Set membership state and parameter estimation for systems described by nonlinear differential equations

This paper investigates the use of guaranteed methods to perform state and parameter estimation for nonlinear continuous-time systems, in a bounded-error context. A state estimator based on a prediction-correction approach is given, where the prediction step consists in a validated integration of an initial value problem for an ordinary differential equation (IVP for ODE) using interval analysis and high-order Taylor models, while the correction step uses a set inversion technique. The state estimator is extended to solve the parameter estimation problem. An illustrative example is presented for each part.     

Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models

A method is developed that allows the verified integration of ODEs based on local modeling with high-order Taylor polynomials with remainder bound. The use of such Taylor models of order n allows convenient automated verified inclusion of functional dependencies with an accuracy that scales with the (n + 1)-st order of the domain and substantially reduces blow-up.Utilizing Schauder's fixed point theorem on certain suitable compact and convex sets of functions, we show how explicit nth order integrators can be developed that provide verified nth order inclusions of a solution of the ODE. The method can be used not only for the computation of solutions through a single initial condition, but also to establish the functional dependency between initial and final conditions, the so-called flow of the ODE. The latter can be used efficiently for a substantial reduction of the wrapping effect.Examples of the application of the method to conventional initial value problems as well as flows are given. The orders of the integration range up to twelve, and the verified inclusions of up to thirteen digits of accuracy have been demanded and obtained.     

Wrapping Function of the Initial Value Problem for ODE: Applications

We discuss applications of the concept of wrapping function in studying the wrapping effect associated with validated (interval) methods for numerical solution of the initial value problem for ordinary differential equations. Initial value problems are characterized with respect to the occurrence of wrapping effect using that there is no wrapping effect if and only if the wrapping function equals the optimal interval enclosure of the solution. Particular attention is paid to linear systems of ODEs where the functions quantifying the wrapping effect can be presented in a simpler form. A necessary condition for no wrapping effect is proved for the general case.     

Taylor Forms—Use and Limits

This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch, and Wittwer, and independently studied and popularized since 1996 by Berz, Makino, and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001.Apart from summarizing what Taylor forms are and do, this review puts them into the perspective of more traditional methods, in particular centered forms, discusses the major applications, and analyzes some of their elementary properties. Particular emphasis is given to overestimation properties and the wrapping effect. A deliberate attempt has been made to offer value statements with appropriate justifications; but all opinions given are my own and might be controversial.     

Limit Values of Derivatives of the Cauchy Integrals and Computation of the Logarithmic Potentials

We derive limit values of high-order derivatives of the Cauchy integrals, which are extensions of the Plemelj-Sokhotskyi formula. We then use them to develop the Taylor expansion of the logarithmic potentials at the normal direction. Based on the Taylor expansion and numerical integration methods for weekly singular functions using grid points, we design fast algorithms for computing the logarithmic potentials. We prove that these methods have an optimal order of convergence with a linear computational complexity. Numerical examples are included to confirm the theoretical estimates for the methods.     

Wrapping Effect and Wrapping Function

We study the wrapping effect associated with validated interval methods for numerical solution of the initial value problem for ordinary differential equations by introducing a new concept of wrapping function. The wrapping function is proved to be the limit of the enclosures of the solution produced by methods of certain type. There is no wrapping effect if and only if the wrapping function equals the optimal interval enclosure of the solution.     

A unified wavelet-based modelling framework for non-linear system identification: the WANARX model structure

A new unified modelling framework based on the superposition of additive submodels, functional components, and wavelet decompositions is proposed for non-linear system identification. A non-linear model, which is often represented using a multivariate non-linear function, is initially decomposed into a number of functional components via the well-known analysis of variance (ANOVA) expression, which can be viewed as a special form of the NARX (non-linear autoregressive with exogenous inputs) model for representing dynamic input–output systems. By expanding each functional component using wavelet decompositions including the regular lattice frame decomposition, wavelet series and multiresolution wavelet decompositions, the multivariate non-linear model can then be converted into a linear-in-the-parameters problem, which can be solved using least-squares type methods. An efficient model structure determination approach based upon a forward orthogonal least squares (OLS) algorithm, which involves a stepwise orthogonalization of the regressors and a forward selection of the relevant model terms based on the error reduction ratio (ERR), is employed to solve the linear-in-the-parameters problem in the present study. The new modelling structure is referred to as a wavelet-based ANOVA decomposition of the NARX model or simply WANARX model, and can be applied to represent high-order and high dimensional non-linear systems.     

基于线性状态反馈和高阶泰勒逼近的自抗扰控制方法

针对对象模型不确定性和输入扰动问题, 设计扩张状态观测器. 提出利用高阶泰勒多项式构造综合扰动的内部模型, 将其作为系统的扩张状态, 由Luenberger 状态观测器对其进行估计. 运用线性状态反馈法, 将原系统状态估值反馈至参考输入, 再结合极点配置法和扩张状态估值得到最终的控制作用. 由于将原系统转化为积分串联型, 实现了系统线性化, 并对干扰进行了有效补偿, 使系统抗扰性能大为增强. 通过数例分析验证了所提出方法的有效性.     

