An adaptive <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>IPG method for the Helmholtz transmission eigenvalue problem

The interior penalty methods using C⁰ Lagrange elements (C⁰IPG) developed in the recent decade for the fourth order problems are an interesting topic in academia at present. In this paper, we discuss the adaptive fashion of C⁰IPG method for the Helmholtz transmission eigenvalue problem. We give the a posteriori error indicators for primal and dual eigenfunctions, and prove their reliability and efficiency. We also give the a posteriori error indicator for eigenvalues and design a C⁰IPG adaptive algorithm. Numerical experiments show that this algorithm is efficient and can get the optimal convergence rate.     

Fast computation in adaptive tree approximation

Summary. Adaptive methods of approximation arise in many settings including numerical methods for PDEs and image processing. They can usually be described by a tree which records the adaptive decisions. This paper is concerned with the fast computation of near optimal trees based on n adaptive decisions. The best tree based on n adaptive decisions could be found by examining all such possibilities. However, this is exponential in n and could be numerically prohibitive. The main result of this paper is to show that it is possible to find near optimal trees using computations linear in n.Mathematics Subject Classification (2000): 65Y20, 65N50, 41A63, 41A15, 68W40, 68W25This work has been supported in part by the Office of Naval Research contracts 03-10051, (N00014-00-1-0470), the Army Research Office contract DAAD 19-02-1-0028, and the National Science Foundation grants DMS 9872890, DMS 0221642.     

Controlling chaos based on an adaptive adjustment mechanism

In this paper, we extend the ideas and techniques developed by Huang [Huang W. Stabilizing nonlinear dynamical systems by an adaptive adjustment mechanism. Phys Rev E 2000;61:R1012–5] for controlling discrete-time chaotic system using adaptive adjustment mechanism to continuous-time chaotic system. Two control approaches, namely adaptive adjustment mechanism (AAM) and modified adaptive adjustment mechanism (MAAM), are investigated. In both case sufficient conditions for the stabilization of chaotic systems are given analytically. The simulation results on Chen chaotic system have verified the effectiveness of the proposed techniques.     

Controlling chaos based on an adaptive adjustment mechanism

In this paper, we extend the ideas and techniques developed by Huang [Huang W. Stabilizing nonlinear dynamical systems by an adaptive adjustment mechanism. Phys Rev E 2000;61:R1012–5] for controlling discrete-time chaotic system using adaptive adjustment mechanism to continuous-time chaotic system. Two control approaches, namely adaptive adjustment mechanism (AAM) and modified adaptive adjustment mechanism (MAAM), are investigated. In both case sufficient conditions for the stabilization of chaotic systems are given analytically. The simulation results on Chen chaotic system have verified the effectiveness of the proposed techniques.     

An adaptive C~0IPG method for the Helmholtz transmission eigenvalue problem

The interior penalty methods using C~0 Lagrange elements(C~0 IPG) developed in the recent decade for the fourth order problems are an interesting topic at present. In this paper, we discuss the adaptive proporty of C~0 IPG method for the Helmholtz transmission eigenvalue problem. We give the a posteriori error indicators for primal and dual eigenfunctions, and prove their reliability and efficiency. We also give the a posteriori error indicator for eigenvalues and design a C~0 IPG adaptive algorithm. Numerical experiments show that this algorithm is efficient and can get the optimal convergence rate.     

Basis pursuit for frequency‐domain identification

W. Sprößig In this paper, we propose a new adaptive method for frequency‐domain identification problem of discrete LTI systems. It is based on a dictionary that is consisting of normalized reproducing kernels. We prove that the singular values of the matrix generated by this dictionary converge to zero rapidly; this makes it quite efficient in representing the original systems with only a few elements. For different systems, it results in different selected sequences from the dictionary, that is, its adaptivity. Meanwhile, the stability of results is automatically guaranteed according to the structure of the dictionary. Two examples are presented to illustrate the idea. Copyright © 2016 John Wiley & Sons, Ltd.     

Topology identification and adaptive synchronization of uncertain complex networks with adaptive double scaling functions

This paper discusses topology identification and adaptive synchronization of uncertain complex networks with adaptive the scaling functions. In comparison with those of the existing scaling function synchronization, the scaling function can be identified by adaptive laws in this paper. Moreover, the uncertain network topological structure are identified simultaneously in the process of synchronization. Illustrative examples are presented to demonstrate the application of the theoretical results.     

An adaptive estimation of MAVE

Minimum average variance estimation (MAVE, Xia et al. (2002) [29]) is an effective dimension reduction method. It requires no strong probabilistic assumptions on the predictors, and can consistently estimate the central mean subspace. It is applicable to a wide range of models, including time series. However, the least squares criterion used in MAVE will lose its efficiency when the error is not normally distributed. In this article, we propose an adaptive MAVE which can be adaptive to different error distributions. We show that the proposed estimate has the same convergence rate as the original MAVE. An EM algorithm is proposed to implement the new adaptive MAVE. Using both simulation studies and a real data analysis, we demonstrate the superior finite sample performance of the proposed approach over the existing least squares based MAVE when the error distribution is non-normal and the comparable performance when the error is normal.     

