Showing papers on "Piecewise published in 2013"

On Event Triggered Tracking for Nonlinear Systems

Abstract: In this technical note, we study an event-based control algorithm for trajectory tracking in nonlinear systems. The desired trajectory is modelled as the solution of a reference system with an exogenous input and it is assumed that the desired trajectory and the exogenous input to the reference system are uniformly bounded. Given a continuous-time control law that guarantees global uniform asymptotic tracking of the desired trajectory, our algorithm provides an event-based controller that not only guarantees uniform ultimate boundedness of the tracking error, but also ensures non-accumulation of inter-execution times. In the case that the derivative of the exogenous input to the reference system is also uniformly bounded, an arbitrarily small ultimate bound can be designed. If the exogenous input to the reference system is piecewise continuous and not differentiable everywhere then the achievable ultimate bound is constrained and the result is local, though with a known region of attraction. The main ideas in the technical note are illustrated through simulations of trajectory tracking by a nonlinear system.

Adaptive piecewise polynomial estimation via trend filtering

Abstract: We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339-360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute $k$th order discrete derivatives over the input points. Perhaps not surprisingly, trend filtering estimates appear to have the structure of $k$th degree spline functions, with adaptively chosen knot points (we say ``appear'' here as trend filtering estimates are not really functions over continuous domains, and are only defined over the discrete set of inputs). This brings to mind comparisons to other nonparametric regression tools that also produce adaptive splines; in particular, we compare trend filtering to smoothing splines, which penalize the sum of squared derivatives across input points, and to locally adaptive regression splines [Ann. Statist. 25 (1997) 387-413], which penalize the total variation of the $k$th derivative. Empirically, we discover that trend filtering estimates adapt to the local level of smoothness much better than smoothing splines, and further, they exhibit a remarkable similarity to locally adaptive regression splines. We also provide theoretical support for these empirical findings; most notably, we prove that (with the right choice of tuning parameter) the trend filtering estimate converges to the true underlying function at the minimax rate for functions whose $k$th derivative is of bounded variation. This is done via an asymptotic pairing of trend filtering and locally adaptive regression splines, which have already been shown to converge at the minimax rate [Ann. Statist. 25 (1997) 387-413]. At the core of this argument is a new result tying together the fitted values of two lasso problems that share the same outcome vector, but have different predictor matrices.

On Event Triggered Tracking for Nonlinear Systems

Abstract: In this paper we study an event based control algorithm for trajectory tracking in nonlinear systems. The desired trajectory is modelled as the solution of a reference system with an exogenous input and it is assumed that the desired trajectory and the exogenous input to the reference system are uniformly bounded. Given a continuous-time control law that guarantees global uniform asymptotic tracking of the desired trajectory, our algorithm provides an event based controller that not only guarantees uniform ultimate boundedness of the tracking error, but also ensures non-accumulation of inter-execution times. In the case that the derivative of the exogenous input to the reference system is also uniformly bounded, an arbitrarily small ultimate bound can be designed. If the exogenous input to the reference system is piecewise continuous and not differentiable everywhere then the achievable ultimate bound is constrained and the result is local, though with a known region of attraction. The main ideas in the paper are illustrated through simulations of trajectory tracking by a nonlinear system.

Robust State-Dependent Switching of Linear Systems With Dwell Time

Abstract: A state-dependent switching law that obeys a dwell time constraint and guarantees the stability of a switched linear system is designed. Sufficient conditions are obtained for the stability of the switched systems when the switching law is applied in presence of polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, is assigned to each subsystem such that it is non-increasing at the switching instants. During the dwell time, this function varies piecewise linearly in time. After the dwell, the system switches if the switching results in a decrease in the value of the LF. The method proposed is also applicable to robust stabilization via state-feedback. It is further extended to guarantee a bound on the L2-gain of the switching system; it is also used in deriving state-feedback control law that robustly achieves a prescribed L2 -gain bound.

Reinitialization-Free Level Set Evolution via Reaction Diffusion

Abstract: This paper presents a novel reaction-diffusion (RD) method for implicit active contours that is completely free of the costly reinitialization procedure in level set evolution (LSE). A diffusion term is introduced into LSE, resulting in an RD-LSE equation, from which a piecewise constant solution can be derived. In order to obtain a stable numerical solution from the RD-based LSE, we propose a two-step splitting method to iteratively solve the RD-LSE equation, where we first iterate the LSE equation, then solve the diffusion equation. The second step regularizes the level set function obtained in the first step to ensure stability, and thus the complex and costly reinitialization procedure is completely eliminated from LSE. By successfully applying diffusion to LSE, the RD-LSE model is stable by means of the simple finite difference method, which is very easy to implement. The proposed RD method can be generalized to solve the LSE for both variational level set method and partial differential equation-based level set method. The RD-LSE method shows very good performance on boundary antileakage. The extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed RD-LSE approach.