Optimal control of nonlinear power systems by an imbedding method

Realistic models of power systems involve high-order non-linear differential equations. The application of optimal control theory in optimizing such systems involves a high-order nonlinear 2-point boundary-value problem whose solution is very cumbersome. In this paper the imbedding method of solving a free-end, fixed-time optimal control problem is applied to a sixth-order nonlinear power system. The optimal designs for both open and closed loop plants are presented. The results of this application are compared with those by the hybrid linearization method considered by other authors. The results indicate that the 2 methods follow quite closely.     

Uncertain Method for Optimal Control Problems With Uncertainties Using Chebyshev Inclusion Functions

In this paper, a new uncertain analysis method is developed for optimal control problems, including interval variables (uncertainties) based on truncated Chebyshev polynomials. The interval arithmetic in this research is employed for analyzing the uncertainties in optimal control problems comprising uncertain‐but‐bounded parameters with only lower and upper bounds of uncertain parameters. In this research, the Chebyshev method is utilized because it generates sharper bounds for meaningful solutions of interval functions, rather than the Taylor inclusion function, which is efficient in handling the overestimation derived from the wrapping effect due to interval computations. For utilizing the proposed interval method on the optimal control problems with uncertainties, the Lagrange multiplier method is first applied to achieve the necessary conditions and then, by using some algebraic manipulations, they are converted into the ordinary differential equation. Afterwards, the Chebyshev inclusion method is employed to achieve the solution of the system. The final results of the Chebyshev inclusion method are compared with the interval Taylor method. The results show that the proposed Chebyshev inclusion function based method better handle the wrapping effect than the interval Taylor method.     

Verified High-Order Inversion of Functional Depedencies and Interval Newton Methods

A new method for computing verified enclosures of the inverses of given functions over large domains is presented. The approach is based on Taylor Model methods, and the sharpness of the enclosures scales with a high order of the domain. These methods have applications in the solution of implicit equations and the Taylor Model based integration of Differential Algebraic Equations (DAE) as well as other tasks where obtaining verified high-order models of inverse functions is required. The accuracy of Taylor model methods has been shown to scale with the (n + 1)-st order of the underlying domain, and as a consequence, they are particularly well suited to model functions over relatively large domains. Moreover, since Taylor models can control the cancellation and dependency problems (see Makino, K. and Berz, M.: Efficient Control of the Dependency Problem Based on Taylor Model Methods, Reliable Computing 5(1) (1999)) that often affect regular interval techniques, the new method can successfully deal with complicated multidimensional problems. As an application of these new methods, a high-order extension of the standard Interval Newton method that converges approximately with the (n + 1)-st order of the underlying domain is developed. Several examples showing various aspects of the practical behavior of the methods are given. This revised version was published online in August 2006 with corrections to the Cover Date.     

A kernel autoassociator approach to pattern classification.

Autoassociators are a special type of neural networks which, by learning to reproduce a given set of patterns, grasp the underlying concept that is useful for pattern classification. In this paper, we present a novel nonlinear model referred to as kernel autoassociators based on kernel methods. While conventional non-linear autoassociation models emphasize searching for the non-linear representations of input patterns, a kernel autoassociator takes a kernel feature space as the nonlinear manifold, and places emphasis on the reconstruction of input patterns from the kernel feature space. Two methods are proposed to address the reconstruction problem, using linear and multivariate polynomial functions, respectively. We apply the proposed model to novelty detection with or without novelty examples and study it on the promoter detection and sonar target recognition problems. We also apply the model to mclass classification problems including wine recognition, glass recognition, handwritten digit recognition, and face recognition. The experimental results show that, compared with conventional autoassociators and other recognition systems, kernel autoassociators can provide better or comparable performance for concept learning and recognition in various domains.     

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.     

Nonlinear Upwind-Biased Free-Stream-Preserving Schemes for Compressible Euler Equations

The focus of the present research is on the analysis of local energy stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local energy stability, i.e., the numerical growth rate does not exceed the growth rate of the continuous problem, is not guaranteed even when the scheme is non-linearly stable and that this may have adverse implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally energy unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we further demonstrate numerically that such schemes cause exponential growth of errors during the simulation. Furthermore, we encounter a similar abnormal behaviour for the compressible Euler equations, for a smooth exact solution of a density wave. Finally, for the same case, we demonstrate numerically that other commonly known split-forms, such as the Kennedy and Gruber splitting, are also locally energy unstable.