Developing two efficient adaptive Newton‐type methods with memory

In this paper, we derive two general adaptive methods with memory in the class of Newton‐type methods by modifying and introducing one and two self accelerators over a variant of Ostrowski's method. The idea of introducing adaptive self‐accelerator (via all the old information for Newton‐type methods) is new and efficient in order to obtain a higher high efficiency index. Finally, we provide convergence analysis and numerical implementations to show the feasibility and applicability of the proposed methods.     

Mapping complexity/Human knowledge as a complex adaptive system

Cartography is the art of map‐making that integrates science, technology, and visual aesthetics for the purpose of rendering the domain of interest, navigable. The science could aid the cartographer if it were to inform about the underlying process. Thus, Mendeleev's periodic table was informed by insights about the atomic mass periodicity. Likewise, Harvey's work on the circulatory system map was informed by his theoretical insights on Galen's errors. Mapping of human knowledge dates back at least to Porphyry who laid out the first tree‐of‐knowledge. Modern knowledge‐cartographers use a wide array of scientometric techniques capable of rendering appealing visuals of massive scientific corpuses. But what has perhaps been lacking is a sound theoretical basis for rendering legible the adaptive dynamics of knowledge creation and accumulation. Proposed is a theoretical framework, knowledge as a complex adaptive system (CAS) patterned on Holland's work on CAS, as well as the view that knowledge is a hierarchically heterarchic dynamical system. As a first leg in the conjoining experimental phase, we extract terms from approximately 1400 complexity science papers published at the Santa Fe Institute, deduce the topic distribution using Latent Dirichlet Allocation, capture the underlying dynamics, and show how to navigate the corpus visually. © 2016 Wiley Periodicals, Inc. Complexity 21: 207–234, 2016     

Two‐dimensional adaptive Fourier decomposition

One‐dimensional adaptive Fourier decomposition, abbreviated as 1‐D AFD, or AFD, is an adaptive representation of a physically realizable signal into a linear combination of parameterized Szegö and higher‐order Szegö kernels of the context. In the present paper, we study multi‐dimensional AFDs based on multivariate complex Hardy spaces theory. We proceed with two approaches of which one uses Product‐TM Systems; and the other uses Product‐Szegö Dictionaries. With the Product‐TM Systems approach, we prove that at each selection of a pair of parameters, the maximal energy may be attained, and, accordingly, we prove the convergence. With the Product‐Szegö dictionary approach, we show that pure greedy algorithm is applicable. We next introduce a new type of greedy algorithm, called Pre‐orthogonal Greedy Algorithm (P‐OGA). We prove its convergence and convergence rate estimation, allowing a weak‐type version of P‐OGA as well. The convergence rate estimation of the proposed P‐OGA evidences its advantage over orthogonal greedy algorithm (OGA). In the last part, we analyze P‐OGA in depth and introduce the concept P‐OGA‐Induced Complete Dictionary, abbreviated as Complete Dictionary. We show that with the Complete Dictionary P‐OGA is applicable to the Hardy H² space on 2‐torus. Copyright © 2016 John Wiley & Sons, Ltd.     

An adaptive robust controller for time delay maglev transportation systems

For engineering systems, uncertainties and time delays are two important issues that must be considered in control design. Uncertainties are often encountered in various dynamical systems due to modeling errors, measurement noises, linearization and approximations. Time delays have always been among the most difficult problems encountered in process control. In practical applications of feedback control, time delay arises frequently and can severely degrade closed-loop system performance and in some cases, drives the system to instability. Therefore, stability analysis and controller synthesis for uncertain nonlinear time-delay systems are important both in theory and in practice and many analytical techniques have been developed using delay-dependent Lyapunov function. In the past decade the magnetic and levitation (maglev) transportation system as a new system with high functionality has been the focus of numerous studies. However, maglev transportation systems are highly nonlinear and thus designing controller for those are challenging. The main topic of this paper is to design an adaptive robust controller for maglev transportation systems with time-delay, parametric uncertainties and external disturbances. In this paper, an adaptive robust control (ARC) is designed for this purpose. It should be noted that the adaptive gain is derived from Lyapunov–Krasovskii synthesis method, therefore asymptotic stability is guaranteed.     

Synchronization of two hyperchaotic systems via adaptive control

This letter presents chaos synchronization problem of two different hyperchaotic systems when the parameters of drive and response systems are fully unknown or uncertain. Based on Lyapunov stability theory, an adaptive control law and a parameter update rule for unknown parameters are derived such that two different high dimensional chaotic systems are to be synchronized. Hyperchaotic Chen system and Second-harmonic generation (SHG) system are taken as an illustrative example to show the effectiveness of the proposed method.     

