Coefficients perturbation methods for higher-order differential equations

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbation methods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in 1984. The basic principle of perturbation methods is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as any supplementary condition associated to it. Recently, the first author obtained practical formulae for calculating tight error bounds for the perturbation methods when this technique is applied to second-order ODEs. This paper extends those results to the case of differential equations of arbitrary order, subjected to some specified initial or boundary conditions. The results of this paper apply to any perturbation-based numerical technique such as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focus on the Tau method and present numerical examples that illustrate the accuracy of our results.     

两步H∞辨识算法的一个近似最优的误差上界

利用逼近理论中的n-宽度和Bernstain不等式,以一般性的窗口系数为变量,对鲁棒 辨识中的两步H∞辨识算法,建立了一个近似最优的误差上界函数.该函数是窗口系数的凸 函数,它不仅可用于计算任意窗口系数对应的辨识误差上界,还为优化选择两步H∞ 辨识算 法的窗口系数提供了可行途径.     

An Interval Hermite-Obreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation

To date, the only effective approach for computing guaranteed bounds on the solution of an initial value problem (IVP) for an ordinary differential equation (ODE) has been interval methods based on Taylor series. This paper derives a new approach, an interval Hermite-Obreschkoff (IHO) method, for computing such enclosures. Compared to interval Taylor series (ITS) methods, for the same stepsize and order, our IHO scheme has a smaller truncation error, better stability, and requires fewer Taylor coefficients and high-order Jacobians.The stability properties of the ITS and IHO methods are investigated. We show as an important by-product of this analysis that the stability of an interval method is determined not only by the stability function of the underlying formula, as in a standard method for an IVP for an ODE, but also by the associated formula for the truncation error.     

TSK fuzzy function approximators: design and accuracy analysis

Fuzzy systems are excellent approximators of known functions or for the dynamic response of a physical system. We propose a new approach to approximate any known function by a Takagi-Sugeno-Kang fuzzy system with a guaranteed upper bound on the approximation error. The new approach is also used to approximately represent the behavior of a dynamic system from its input-output pairs using experimental data with known error bounds. We provide sufficient conditions for this class of fuzzy systems to be universal approximators with specified error bounds. The new conditions require a smaller number of membership functions than all previously published conditions. We illustrate the new results and compare them to published error bounds through numerical examples.     

Computation of travelling wave solutions of scalar conservation laws with a stiff source term

In this paper we propose a nonoscillatory numerical technique to compute the travelling wave solution of scalar conservation laws with a stiff source term. This procedure is based on the dynamical behavior described by the associated stationary ODE and it reduces/avoids numerical errors usually encountered with these problems, i.e., spurious oscillations and incorrect wave propagation speed. We combine this treatment with either the first order Lax-Friedrichs scheme or the second order Nessyahu-Tadmor scheme. We have tested several model problems by LeVeque and Yee for which the stiffness coefficient can be increased. We have also tested a problem with a nonlinear flux and a discontinuous source term.     

Adaptive Techniques for Spline Collocation

We integrate optimal quadratic and cubic spline collocation methods for second-order two-point boundary value problems with adaptive grid techniques, and grid size and error estimators. Some adaptive grid techniques are based on the construction of a mapping function that maps uniform to non-uniform points, placed appropriately to minimize a certain norm of the error. One adaptive grid technique for cubic spline collocation is mapping-free and resembles the technique used in COLSYS (COLNEW) [2], [4]. Numerical results on a variety of problems, including problems with boundary or interior layers, and singular perturbation problems indicate that, for most problems, the cubic spline collocation method requires less computational effort for the same error tolerance, and has equally reliable error estimators, when compared to Hermite piecewise cubic collocation. Comparison results with quadratic spline collocation are also presented.     

Adaptive control of nonlinear systems via approximate linearization

This paper presents an adaptive control scheme for nonlinear systems that violates some of the common regularity and structural conditions of current nonlinear adaptive schemes such as involutivity, existence of a well-defined relative degree, and minimum phase property. While the controller is designed using an approximate model with suitable properties, the parameter update law is derived from an observation error based on the exact model described in suitable coordinates. The authors show that this approach results in a stable, closed-loop system and achieves adaptive tracking with bounds on the tracking error and parameter estimates. The authors also present a constructive procedure for adaptive state regulation which is based on the quadratic linearization technique via dynamic state feedback. This regulation scheme does not impose any restriction on the location of the unknown parameters and is applicable to any linearly controllable nonlinear system     