Martingale Optimal Transport and Robust Hedging in Continuous Time

Abstract: The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fi xed maturity. The dual is a Monge-Kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that has the given marginal at maturity. In addition to duality, a family of simple, piecewise constant super-replication portfolios that asymptotically achieve the minimal super-replication cost is constructed.

Nonsynchronized Robust Filtering Design for Continuous-Time T–S Fuzzy Affine Dynamic Systems Based on Piecewise Lyapunov Functions

Abstract: This paper investigates the problem of robust H∞ state estimation for a class of continuous-time nonlinear systems via Takagi-Sugeno (T-S) fuzzy affine dynamic models. Attention is focused on the analysis and design of an admissible full-order filter such that the resulting filtering error system is asymptotically stable with a guaranteed H∞ disturbance attenuation level. It is assumed that the plant premise variables, which are often the state variables or their functions, are not measurable so that the filter implementation with state-space partition may not be synchronous with the state trajectories of the plant. Based on piecewise quadratic Lyapunov functions combined with S-procedure and some matrix inequality linearization techniques, some new results are presented for the filtering design of the underlying continuous-time T-S fuzzy affine systems. Illustrative examples are given to validate the effectiveness and application of the proposed design approaches.

Finite element analysis for a coupled bulk-surface partial differential equation

Abstract: In this paper, we define a new finite element method for numerically approximating the solution of a partial differential equation in a bulk region coupled with a surface partial differential equation posed on the boundary of the bulk domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface. Two finite element spaces are defined, one in the bulk region and one on the surface, by taking the set of all continuous functions which are also piecewise polynomial on each bulk simplex or boundary face. We study this method in the context of a model elliptic problem; in particular, we look at well-posedness of the system using a variational formulation, derive perturbation estimates arising from domain approximation and apply these to find the optimal-order error estimates. A numerical experiment is described which demonstrates the order of convergence.

Global optimization of bilinear programs with a multiparametric disaggregation technique

Abstract: In this paper, we present the derivation of the multiparametric disaggregation technique (MDT) by Teles et al. (J. Glob. Optim., 2011) for solving nonconvex bilinear programs. Both upper and lower bounding formulations corresponding to mixed-integer linear programs are derived using disjunctive programming and exact linearizations, and incorporated into two global optimization algorithms that are used to solve bilinear programming problems. The relaxation derived using the MDT is shown to scale much more favorably than the relaxation that relies on piecewise McCormick envelopes, yielding smaller mixed-integer problems and faster solution times for similar optimality gaps. The proposed relaxation also compares well with general global optimization solvers on large problems.

Stabilization of a Class of Switched Linear Neutral Systems Under Asynchronous Switching

Abstract: This technical note concerns the stabilization problem for a class of switched linear neutral systems in which time delays appear in both the state and the state derivatives. In addition, the switching signal of the switched controller also involves time delays, which makes the switching between the controller and the system asynchronous. Based on a new integral inequality and the piecewise Lyapunov-Krasovskii functional technique, a condition for global uniform exponential stability of the switched neutral system under an average dwell time (ADT) scheme is proposed. Then, the corresponding solvability condition for the controller is established. Finally, a numerical example is given to illustrate the effectiveness of the proposed theory.

Dynamics of an infectious diseases with media/psychology induced non-smooth incidence.

Abstract: This paper proposes and analyzes a mathematical model on an infectious disease system with a piecewise smooth incidence rate concerning media/psychological effect. The proposed models extend the classic models with media coverage by including a piecewise smooth incidence rate to represent that the reduction factor because of media coverage depends on both the number of cases and the rate of changes in case number. On the basis of properties of Lambert W function the implicitly defined model has been converted into a piecewise smooth system with explicit definition, and the global dynamic behavior is theoretically examined. The disease-free is globally asymptotically stable when a certain threshold is less than unity, while the endemic equilibrium is globally asymptotically stable for otherwise. The media/psychological impact although does not affect the epidemic threshold, delays the epidemic peak and results in a lower size of outbreak (or equilibrium level of infected individuals).

