Multistep Approximation Research Articles

Alternated and multi-step inertial approximation methods for solving convex bilevel optimization problems

In this paper, we propose three kinds of inertial approximation methods based on the proximal gradient algorithm to accelerate the convergence of the algorithm for solving convex bilevel optimization problems. Under reasonable parameters, we prove that our algorithms converge strongly to some solution of the problem, which is the unique solution of a variational inequality problem. Firstly, two alternated inertial methods are presented. Secondly, a multi-step inertial method is proposed to accelerate the convergence of the algorithm. Thirdly, an alternated multi-step inertial method is further introduced. In addition, we also consider that the inertial parameters can be chosen as positive or negative parameters, and we get better results. Our numerical results illustrate the performances of our algorithms and present some comparisons with related algorithms. Our results improve and extend the corresponding results reported by some authors recently.

A Third Order Exponential Time Differencing Numerical Scheme for No-Slope-Selection Epitaxial Thin Film Model with Energy Stability

In this paper we propose and analyze a (temporally) third order accurate exponential time differencing (ETD) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. A linear splitting is applied to the physical model, and an ETD-based multistep approximation is used for time integration of the corresponding equation. In addition, a third order accurate Douglas-Dupont regularization term, in the form of \(-A {\Delta t}^2 \phi _0 (L_N) \Delta _N^2 ( u^{n+1} - u^n)\), is added in the numerical scheme. A careful Fourier eigenvalue analysis results in the energy stability in a modified version, and a theoretical justification of the coefficient A becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the \(\ell ^\infty (0,T; H_h^1) \cap \ell ^2 (0,T; H_h^3)\) norm, with the help of a careful eigenvalue bound estimate, combined with the nonlinear analysis for the NSS model. This convergence estimate is the first such result for a third order accurate scheme for a gradient flow. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for \(\varepsilon =0.02\) (up to \(T=3 \times 10^5\)) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization

In this paper, we aim to propose some spectral gradient methods via variational technique under log-determinant norm. The spectral parameters satisfy the modified weak secant relations that inspired by the multistep approximation for solving large scale unconstrained optimization. An executable code is developed to test the efficiency of the proposed method with spectral gradient method using standard weak secant relation as constraint. Numerical results are presented which suggest a better performance has been achieved.

From Circuit Theory, Simulation to SPICE<sup>Diego<\/sup>: A Matrix Exponential Approach for Time-Domain Analysis of Large-Scale Circuits

SPICE (Simulation Program with Integrated Circuit Emphasis) is a widely used circuit simulation framework for integrated circuit designs. The basic skeleton of SPICE time domain simulation was derived from the versions developed in UC Berkeley during the 1970s. In this paper, we report most recent numerical integration methods to improve traditional SPICE time integration schemes, which are based on linear multi-step and low order approximation for the circuit differential equation system. Recently, matrix exponential based time domain simulation algorithms are being developed to address long-term issues in the standard numerical integration methods. We review the related techniques in matrix exponential based approaches and state several distinguished features in challenging simulation problems, such as linear power network analysis and nonlinear circuit system simulation (SPICEDiego). We believe that the matrix exponential approaches can shed new light on the research and development of future circuit simulation algorithmic systems.

Exponential stability of time-varying linear systems

This paper considers the stability of both continuous and discrete time-varying linear systems. Stability estimates are obtained in either case in terms of the Lipschitz constant for the governing matrices and the assumed uniform decay rate of the corresponding frozen time linear systems. The main techniques used in the analysis are comparison methods, scaling and the application of continuous stability estimates to the discrete case. Counterexamples are presented to show the necessity of the stability hypotheses. The discrete results are applied to derive sufficient conditions for the stability of a backward Euler approximation of a time-varying system and a one-leg linear multistep approximation of a scalar system.

Evaluation of eigenvalues of a smooth potential via Schroedinger transmission across multi-step potential

In a one-dimensional quantal solution of Schroedinger equation, the general expressions for reflection and transmission coefficients are derived for a potential constituting n number of rectangular wells and barriers. These expressions are readily used for the estimation of eigenvalues of a smooth potential which is simulated by a multi-step potential. The applicability of this method is demonstrated with success in potentials with different forms including the most versatile Ginocchio potential where the widely used numerical method like Runge-Kutta integration algorithm fails to yield the result. Accurate evaluation of eigenvalues free from numerical problem for any form of potentials, whether analytically solvable or not, is the highlight of the present multi-step approximation method in the theory of potential scattering.

The Pathwise Convergence of Approximation Schemes for Stochastic Differential Equations

AbstractThe authors of this paper study approximation methods for stochastic differential equations, and point out a simple relation between the order of convergence in the pth mean and the order of convergence in the pathwise sense: Convergence in the pth mean of order α for all p ≥ 1 implies pathwise convergence of order α – ε for arbitrary ε > 0. The authors then apply this result to several one-step and multi-step approximation schemes for stochastic differential equations and stochastic delay differential equations. In addition, they give some numerical examples.