Approximation of infinite-dimensional systems

A Fourier series-based method for approximation of stable infinite-dimensional linear time-invariant system models is discussed. The basic idea is to compute the Fourier series coefficients of the associated transfer function T_d(Z) and then take a high-order partial sum. Two results on H^∞ convergence and associated error bounds of the partial sum approximation are established. It is shown that the Fourier coefficients can be replaced by the discrete Fourier transform coefficients while maintaining H^∞ convergence. Thus, a fast Fourier transform algorithm can be used to compute the high-order approximation. This high-order finite-dimensional approximation can then be reduced using balanced truncation or optimal Hankel approximation leading to the final finite-dimensional approximation to the original infinite-dimensional model. This model has been tested on several transfer functions of the time-delay type with promising results     

Fast evaluation of vector splines in three dimensions

Vector spline techniques have been developed as general-purpose methods for vector field reconstruction. However, such vector splines involve high computational complexity, which precludes applications of this technique to many problems using large data sets. In this paper, we develop a fast multipole method for the rapid evaluation of the vector spline in three dimensions. The algorithm depends on a tree-data structure and two hierarchical approximations: an upward multipole expansion approximation and a downward local Taylor series approximation. In comparison with the CPU time of direct calculation, which increases at a quadratic rate with the number of points, the presented fast algorithm achieves a higher speed in evaluation at a linear rate. The theoretical error bounds are derived to ensure that the fast method works well with a specific accuracy. Numerical simulations are performed in order to demonstrate the speed and the accuracy of the proposed fast method.     

A comparison of some methods for bounding connected and disconnected solution sets of interval linear systems

Summary Finding bounding sets to solutions to systems of algebraic equations with uncertainties in the coefficients, as well as rapidly but rigorously locating all solutions to nonlinear systems or global optimization problems, involves bounding the solution sets to systems of equations with wide interval coefficients. In many cases, singular systems are admitted within the intervals of uncertainty of the coefficients, leading to unbounded solution sets with more than one disconnected component. This, combined with the fact that computing exact bounds on the solution set is NP-hard, limits the range of techniques available for bounding the solution sets for such systems. However, the componentwise nature and other properties make the interval Gauss–Seidel method suited to computing meaningful bounds in a predictable amount of computing time. For this reason, we focus on the interval Gauss–Seidel method. In particular, we study and compare various preconditioning techniques we have developed over the years but not fully investigated, comparing the results. Based on a study of the preconditioners in detail on some simple, specially–designed small systems, we propose two heuristic algorithms, then study the behavior of the preconditioners on some larger, randomly generated systems, as well as a small selection of systems from the Matrix Market collection.      

L1 adaptive control for general partial differential equation (PDE) systems

This paper addresses the L₁ adaptive control problem for general Partial Differential Equation (PDE) systems. Since direct computation and analysis on PDE systems are difficult and time-consuming, it is preferred to transform the PDE systems into Ordinary Differential Equation (ODE) systems. In this paper, a polynomial interpolation approximation method is utilized to formulate the infinite dimensional PDE as a high-order ODE first. To further reduce its dimension, an eigenvalue-based technique is employed to derive a system of low-order ODEs, which is incorporated with unmodeled dynamics described as bounded-input, bounded-output (BIBO) stable. To establish the equivalence with original PDE, the reduced-order ODE system is augmented with nonlinear time-varying uncertainties. On the basis of the reduced-order ODE system, a dynamic state predictor consisting of a linear system plus adaptive estimated parameters is developed. An adaptive law will update uncertainty estimates such that the estimation error between predicted state and real state is driven to zero at each time-step. And a control law is designed for uncertainty handling and good tracking delivery. Simulation results demonstrate the effectiveness of the proposed modeling and control framework.     

Explicit A-Priori error bounds and Adaptive error control for approximation of nonlinear initial value differential systems

This paper develops and demonstrates a guaranteed a-priori error bound for the Taylor polynomial approximation of any degree to the solution of initial value ordinary differential equations. The error bound is explicit and does not require upper bounds on the potentially complicated and intrinsically unknown right-hand side nor on any of its higher-order derivatives as with existing bounds, and thus it provides a valuable tool for the numerous applications in which initial value ode problems arise and for which approximate solutions must be sought.     

