Showing papers on "Piecewise linear function published in 2007"

Piecewise linear regularized solution paths

Abstract: We consider the generic regularized optimization problem β(λ) = argminβ L(y, Xβ) + λJ(β). Efron, Hastie, Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407-499] have shown that for the LASSO-that is, if L is squared error loss and J(β) = ∥β∥ 1 is the l 1 norm of β-the optimal coefficient path is piecewise linear, that is, ∂β(λ)/∂λ. is piecewise constant. We derive a general characterization of the properties of (loss L, penalty J) pairs which give piecewise linear coefficient paths. Such pairs allow for efficient generation of the full regularized coefficient paths. We investigate the nature of efficient path following algorithms which arise. We use our results to suggest robust versions of the LASSO for regression and classification, and to develop new, efficient algorithms for existing problems in the literature, including Mammen and van de Geer's locally adaptive regression splines.

Optimality of a Standard Adaptive Finite Element Method

Abstract: In this paper an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever, for some s > 0, the solution can be approximated within a tolerance e > 0 in energy norm by a continuous piecewise linear function on some partition with O(e-1/s) triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate with piecewise constants, then the adaptive method produces approximations that converge with this rate, taking a number of operations that is of the order of the number of triangles in the output partition. The method is similar in spirit to that from [SINUM, 38 (2000), pp. 466-488] by Morin, Nochetto, and Siebert, and so in particular it does not rely on a recurrent coarsening of the partitions. Although the Poisson equation in two dimensions with piecewise linear approximation is considered, the results generalize in several respects.

Mixed finite element methods for linear elasticity with weakly imposed symmetry

Abstract: In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approxima- tions to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

Universal Piecewise Linear Prediction Via Context Trees

Abstract: This paper considers the problem of piecewise linear prediction from a competitive algorithm approach. In prior work, prediction algorithms have been developed that are "universal" with respect to the class of all linear predictors, such that they perform nearly as well, in terms of total squared prediction error, as the best linear predictor that is able to observe the entire sequence in advance. In this paper, we introduce the use of a "context tree," to compete against a doubly exponential number of piecewise linear (affine) models. We use the context tree to achieve the total squared prediction error performance of the best piecewise linear model that can choose both its partitioning of the regressor space and its real-valued prediction parameters within each region of the partition, based on observing the entire sequence in advance, uniformly, for every bounded individual sequence. This performance is achieved with a prediction algorithm whose complexity is only linear in the depth of the context tree per prediction. Upper bounds on the regret with respect to the best piecewise linear predictor are given for both the scalar and higher order case, and lower bounds on the regret are given for the scalar case. An explicit algorithmic description and examples demonstrating the performance of the algorithm are given.

Direct Search Approaches Using Genetic Algorithms for Optimization of Water Reservoir Operating Policies

Abstract: The direct search approach to determine optimal reservoir operating policies is proposed with a real coded genetic algorithm (GA) as the optimization method. The parameters of the policies are optimized using the objective values obtained from system simulations. Different reservoir release rules or forms, such as linear, piecewise linear, fuzzy rule base, and neural network, are applied to a single reservoir system and compared with conventional models such as stochastic dynamic programming and dynamic programming and regression. The results of historical and artificial time series simulations show that the GA models are generally superior in identifying better expected system performance. Parsimony of policy parameters is inferred as a principle for selecting the structure of the policy, and Fourier series can be helpful for reducing the number of parameters by defining the time variations of coefficients. The proposed method has shown to be flexible and robust in optimizing various types of policies, e...

A Piecewise-Linear Moment-Matching Approach to Parameterized Model-Order Reduction for Highly Nonlinear Systems

Abstract: This paper presents a parameterized reduction technique for highly nonlinear systems. In our approach, we first approximate the nonlinear system with a convex combination of parameterized linear models created by linearizing the nonlinear system at points along training trajectories. Each of these linear models is then projected using a moment-matching scheme into a low-order subspace, resulting in a parameterized reduced-order nonlinear system. Several options for selecting the linear models and constructing the projection matrix are presented and analyzed. In addition, we propose a training scheme which automatically selects parameter-space training points by approximating parameter sensitivities. Results and comparisons are presented for three examples which contain distributed strong nonlinearities: a diode transmission line, a microelectromechanical switch, and a pulse-narrowing nonlinear transmission line. In most cases, we are able to accurately capture the parameter dependence over the parameter ranges of plusmn50% from the nominal values and to achieve an average simulation speedup of about 10x.

