) scheme. Numerical results show that the NSFD schemes are accurate and efficient in the numerical simulation of the kink-wave solution of the B(2,2) equation. We see that although the SFD scheme yields numerical instability for large step sizes, NSFD schemes provide reliable results for long time integration. Local truncation errors show that the NSFD schemes are consistent with the B(2,2) equation.

Iteration complexity analysis of a partial LQP-based alternating direction method of multipliers

In this paper, we consider a prototypical convex optimization problem with multi-block variables and separable structures. By adding the Logarithmic Quadratic Proximal (LQP) regularizer with suitable proximal parameter to each of the first grouped subproblems, we develop a partial LQP-based Alternating Direction Method of Multipliers (ADMM-LQP). The dual variable is updated twice with relatively larger stepsizes than the classical region (0,1+52). Using a prediction-correction approach to analyze properties of the iterates generated by ADMM-LQP, we establish its global convergence and sublinear convergence rate of O(1/T) in the new ergodic and nonergodic senses, where T denotes the iteration index. We also extend the algorithm to a nonsmooth composite convex optimization and establish similar convergence results as our ADMM-LQP.

Fast implicit active contour model for inverse lithography.

We combine the ideas from level-set methods in computer vision and inverse imaging to derive a generalized active contour model for inverse lithography problems endowed with a locally implemented semi-implicit difference scheme. We introduce a cognitive analogy to move an initial guess of the interesting pattern contour by image-driven forces to the boundaries of the desired layout pattern. We develop an efficient semi-implicit numerical scheme implemented in the vicinity of the zero level-set and apply additive operator splitting (AOS) with respect to coordinate axes to solve consecutive one-dimensional linear systems of equations with the Thomas method. We demonstrate with simulation results that computation and convergence efficiency are jointly improved with reduced optimization dimensionality and a sufficient large step-size.

Analysis of the Energy Stability for Stabilized Semi-implicit Schemes of the Functionalized Cahn-Hilliard Mass-conserving Gradient Flow Equation

A stabilized semi-implicit scheme was designed in [3] to solve the Functionalized Cahn-Hilliard (CCH) equation, but there is a lack of theoretical analysis of the energy stability. In this paper, we generalize this scheme to solve the general FCH mass-conserving gradient flow (FCH-MCGF) equation and show the theoretical analysis results about the unique solvability and energy stability. We successfully prove that this scheme is uniquely solvable and energy stable in theory by rewriting the double-well potential function to satisfy the Lipschitz-type condition. The range of stabilization parameters is theoretically given as well. In addition, another similar energy stable scheme is proposed, which slightly widens the range of stabilization parameters in theory and has almost the same precision as the previous one. Both the detailed numerical procedure and the selection of stabilization parameters are presented. Finally, several numerical experiments are performed for the FCH-MCGF equation based on these schemes. Specially, the adaptive time step size is considered in the scheme for the simulations of the phase separation in 2D and 3D, since any time step size can be used according to our theoretical results. Numerical results show that these schemes are energy stable and the large time step size indeed can be used in computations. Moreover, by comprehensive comparisons of stability and accuracy among the stabilized semi-implicit scheme, the convex splitting scheme, and the fully implicit scheme, we conclude that the performance of the stabilized semi-implicit scheme is the best, and the convex splitting scheme performs better than the fully implicit scheme.

Highly accurate, linear, and unconditionally energy stable large time-stepping schemes for the Functionalized Cahn–Hilliard gradient flow equation

This paper aims to develop a class of large time-stepping schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation. By using the Invariant Energy Quadratization (IEQ) approach, we successfully construct several highly efficient linear schemes that allow large time step sizes to be used in computations. All the developed schemes with a fixed time step size are proved to be unconditionally uniquely solvable and unconditionally energy stable in theory. Moreover, we show that the obtained semi-discrete systems can be solved efficiently by using the preconditioned Krylov subspace methods, since all the linear operators of these systems are proved to be symmetric and positive definite. Finally, several numerical experiments are carried out to verify the accuracy, efficiency and energy stability of the developed IEQ schemes. We observe that our IEQ schemes are more accurate than the Scalar Auxiliary Variable (SAV) schemes. This can be attributed to the fact that the IEQ method can preserve the operator structures of the original equation as much as possible.

Moment extract method for drift kinetic simulation of magnetized plasma

A novel time integration method of the drift kinetic equation is proposed, where low order fluid moment equations including the lowest order terms for the dispersive Alfvén wave propagation are extracted from the electron drift kinetic equations. The moment extract method enables ones to implement an implicit solver for wave propagation by means of the operator splitting, while the kinetic part is solved explicitly. Numerical tests of the new scheme have verified stable and accurate time-integration of the drift kinetic equation and enabled to take a large time step size when the electron β value is smaller than the electron-to-ion mass ratio.

New convergence analysis of a primal-dual algorithm with large stepsizes

We consider a primal-dual algorithm for minimizing $f(x)+h\square l(Ax)$ with Fr\'echet differentiable $f$ and $l^*$. This primal-dual algorithm has two names in literature: Primal-Dual Fixed-Point algorithm based on the Proximity Operator (PDFP$^2$O) and Proximal Alternating Predictor-Corrector (PAPC). In this paper, we prove its convergence under a weaker condition on the stepsizes than existing ones. With additional assumptions, we show its linear convergence. In addition, we show that this condition (the upper bound of the stepsize) is tight and can not be weakened. This result also recovers a recently proposed positive-indefinite linearized augmented Lagrangian method. In addition, we apply this result to a decentralized consensus algorithm PG-EXTRA and derive the weakest convergence condition.