Variable Step-size Strategy Research Articles

Variable matrix-type step-size affine projection algorithm with orthogonalized input vectors

In this paper, we propose a variable matrix-type step-size affine projection algorithm (APA) with orthogonalized input vectors. We generate orthogonalized input vectors using the Gram–Schmidt process to implement the weight update equation of the APA using the sum of normalized least mean squares (NLMS)-like updating equations. This method allows us to use individual step sizes corresponding to each NLMS-like equation, which is equivalent to adopting the step size in the form of a diagonal matrix in the APA. We adopt a variable step-size scheme, in which the individual step sizes are determined to minimize the mean square deviation of the APA in order to achieve the fastest convergence on every iteration. Furthermore, because of the weight vector updated successively only along each innovative one among the reused inputs and effect of the regularization absorbed into the derived step size, the algorithm works well even for badly excited input signals. Experimental results show that our proposed algorithm has almost optimal performance in terms of convergence rate and steady-state estimation error, and these results are remarkable especially for badly excited input signals.

A variable step-size strategy for distributed estimation over adaptive networks

AbstractA lot of work has been done recently to develop algorithms that utilize the distributed structure of anad hocwireless sensor network to estimate a certain parameter of interest. One such algorithm is called diffusion least-mean squares (DLMS). This algorithm estimates the parameter of interest using the cooperation between neighboring sensors within the network. The present work proposes an improvement on the DLMS algorithm by using a variable step-size LMS (VSSLMS) algorithm. In this work, first, the well-known variants of VSSLMS algorithms are compared with each other in order to select the most suitable algorithm which provides the best trade-off between performance and complexity. Second, the detailed convergence and steady-state analyses of the selected VSSLMS algorithm are performed. Finally, extensive simulations are carried out to test the robustness of the proposed algorithm under different scenarios. Moreover, the simulation results are found to corroborate the theoretical findings very well.

A variable step-size gradient adaptive lattice algorithm for multiple sinusoidal interference cancelation

Abstract In passive sonar, adaptive algorithms can be used to cancel strong sinusoidal self-interferences. In order to correctly recover low-power target signals during the early stages of processing, these adaptive algorithms must provide fast convergence and, at the same time, narrow notches at the frequencies of the sinusoids. In this respect, the gradient adaptive lattice (GAL) algorithm is a very attractive choice. However, the GAL algorithm with a constant step-size parameter has to compromise between the convergence rate and notch bandwidths. Therefore, in this article, we propose a variable step-size scheme for the GAL algorithm that can achieve both a fast convergence rate and narrow notches. Simulation results demonstrate the efficiency of the proposed algorithm compared to both the conventional GAL algorithm and transversal adaptive filter combined with the variable step-size scheme.

New Algorithm of Two-Point Block Method for Solving Boundary Value Problem with Dirichlet and Neumann Boundary Conditions

Two-point block method with variable step-size strategy is presented to obtain the solutions for boundary value problems directly. Dirichlet type and Neumann type of boundary conditions are studied in this paper. Multiple shooting techniques adapted with the three-step iterative method are employed for generating the guessing value. Six boundary value problems are solved using the proposed method, and the numerical results are compared to the existing methods. The results suggest a significant improvement in the efficiency of the proposed methods in terms of the number of steps, execution time, and accuracy.

Retrieval of sparse solutions of multiple-measurement vectors via zero-point attracting projection

A new sparse signal recovery algorithm for multiple-measurement vectors (MMV) problem is proposed in this paper. The sparse representation is iteratively drawn based on the idea of zero-point attracting projection (ZAP). In each iteration, the solution is first updated along the negative gradient direction of an approximate ℓ2,0 norm to encourage sparsity, and then projected to the solution space to satisfy the under-determined equation. A variable step size scheme is adopted further to accelerate the convergence as well as to improve the recovery accuracy. Numerical simulations demonstrate that the performance of the proposed algorithm exceeds the references in various aspects, as well as when applied to the modulated wideband converter, where recovering MMV problem is crucial to its performance.

Improved interframe registration based nonuniformity correction for focal plane arrays

In this paper, an improved interframe registration based nonuniformity correction algorithm for focal plane arrays is proposed. The method simultaneously estimates detector parameters and carries out the nonuniformity correction by minimizing the mean square error between the two properly registered image frames. A new masked phase correlation algorithm is introduced to obtain reliable shift estimates in the presence of fixed pattern noise. The use of an outliers exclusion scheme, together with a variable step size strategy, could not only promote the correction precision considerably, but also eliminate ghosting artifacts effectively. The performance of the proposed algorithm is evaluated with clean infrared image sequences with simulated nonuniformity and real pattern noise. We also apply the method to a real-time imaging system to show how effective it is in reducing noise and the ghosting artifacts.