Iterative Solution of Piecewise Linear Systems

Abstract: The correct formulation of numerical models for free-surface hydrodynamics often requires the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In so doing one may prevent the development of unrealistic negative water depths. The resulting piecewise linear systems are equivalent to particular linear complementarity problems whose solutions could be obtained by using, for example, interior point methods. These methods may have a favorable convergence property, but they are purely iterative and convergence to the exact solution is proven only in the limit of an infinite number of iterations. In the present paper a simple Newton-type procedure for certain piecewise linear systems is derived and discussed. This procedure is shown to have a finite termination property, i.e., it converges to the exact solution in a finite number of steps, and, actually, it converges very quickly, as confirmed by a few numerical tests.

Variable Disaggregation in Network Flow Problems with Piecewise Linear Costs

Abstract: We study mixed-integer programming formulations, based upon variable disaggregation, for generic multicommodity network flow problems with nonconvex piecewise linear costs, a problem class that arises frequently in many application domains in telecommunications, transportation, and logistics. We present several structural results for these formulations, and we analyze the results of extensive experiments on a large set of instances with various characteristics. In particular, we show that the linear programming relaxation of an extended disaggregated model approximates the objective function by its lower convex envelope in the space of commodity flows. Together, the theoretical and computational results allow us to suggest which formulation might be the most appropriate, depending on the characteristics of the problem instances.

Computational Aspects of Approximate Explicit Nonlinear Model Predictive Control

Abstract: It has recently been shown that the feedback solution to linear and quadratic constrained Model Predictive Control (MPC) problems has an explicit representation as a piecewise linear (PWL) state feedback. For nonlinear MPC the prospects of explicit solutions are even higher than for linear MPC, since the benefits of computational efficiency and verifiability are even more important. Preliminary studies on approximate explicit PWL solutions of convex nonlinear MPC problems, based on multi-parametric Nonlinear Programming (mp-NLP) ideas show that sub-optimal PWL controllers of practical complexity can indeed be computed off-line. However, for non-convex problems there is a need to investigate practical computational methods that not necessarily lead to guaranteed properties, but when combined with verification and analysis methods will give a practical tool for development and implementation of explicit NMPC. The present paper focuses on the development of such methods. As a case study, the application of the developed approaches to compressor surge control is considered.

Input-to-state stability and interconnections of discontinuous dynamical systems

Abstract: In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov's solution concept, that is appropriate for 'open' systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a piecewise smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.

Observer design for a class of piecewise linear systems

Abstract: SUMMARY In this paper we present observer design procedures for a class of bi-modal piecewise linear systems in both continuous and discrete time. We propose Luenberger-type observers, and derive sufficient conditions for the observation error dynamics to be globally asymptotically stable, in the case when the system dynamics is continuous over the switching plane. When the dynamics is discontinuous, we derive conditions that guarantee that the relative estimation error with respect to the state of the observed system will be asymptotically small. The presented theory is illustrated with several examples. Copyright # 2007 John Wiley & Sons, Ltd.

Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems

Abstract: We study the numerical approximation of distributed optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the L 2-error estimates are of order o(h), which is optimal according with the $C^{0,1}(\overline{\Omega})$ -regularity of the optimal control.

Penalized linear unbiased selection

Abstract: We introduce MC+, a fast, continuous, nearly unbiased, and accurate method of penalized variable selection in high-dimensional linear regression. The LASSO is fast and continuous, but biased. The bias of the LASSO interferes with variable selection. Subset selection is unbiased but computationally costly. The MC+ has two elements: a minimax concave penalty (MCP) and a penalized linear unbiased selection (PLUS) algorithm. The MCP provides the minimum non-convexity of the penalized loss given the level of bias. The PLUS computes multiple local minimizers of a possibly non-convex penalized loss function in certain main branch of the graph of such solutions. Its output is a continuous piecewise linear path encompassing from the origin to an optimal solution for zero penalty. We prove that for a universal penalty level, the MC+ has high probability of correct selection under much weaker conditions compared with existing results for the LASSO for large n and p, including the case of p ≫ n. We provide estimates of the noise level for proper choice of the penalty level. We choose the sparsest solution within the PLUS path for a given penalty level. We derive degrees of freedom and Cp-type risk estimates for general penalized LSE, including the LASSO estimator, and prove their unbiasedness. We provide necessary and sufficient conditions for the continuity of the penalized LSE under general sub-square penalties. Simulation results overwhelmingly support our claim of superior variable selection properties and demonstrate the computational efficiency of the proposed method.