An hp‐adaptive Newton‐Galerkin finite element procedure for semilinear boundary value problems

In this paper, we develop an hp‐adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction‐type adaptive Newton method and an hp‐version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully hp‐adaptive Newton–Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Matrices that commute with their derivative. On a letter from Schur to Wielandt

We examine when a matrix whose elements are differentiable functions in one variable commutes with its derivative. This problem was discussed in a letter from Schur to Wielandt written in 1934, which we found in Wielandt’s Nachlass. We present this letter and its translation into English. The topic was rediscovered later and partial results were proved. However, there are many subtle observations in Schur’s letter which were not obtained in later years. Using an algebraic setting, we put these into perspective and extend them in several directions. We present in detail the relationship between several conditions mentioned in Schur’s letter and we focus in particular on the characterization of matrices called Type 1 by Schur. We also present several examples that demonstrate Schur’s observations.     

Forecasting models: a comparison of several adaptive forecasting procedures

The object of the present investigation is to study and compare several adaptive forecasting methods. The present study consists of two parts. The adaptive forecasting models were selected under strict constraints on storage complexity. Included in the present study is the identification of the most robust and efficient adaptive forecasting procedures. The first part of the study consists of four methods which can be divided into three general classes: exponential smoothing, parameter switching and Kalman filtering

The results presented in this paper and the sequel, Part II, extend and correct the findings given by Bunn, [1]. The published results included several errors , including incorrect formulae, imprecise specification of initial conditions, and use of inadequate sample size. The present study considers data simultaneous from a very large set of underlying processes, including several for which exponential smoothing methods are not even quasi-optimal, in order to enhance the utility of the results in selecting suitable methods for forecasting

Part II of the present evaluations considers six additional forecasting procedures, including three which contain explicit corrections for first order autoregressive effects     

Identification and adaptive control: An application to flight control systems

This paper presents the first results obtained in the application of stochastic control theory to flight control problems. It involves identification and adaptive control of an aircraft which operates over a wide range of environmental conditions that affect its dynamic characteristics. The bulk of the paper deals with theidentification problem of estimating stability derivatives in the presence of turbulence. Simulation results are presented both for identification and control (windgust alleviation and desired response to pilot input). While no practical implementation is reported, the implications for such implementation appear to be promising.The author is indebted to Mr. K. Iliff for all the simulation results using the IBM 360-91 at UCLA. The many helpful discussions with Dr. H. Rediess are gratefully acknowledged. Thanks are also due to Mr. Jiri Ruzicka for lending a helping hand throughout.     

Algebraic modular forms

In this paper, we develop an algebraic theory of modular forms, for connected, reductive groupsG overQ with the property that every arithmetic subgroup Γ ofG(Q) is finite. This theory includes our previous work [15] on semi-simple groupsG withG(R) compact, as well as the theory of algebraic Hecke characters for Serre tori [20]. The theory of algebraic modular forms leads to a workable theory of modular forms (modp), which we hope can be used to parameterize odd modular Galois representations. The theory developed here was inspired by a letter of Serre to Tate in 1987, which has appeared recently [21]. I want to thank Serre for sending me a copy of this letter, and for many helpful discussions on the topic.     

Priors for Bayesian adaptive spline smoothing

Adaptive smoothing has been proposed for curve-fitting problems where the underlying function is spatially inhomogeneous. Two Bayesian adaptive smoothing models, Bayesian adaptive smoothing splines on a lattice and Bayesian adaptive P-splines, are studied in this paper. Estimation is fully Bayesian and carried out by efficient Gibbs sampling. Choice of prior is critical in any Bayesian non-parametric regression method. We use objective priors on the first level parameters where feasible, specifically independent Jeffreys priors (right Haar priors) on the implied base linear model and error variance, and we derive sufficient conditions on higher level components to ensure that the posterior is proper. Through simulation, we demonstrate that the common practice of approximating improper priors by proper but diffuse priors may lead to invalid inference, and we show how appropriate choices of proper but only weakly informative priors yields satisfactory inference.     

Hesitant adaptive search: the distribution of the number of iterations to convergence

Hesitant adaptive search is a stochastic optimisation procedure which accommodates hesitation, or pausing, at objective function values. It lies between pure adaptive search (which strictly improves at each iteration) and simulated annealing with constant temperature (which allows backtracking, or the acceptance of worse function values). In this paper we build on an earlier work and make two contributions; first, we link hesitant adaptive search to standard counting process theory, and second, we use this to derive the exact distribution of the number of iterations of hesitant adaptive search to termination. Received: November 17, 1997 / Accepted: July 9, 1999?Published online December 15, 2000