A Variable Step Size Riemannian Sum for an Itô Integral

We investigate the problem of using a Riemannian sum with random subintervals to approximate the iterated Itô integral ∫wdw - or, equivalently, solving the corresponding stochastic differential equation by Euler's method with variable step sizes. In the past this task has been used as a counterexample to illustrate that variable step sizes must be used with extreme caution in stochastic numerical analysis. This article establishes a class of variable step size schemes which do work.

A Variable Step Size Riemannian Sum for an Itô Integral

We investigate the problem of using a Riemannian sum with random subintervals to approximate the iterated Itô integral ∫wdw - or, equivalently, solving the corresponding stochastic differential equation by Euler's method with variable step sizes. In the past this task has been used as a counterexample to illustrate that variable step sizes must be used with extreme caution in stochastic numerical analysis. This article establishes a class of variable step size schemes which do work.

An Adaptive Algorithm with Variable Step-Size for Parallel Notch Filter

A parallel notch filter (PNF) for eliminating a sinusoidal signal whose frequency and phase are unknown, has been proposed previously. The PNF achieves both fast convergence and high estimation accuracy when the step-size for adaptation is appropriately determined. However, there has been no discussion of how to determine the appropriate step-size. In this paper, we derive the convergence condition on the step-size, and propose an adaptive algorithm with variable step-size so that convergence of the PNF is automatically satisfied. Moreover, we present a new filtering structure of the PNF that increases the convergence speed while keeping the estimation accuracy. We also derive a variable step-size scheme for the new PNF to guarantee the convergence. Simulation results show the effectiveness of the proposed method.

Stability of one-leg -methods for the variable coefficient pantograph equation on the quasi-geometric mesh

 where a(t) and b(t) are continuous complex-valued functions and q ∈ (0, 1) is a fixed constant. We also extend our analysis to the neutral equation y � (t) = a(t) y(t) + b(t) y( qt ) + c(t) y � ( qt ), where c(t) is also a continuous complex-valued function. In recent years, stability properties of numerical methods for this kind of equation have been studied by numerous authors who have mainly considered the constant coefficient case and integration over meshes with constant stepsize. In general, the developed techniques give rise to non-standard recurrence relations. Instead, in the present paper we study constrained variable stepsize schemes. In Bellen et al. (1997, Appl. Numer. Math.,24, 279–293) the behaviour of the class of Θ-methods was analysed with relevance for the constant coefficient version of the above equation. As with the constant coefficient case, we show here that the methods in this class are stable if and only if 1 Θ > 1/2. The basic tool we use in our analysis is the spectral radius of a family of matrices, computed by means of polytope extremal norms. An extension of the stability results to a more general class of equations is also discussed in the last section.

Explicit pseudo two-step RKN methods with stepsize control

This paper is devoted to variable stepsize strategy implementations of a class of explicit pseudo two-step Runge–Kutta–Nyström methods of arbitrarily high order for solving nonstiff problems for systems of special second-order differential equations. The constant stepsize explicit pseudo two-step Runge–Kutta–Nyström methods are developed into variable stepsize ones and equipped with embedded formulas giving a cheap error estimate for stepsize control. By two examples of widely-used test problems, a pseudo two-step Runge–Kutta–Nyström method of order 8 implemented with variable stepsize strategy is shown to be much more efficient than parallel and sequential codes available in the literature. With stringent error tolerances, this new explicit pseudo two-step Runge–Kutta–Nyström method is even superior to sequential codes in a sequential computer.

Variable stepsize schemes for effective order methods and enhanced order composition methods

The interest in the concept of “effective order” has been revived by its rediscovery in applications to symplectic problems. In this paper we revert to the original application, the construction of explicit Runge–Kutta methods. Changing stepsize is a characteristic difficulty with effective order methods and we propose a way of overcoming this difficulty. We also consider the possible cancellation of local truncation errors of two methods over two successive steps. Using the algebraic approach for deriving these results gives us further insight into these methods and compositions of methods. A particular sixth stage Runge–Kutta pair is derived in the paper and is shown to be competitive.

Reversible adaptive regularization methods for atomic N-body problems in applied fields

Reversible and adaptive integration methods based on Kustaanheimo-Stiefel regularization and modified Sundman transformations have recently been proposed for perturbed Kepler motion and certain types of N-body problems including atomic scattering and gravitation. These methods are explicit, and demonstrate superior conservation properties compared to standard explicit integrators applied with or without variable stepsize schemes. In this paper, reversible adaptive regularization is applied to treat (classical) atomic systems in applied electrical and/or magnetic fields.