A Weak Galerkin Finite Element Method with Polynomial Reduction

Abstract: The novel idea of weak Galerkin (WG) finite element methods is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different weak Galerkin finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. For illustrative purpose, the authors use second order elliptic problems to demonstrate the basic idea of polynomial reduction. A new weak Galerkin finite element method is proposed and analyzed. This new finite element scheme features piecewise polynomials of degree $k\ge 1$ on each element plus piecewise polynomials of degree $k-1\ge 0$ on the edge or face of each element. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^1$ norm and the standard $L^2$ norm. In addition, the paper presents a great deal of numerical experiments to demonstrate the power of the WG method in dealing with finite element partitions consisting of arbitrary polygons in two dimensional spaces or polyhedra in three dimensional spaces. The numerical examples include various finite element partitions such as triangular mesh, quadrilateral mesh, honey comb mesh in 2d and mesh with deformed cubes in 3d. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.

Piecewise linearity of approximate density functionals revisited: implications for frontier orbital energies.

Abstract: In the exact Kohn-Sham density-functional theory, the total energy versus the number of electrons is a series of linear segments between integer points. However, commonly used approximate density functionals produce total energies that do not exhibit this piecewise-linear behavior. As a result, the ionization potential theorem, equating the highest occupied eigenvalue with the ionization potential, is grossly disobeyed. Here, we show that, contrary to conventional wisdom, most of the required piecewise linearity of an arbitrary approximate density functional can be restored by careful consideration of the ensemble generalization of density-functional theory. Furthermore, the resulting formulation introduces the desired derivative discontinuity to any approximate exchange-correlation functional, even one that is explicitly density dependent. This opens the door to calculations of the ionization potential and electron affinity, even without explicit electron removal or addition. All these advances are achieved while neither introducing empiricism nor changing the underlying functional form. The power of the approach is demonstrated on benchmark systems using the local density approximation as an illustrative example.

Dynamical Steps that Bridge Piecewise Adiabatic Shapes in the Exact Time-Dependent Potential Energy Surface

Abstract: We study the exact time-dependent potential energy surface (TDPES) in the presence of strong nonadiabatic coupling between the electronic and nuclear motion. The concept of the TDPES emerges from the exact factorization of the full electron-nuclear wave function [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)]. Employing a one-dimensional model system, we show that the TDPES exhibits a dynamical step that bridges between piecewise adiabatic shapes. We analytically investigate the position of the steps and the nature of the switching between the adiabatic pieces of the TDPES.

Global error bounds for piecewise convex polynomials

Abstract: In this paper, by examining the recession properties of convex polynomials, we provide a necessary and sufficient condition for a piecewise convex polynomial to have a Holder-type global error bound with an explicit Holder exponent. Our result extends the corresponding results of Li (SIAM J Control Optim 33(5):1510–1529, 1995) from piecewise convex quadratic functions to piecewise convex polynomials.

Groups of piecewise projective homeomorphisms

Abstract: The group of piecewise projective homeomorphisms of the line provides straightforward torsion-free counterexamples to the so-called von Neumann conjecture. The examples are so simple that many additional properties can be established.

Exponential ergodicity for Markov processes with random switching

Abstract: We study a Markov process with two components: the first component evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of the second component. The second component is discrete and its jump rates may depend on the position of the whole process. Under regularity assumptions on the jump rates and Wasserstein contraction conditions for the underlying dynamics, we provide a concrete criterion for the convergence to equilibrium in terms of Wasserstein distance. The proof is based on a coupling argument and a weak form of the Harris theorem. In particular, we obtain exponential ergodicity in situations which do not verify any hypoellipticity assumption, but are not uniformly contracting either. We also obtain a bound in total variation distance under a suitable regularising assumption. Some examples are given to illustrate our result, including a class of piecewise deterministic Markov processes.

Solving integral equations on piecewise smooth boundaries using the RCIP method: a tutorial

Abstract: Recursively compressed inverse preconditioning (RCIP) is a numerical method for obtaining highly accurate solutions to integral equations on piecewise smooth surfaces. The method originated in 2008 as a technique within a scheme for solving Laplace’s equation in two-dimensional domains with corners. In a series of subsequent papers, the technique was then refined and extended as to apply to integral equation formulations of a broad range of boundary value problems in physics and engineering. The purpose of the present paper is threefold: first, to review the RCIP method in a simple setting; second, to show how easily the method can be implemented in MATLAB; third, to present new applications of RCIP to integral equations of scattering theory on planar curves with corners.