     

A mixed corrected symmetric SPH (MC-SSPH) method for computational dynamic problems

In this work, a mixed corrected symmetric smoothed particle hydrodynamics (MC-SSPH) method is proposed for solving the non-linear dynamic problems, and is extended to simulate the fluid dynamic problems. The proposed method is achieved by improving the conventional SPH, in which the constructed process is based on decomposing the high-order partial differential equation into multi-first-order partial differential equations (PDEs), correcting the particle approximations of the kernel and first-order kernel gradient of SPH under the concept of Taylor series, and finally making the obtained local matrix symmetric. For the purpose of verifying the validity and capacity of the proposed method, the Burgers? and modified KdV–Burgers? equations are solved using MC-SSPH and compared with other mesh-free methods. Meanwhile, the proposed MC-SSPH is further extended and applied to simulate free surface flows for better illustrating the special merit of particle method. All the numerical results agree well with available data, and demonstrate that the MC-SSPH method possesses the higher accuracy and better stability than the conventional SPH method, and the better flexibility and extended application than the other mesh-free methods.     

Telescopic Projective Integration for Linear Kinetic Equations with Multiple Relaxation Times

We study a general, high-order, fully explicit numerical method for simulating kinetic equations with a BGK-type collision model with multiple relaxation times. In that case, the problem is stiff and its spectrum consists of multiple separated eigenvalue clusters. Projective integration methods are explicit integration schemes that first take a few small (inner) steps with a simple, explicit method, after which the solution is extrapolated forward in time over a large (outer) time step. These are very efficient schemes, provided there are only two clusters of eigenvalues, one corresponding to a single fast relaxation time scale, and one corresponding to the slow macroscopic dynamics. Here, we show how telescopic projective integration can be used to efficiently integrate kinetic equations with multiple relaxation times. Telescopic projective integration generalizes the idea of projective integration by constructing a hierarchy of projective levels. The main idea is to adjust the size of the inner time step at each level to one of the relaxation time scales. We show that the size of the outer time step, as well as the required number of inner steps at each level, does not depend on the stiffness of the problem. The computational cost of the method depends on the stiffness of the problem only via the number of projective levels. For problems with a fixed number of well-separated spectral clusters, the number of projective levels is independent of the stiffness, and the computational cost of telescopic projective integration is independent of the stiffness. For problems with a time-varying spectrum, the number of projective levels grows logarithmically with the stiffness. We illustrate numerically that, also in that case, the resulting computational cost is acceptable.     

An Entropy Stable <Emphasis Type="Italic">h</Emphasis> / <Emphasis Type="Italic">p</Emphasis> Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property

This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre–Gauss–Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between non-conforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general non-linear hyperbolic systems on h / p non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples.     

数值求解优化问题在活动轮廓模型上的应用

针对活动轮廓模型图像分割过程中迭代次数多,分割速度慢的问题,提出一种高阶的数值求解方法。分析活动轮廓模型中基于全局信息的CV模型,以及基于局部信息的LBF模型,LIF模型。使用二阶、三阶Runge-Kutta方法,原始Euler方法对模型进行数值求解实验对比分析。并对LBF模型中平滑项系数,时间步长的选取进行讨论。通过对非同质图像、同质图像的实验结果分析表明,所采用的数值方法能够有效地提高数值收敛精度、减少迭代次数、计算效率高。对不同系数和时间步长,数值方法也能表现出较好的稳定性。     

New Methods for High-Dimensional Verified Quadrature

Conventional verified methods for integration often rely on the verified bounding of analytically derived remainder formulas for popular integration rules. We show that using the approach of Taylor models, it is possible to devise new methods for verified integration of high order and in many variables. Different from conventional schemes, they do not require an a-priori derivation of analytical error bounds, but the rigorous bounds are calculated automatically in parallel to the computation of the integral.The performance of various schemes are compared for examples of up to order ten in up to eight variables. Computational expenses and tightness of the resulting bounds are compared with conventional methods.     

Explorations of a family of stochastic Newmark methods in engineering dynamics

A family of stochastic Newmark methods are explored for direct (path-wise or strong) integrations of stochastically driven dynamical systems of engineering interest. The stochastic excitations are assumed to be modeled by white noise processes or their filters and may be applied additively or multiplicatively. The family of stochastic Newmark maps are developed through a two-parameter, implicit Ito–Taylor expansion of the displacement and velocity vectors associated with the governing stochastic differential equations (SDE-s). Detailed estimates of local and global error orders for the response variables are provided in terms of the given time step size, h. While higher order Newmark methods lead to higher accuracies, far less random variables need to be modeled in the lower order methods to make it much more attractive from a computational point of view. For the specific case of a linear dynamical system, the stochastic Newmark map is used to obtain a closed form map for computing the temporal evolution of the response co-variance matrix. A host of numerical illustrations, covering linear and non-linear, single- and multi-degree-of-freedom dynamical systems, are provided to bring out the advantages and possible weaknesses of the methods proposed.