Continuous variable stepsize explicit pseudo two-step RK methods

The aim of this paper is to apply a class of constant stepsize explicit pseudo two-step Runge-Kutta methods of arbitrarily high order to nonstiff problems for systems of first-order differential equations with variable stepsize strategy. Embedded formulas are provided for giving a cheap error estimate used in stepsize control. Continuous approximation formulas are also considered for use in an eventual implementation of the methods with dense output. By a few widely used test problems, we compare the efficiency of two pseudo two-step Runge-Kutta methods of orders 5 and 8 with the codes DOPRI5, DOP853 and PIRK8. This comparison shows that in terms of ƒ-evaluations on a parallel computer, these two pseudo two-step Runge-Kutta methods are a factor ranging from 3 to 8 cheaper than DOPRI5, DOP853 and PIRK8. Even in a sequential implementation mode, fifth-order new method beats DOPRI5 by a factor more than 1.5 with stringent error tolerances.

Equilibrium states for predictor-corrector methods

A theory is developed that explains the stepsize patterns observed when standard predictor-corrector methods with variable stepsize strategy are used to solve stiff or mildly stiff problems. In some cases an algorithmic steady state occurs with smooth almost constant stepsizes; at other times an oscillating stepsize pattern of stepsizes is observed with the possibility of frequent rejected steps.

Structure Preservation for Constrained Dynamics with Super Partitioned Additive Runge--Kutta Methods

A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the Euler--Lagrange equations is presented. A new class of integrators is defined: the super partitioned additive Runge--Kutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structure-preservation, the class of s-stage Lobatto IIIA-B-C-C* SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.

Computational efficiency of the continuous cell-to-cell mapping technique as a function of integration schemes

The continuous cell-to-cell mapping technique (CCCMT) is a recently proposed Markovian approach which can be used for the dynamic reliability and safety analysis of process control systems, as well as for the global analysis of nonlinear dynamic systems in general. A parametric study is performed on the computational efficiency of CCCMT as a function of various integration schemes. The results show that a variable stepsize scheme is desirable with a sufficient required precision. Among the integration schemes considered, a fourth order Runge-Kutta scheme seems to be preferable for short term simulations and the mid-point implicit scheme seems to be preferable for long term simulations or to obtain the steady-state system behavior.

Variable step-size techniques in continuous Runge-Kutta methods for isospectral dynamical systems

In this paper we consider numerical methods for the dynamical system L′ = [ B( L), L], L(0) = L 0, where L 0 is a n × n symmetric matrix, [if[B(L),L] is the commutator of B( L) and L, and B( L) is a skew-symmetric matrix for each symmetric matrix L. The differential system is isospectral, i.e., L( t) preserves the eigenvalues of L 0, for t⩾0. The matrix B( L) characterizes the flow, and for special B(·), the solution matrix L( t) tends, as t increases, to a diagonal matrix with the same eigenvalues of L 0. In [11] a modification of the MGLRK methods, introduced in [2], has been proposed. These procedures are based on a numerical approximation of the Flaschka formulation of (∗) by Runge-Kutta (RK) methods. Our numerical schemes (denoted by E dGLRK s consist in solving the system (∗) by a continuous explicit Runge-Kutta method (CERK) and then performing a single step of a Gauss-Legendre RK method, for the Flaschka formulation of (∗), in order to convert the approximation of L( t) to an isospectral solution. The problems of choosing a constant time step or a variable time step strategy are both of great importance in the application of these methods. In this paper, we introduce a definition of stability for the isospectral numerical methods. This definition involves a potential function associated to the isospectral flow. For the class E dGLRK s we propose a variable step-size strategy, based on this potential function, and an optimal constant time step h in the stability interval. The variable time step strategy will be compared with a known variable step-size strategy for RK methods applied to these dynamical systems. Numerical tests will be given and a comparison with the QR algorithm will be shown.

Asymptotic stability properties of θ-methods for the pantograph equation

In this paper we consider asymptotic stability properties of θ-methods for the following pantograph equation: ▪ where a, b, c ∈ C . In recent years stability properties of numerical methods for this kind of equation have been studied by numerous authors who have considered ineshes with fixed stepsize. In general the developed techniques give rise to non-ordinary recurrence relations. In this work, instead, we study constrained variable stepsize schemes, suggested by theoretical and computational reasons, which lead to a non-stationary difference equation. For a first insight, we focus our attention on the class of θ-methods and show that asymptotic stability is obtained for θ> 1 2 . Finally, some preliminary considerations are devoted to the non-neutral and non-stationary pantograph equation.

An efficient solution procedure for the ONE-STEP experiment

The ONE-STEP experiment 1 describes an experimental procedure for determining the hydraulic properties of a soil sample. It involves measurements of cumulative outflow with time from a soil core placed in a pressure cell. The water flow in this soil core satisfies the classical Richards equation. The accuracy of numerical schemes for simulating this experiment strongly depends on experimental conditions. In case the pneumatic head applied to the pressure cell is high the solution is characterized by a “steep drying front”. A small spatial discretization step size is then required to obtain accurate solutions. To improve the efficiency of the procedure for determining soil hydraulic properties, we have studied the applicability of a simple variable step size scheme.