Order conditions for ARKN methods solving oscillatory systems

For the perturbed oscillators in one-dimensional case, J.M. Franco designed the so-called Adapted Runge-Kutta-Nyström (ARKN) methods and derived the sufficient order conditions as well as the necessary and sufficient order conditions for ARKN methods based on the B-series theory [J.M. Franco, Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm. 147 (2002) 770-787]. These methods integrate exactly the unperturbed oscillators and are highly efficient when the perturbing function is small. Unfortunately, some critical mistakes have been made in the derivation of order conditions in that paper. On the basis of the results from that paper, Franco extended directly the ARKN methods and the corresponding order conditions to multidimensional case where the perturbed function f does not depend on the first derivative y^′ [J.M. Franco, New methods for oscillatory systems based on ARKN methods, Appl. Numer. Math. 56 (2006) 1040-1053]. In this paper, we present the order conditions for the ARKN methods for the general multidimensional perturbed oscillators where the perturbed function f may depend on only y or on both y and y^′.     

Gaussian quadrature formulae-second peano kernels,nodes, weights and Bessel functions

The central results of this paper are bounds for the second Peano kernels of the Gaussian quadrature formulae. Hence, for these quadrature formulae, we derive asymptotically optimal error constants for classes of functions with a bounded second derivative or with a first derivative of bounded variation, or for a class of convex functions. To obtain these bounds, we first prove inequalities related to MacMahon's expansion and some further results on the Bessel functionJ _o , as well as some “trapezoid theorems for the weights of Gaussian formulae” (cf. Davis and Rabinowitz [6]) with explicit bounds for the error term.     

Multidimensional exponentially fitted simplicial finite elements for convection-diffusion equations with tensor-valued diffusion

Abstract In this paper we propose and analyze an exponentially fitted simplicial finite element method for the numerical approximation of solutions to diffusion-convection equations with tensor-valued diffusion coefficients. The finite element method is first formulated using exponentially fitted finite element basis functions constructed on simplicial elements in arbitrary dimensions. Stability of the method is then proved by showing that the corresponding bilinear form is coercive. Upper error bounds for the approximate solution and the associated flux are established.     

Reliable Approximation of Weight Factors Entering Residual-based Error Bounds for FE-discretisations

In this note we refine strategies of the so called dual-weighted-residual (DWR) approach to a posteriori error control for FE-schemes. We derive rigorous error bounds, especially we control the approximation process of the (unknown) dual solution entering the proposed estimate.     

Cone-bounded nonlinearities and mean-square bounds—Smoothing and prediction

Mean-square performance bounds are derived for smoothing and prediction problems associated with the broad class of nonlinear dynamic systems which, when modeled by Ito differential equations, contain drift (·dt) coefficients which are, to within a uniformly Lipschitz residual, jointly linear in the system state and externally applied control. Included in this paper are lower bounds on the error covariance attainable by any smoother or any predictor, including the optimum, and upper bounds on the performance of some simple, implementable predictors reminiscent of the designs which are optimal in the linear case. The lower bounds on smoothing and prediction performance are established using measure-transformation techniques to relate a version of the nonlinear problem to its linearization. The upper bound on prediction performance is constructed by a direct analysis of the estimation error. All the bounds hold for correlated system and observation noises. All are rigorously derived and independent of control or control law. In each case, the computational effort is comparable to that for the corresponding optimum linear smoothing or prediction problem. The bounds converge with vanishing nonlinearity (vanishing Lipschitz constants) to the known optimum performance for the limiting linear system. Consequently, the bounds are asymptotically tight and the simple designs studied are asymptotically optimal with vanishing nonlinearity.     

Multi-degree reduction of tensor product Bézier surfaces with conditions of corners interpolations

This paper studies the multi-degree reduction of tensor product B(?)zier surfaces with any degree interpolation conditions of four corners, which is urgently to be resolved in many CAD/CAM systems. For the given conditions of corners interpolation, this paper presents one intuitive method of degree reduction of parametric surfaces. Another new approximation algorithm of multi-degree reduction is also presented with the degree elevation of surfaces and the Chebyshev polynomial approximation theory. It obtains the good approximate effect and the boundaries of degree reduced surface can preserve the prescribed continuities. The degree reduction error of the latter algorithm is much smaller than that of the first algorithm. The error bounds of degree reduction of two algorithms are also presented .     

Arbitrarily tight upper and lower bounds on the Bayesianprobability of error

This paper presents new upper and lower bounds on the minimum probability of error of Bayesian decision systems for the two-class problem. These bounds can be made arbitrarily close to the exact minimum probability of error, making them tighter than any previously known bounds