Capacities, Measurable Selection and Dynamic Programming Part II: Application in Stochastic Control Problems

Abstract: We aim to give an overview on how to derive the dynamic programming principle for a general stochastic control/stopping problem, using measurable selection techniques. By considering their martingale problem formulation, we show how to check the required measurability conditions for different versions of control/stopping problem, including in particular the controlled/stopped diffusion processes problem. Further, we also study the stability of the control problem, i.e. the approximation of the control process by piecewise constant control problems. It induces in particular an equivalence result for different versions of the controlled/stopped diffusion processes problem.

Efficient Density Estimation via Piecewise Polynomial Approximation

Abstract: We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is $\tau$-close (in total variation distance) to an unknown probability distribution $q$ that is defined by an unknown partition of $I$ into $t$ intervals and $t$ unknown degree-$d$ polynomials specifying $q$ over each of the intervals. We give an algorithm that draws $\tilde{O}(t ew{(d+1)}/\eps^2)$ samples from $p$, runs in time $\poly(t,d,1/\eps)$, and with high probability outputs a piecewise polynomial hypothesis distribution $h$ that is $(O(\tau)+\eps)$-close (in total variation distance) to $p$. This sample complexity is essentially optimal; we show that even for $\tau=0$, any algorithm that learns an unknown $t$-piecewise degree-$d$ probability distribution over $I$ to accuracy $\eps$ must use $\Omega({\frac {t(d+1)} {\poly(1 + \log(d+1))}} \cdot {\frac 1 {\eps^2}})$ samples from the distribution, regardless of its running time. Our algorithm combines tools from approximation theory, uniform convergence, linear programming, and dynamic programming. We apply this general algorithm to obtain a wide range of results for many natural problems in density estimation over both continuous and discrete domains. These include state-of-the-art results for learning mixtures of log-concave distributions; mixtures of $t$-modal distributions; mixtures of Monotone Hazard Rate distributions; mixtures of Poisson Binomial Distributions; mixtures of Gaussians; and mixtures of $k$-monotone densities. Our general technique yields computationally efficient algorithms for all these problems, in many cases with provably optimal sample complexities (up to logarithmic factors) in all parameters.

On stable piecewise linearization and generalized algorithmic differentiation

Abstract: It is shown how functions that are defined by evaluation programs involving the absolute value function abs besides smooth elementals can be approximated locally by piecewise-linear models in the style of algorithmic or automatic differentiation AD. The model can be generated by a minor modification of standard AD tools and it is Lipschitz continuous with respect to the base point at which it is developed. The discrepancy between the original function, which is piecewise differentiable, and the piecewise linear model is of second order in the distance to the base point. Consequently, successive piecewise linearization yields bundle type methods for unconstrained minimization and Newton-type equation solvers. As a third fundamental numerical task we consider the integration of ordinary differential equations, for which we examine generalizations of the midpoint and the trapezoidal rule for the case of Lipschitz continuous right hand sides RHSs. As a by-product of piecewise linearization, we show how to compute at any base point some generalized Jacobians of the original function, namely those that are conically active as defined by Khan and Barton. This subset of the Clarke Jacobian is never empty, independent of the particular function representation in terms of elementals, and also invariant with respect to linear transformations on domain and range. However, like all generalized derivatives the conically active Jacobians reduce almost everywhere to the singleton formed by the proper Jacobian, which may approximate the original function only in a minuscule neighbourhood. Since the piecewise linearization always reflects kinks in the vicinity, we illustrate how it can be used to approximate generalized Jacobians at nearby points along a user specified preferred direction.

New Stability Conditions Based on Piecewise Fuzzy Lyapunov Functions and Tensor Product Transformations

Abstract: Improvements of recent stability conditions for continuous-time Takagi-Sugeno (T-S) fuzzy systems are proposed. The key idea is to bring together the so-called local transformations of membership functions and new piecewise fuzzy Lyapunov functions. By relying on these special local transformations, the associated linear matrix inequalities that are used to prove the system's stability can be relaxed without increasing the number of conditions. In addition, to enhance the usefulness of the proposed methodology, one can choose between two different sets of conditions characterized by independence or dependence on known bounds of the membership functions time derivatives. A standard example is presented to illustrate that the proposed method is able to provide substantial improvements in some cases.