Multistep approximation of linear sectorial evolution equations

This paper concerns the linear multistep approximation of a linear sectorial evolution equation u t = Au on a complex Banach space X. Given a strictly A(a)-stable q-step method of order p whose stability region includes a sectorial region containing the spectrum of the operator A, the corresponding evolution semigroup for the method is C n (hA), n ≥ 0, defined on X q , where C(z) ∈ L(C q ) denotes the one-step map associated with the method. It is shown that for appropriately chosen V, Y: C → C q , based on the principal right and left eigenvectors of C(z), C n (hA) approximates the semigroup V(hA)e nhA Y H (hA) with optimal order p.

Fractional-Step법을 이용한 비압축성 비정상 Navier-Stokes 방정식의 유한 요소해석

The main objective of the research is to develop a research code solving transient incompressible Navier-Stokes equation. In this research code, Adams-Bashforth method was applied to the convective terms of the navier stokes equation and the splitted equations were discretized spatially by finite element methods to solve the complex geometry problems easily. To reduce the divergence on the boundaries of pressure poisson equation due to the unsuitable pressure boundary conditions, multi step approximation pressure boundary conditions derived from the boundary linear momentum equations were used. Simulations of Lid Driven Flow and Flow over Cylinder were conducted to prove the accuracy by means of the comparison with results of the previous workers.<br/>

On a multistep method approximating a linear sectorial evolution equation

This paper concerns the linear multistep approximation of an abstract dissipative linear sectorial evolution equation on a Banach space X . We study how well the semigroup generated by a sectorial operator A is approximated by the numerical semigroup generated by a q-step, strictly A(α)-stable multistep method. An optimal order error bound is obtained.

Application of random walk theory to the first-order Fermi acceleration in shock waves

We formulate the first-order Fermi acceleration in shock waves in terms of the random walk theory. The formulation is applicable to any value of the shock speed and the particle speed – in particular, to the acceleration in relativistic shocks and to the injection problem, where the particle speed is comparable to the fluid speed – as long as large-angle scattering is suitable for the scattering process of particles. We first show that the trajectory of a particle suffering from large-angle scattering can be treated as a random walk in a moving medium with an absorbing boundary (e.g., the shock front). We derive an integral equation to determine the density of scattering points of the random walk, and by solving it approximately we obtain approximate solutions of the probability density of pitch angle at, and the return probability after, the shock crossing in analytic form. These approximate solutions include corrections of several non-diffusive effects to the conventional diffusion approximation, and we show that they agree well with Monte Carlo results for the isotropic scattering model for any shock speed and particle speed. When we neglect effects of ‘a few-step return’, we obtain ‘the multistep approximation’ which includes only the effect of finite mean free path and which is equivalent to ‘the relativistic diffusion approximation’ used by Peacock if the correct diffusion length is used in his expression. We find that the multistep approximation is not appropriate to describe the probability densities of individual particles for relativistic shocks, but that the pitch angle distribution at the shock front in steady state is in practice quite well approximated by that given by the multistep approximation, because the effects of a finite mean free path and a few-step return compensate each other when averaged over the pitch angle distribution. Finally, we give an analytical expression for the spectral index of accelerated particles in parallel shocks valid for arbitrary shock speed using this approximation.

Treating some solid state problems with the Dirac equation

The ambiguity involved in the definition of effective-mass Hamiltonians for nonrelativistic models is resolved using the Dirac equation. The multistep approximation is extended for relativistic cases allowing the treatment of arbitrary potential and effective-mass profiles without ordering problems. On the other hand, if the Schrodinger equation is supposed to be used, our relativistic approach demonstrate that both results are coincidents if the BenDaniel and Duke prescription for the kinetic-energy operator is implemented. Applications for semiconductor heterostructures are discussed.

Multistep approximation algorithms: Improved convergence rates through postconditioning with smoothing kernels

We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in the context of the numerical solution of PDEs) suffer from a “loss of derivatives”, and that the subsequent linear rate of convergence can be improved to be superlinear using an adaptation of Nash–Moser iteration for numerical analysis purposes; the essence of the adaptation being a splitting of the inversion and the smoothing into two separate steps. We show how these ideas apply to scattered data approximation as well as the numerical solution of partial differential equations. We investigate the use of several radial kernels for the smoothing operation. In our numerical examples we use radial basis functions also in the inversion step.

Approximation of continuous time stochastic processes by a local linearization method

This paper investigates the rate of convergence of an alternative approximation method for stochastic differential equations. The rates of convergence of the one-step and multi-step approximation errors are proved to be O ( ( Δ t ) 2 ) O((\Delta t)^2) and O ( Δ t ) O(\Delta t) in the L p L_p sense respectively, where Δ t \Delta t is discrete time interval. The rate of convergence of the one-step approximation error is improved as compared with methods assuming the value of Brownian motion to be known only at discrete time. Through numerical experiments, the rate of convergence of the multi-step approximation error is seen to be much faster than in the conventional method.

Multistep polygonal approximation techniques

Conventional polygonal approximation techniques have a fixed parameter that makes it difficult to extract the minimum number of critical points that represent faithfully complex contours having various details. We propose multistep polygonal approximation algorithms that integrate line segments detected at different scales or resolutions by means of scale or resolution partitioning of contour patterns. Via a computer simulation, we show that the proposed multistep methods approximate the contours better than conventional methods having a fixed parameter.

On multistep approximation of semigroups in Banach spaces

We consider a multistep rational approximation of a bounded, strongly continuous semigroup on a Banach space. We study the convergence when the time step converges to zero under a weak form of A-stability assumption.