Frequency Coupling Matrix of a Voltage-Source Converter Derived From Piecewise Linear Differential Equations

Abstract: When a power-electronic converter is introduced into a linear network, voltage and current harmonics of differing orders become coupled (through the modulation effect of the converter). The interharmonic coupling introduced by the modulation effect of a converter may be mathematically represented through a frequency coupling matrix (FCM). Given that the source of the coupling is a modulation process, researchers have, in the past, focused on deriving the FCM in the frequency domain - a process that requires the truncation of the harmonic representation of signals. This paper presents an alternate approach to evaluate the FCM based on a time-domain derivation. In contrast to frequency-domain-based methods, it is shown that the time-domain approach avoids truncation. Furthermore, the time-domain approach does not require system linearization about an operating point; thus, the FCM is not limited by small-signal assumptions.

Computing maximum likelihood estimators of a log-concave density function

Abstract: We consider the problem of estimating a density function that is assumed to be log-concave. This semi-parametric model includes many well-known parametric classes; such as Normal, Gamma, Laplace, Logistic, Beta or Extreme value distributions, for specific parameter ranges. It is known that the maximum likelihood estimator for the log-density is always a piecewise linear function with at most as many knots as observations, but typically much less. We show that this property can be exploited to design a linearly constrained optimization problem whose iteratively calculated solution yields the estimator. We compare several standard and one recently proposed algorithm regarding their performance on this problem.

Output Tracking of Piecewise-Linear Systems via Error Feedback Regulator With Application to Synchronization of Nonlinear Chua's Circuit

Abstract: This paper considers the output tracking problem of general piecewise discrete-time linear systems via error feedback scheme. A number of sufficient conditions are obtained based on piecewise-quadratic Lyapunov functions in the framework of output regulation theory. The resulting closed-loop system is guaranteed to be stable and the output tracking is achieved asymptotically. Moreover, the proposed piecewise-linear model based regulator has been successfully applied to the chaos synchronization of general nonlinear Chua's circuits. Simulation results are also given to illustrate the performance and advantages of the proposed approach.

Horseshoes near homoclinic orbits for piecewise linear differential systems in ℝ3

Abstract: For a three-parametric family of continuous piecewise linear differential systems introduced by Arneodo et al. [1981] and considering a situation which is reminiscent of the Hopf-Zero bifurcation, ...

Piecewise Linear Identification for the Rate-Independent and Rate-Dependent Duhem Hysteresis Models

Abstract: We consider the semilinear Duhem model and develop an identification method for rate-independent and rate-dependent hysteresis. For rate-independent hysteresis, we reparameterize the system in terms of the input signal, so that the system has the form of a switching linear time-invariant system with ramp-plus-step forcing. For rate-dependent hysteresis, the system can be viewed as a switching linear time-invariant system for triangle wave inputs. Least-squares-based methods are developed to identify the rate-independent and rate-dependent semilinear Duhem models

Wide-Angle Shift-Map PE for a Piecewise Linear Terrain—A Finite-Difference Approach

Abstract: A wide-angle parabolic wave equation solution is presented using shift-map and finite-difference techniques. The corresponding split-step Fourier solution is well known. The solution using finite-difference technique, where the standard parabolic wave equation is modified into the so-called Claerbout equation allowing propagation angles up to 45deg from the paraxial direction, is also well known. Here, we present an extension to that solution in which the shift-map technique is incorporated into the finite-difference scheme allowing a varying terrain to be considered. The result is a solution that corresponds to the well known split-step solution, which is believed to perform well for terrain slopes up to 10deg-15deg and discontinuous slope changes on the order of 15deg-20deg. This solution is a first-order one with respect to the terrain slope. However, when using the finite-difference technique, it is also possible to find a second-order solution with respect to the terrain slope. This new solution performs well for slopes up to about 15deg and discontinuous slope changes up to about 30deg, which is an improvement.

Accurate Computation of Geodesic Distance Fields for Polygonal Curves on Triangle Meshes

Abstract: We present an algorithm for the efficient and accurate computation of geodesic distance fields on triangle meshes. We generalize the algorithm originally proposed by Surazhsky et al. [1]. While the original algorithm is able to compute geodesic distances to isolated points on the mesh only, our generalization can handle arbitrary, possibly open, polygons on the mesh to define the zero set of the distance field. Our extensions integrate naturally into the base algorithm and consequently maintain all its nice properties. For most geometry processing algorithms, the exact geodesic distance information is sampled at the mesh vertices and the resulting piecewise linear interpolant is used as an approximation to the true distance field. The quality of this approximation strongly depends on the structure of the mesh and the location of the medial axis of the distance field. Hence our second contribution is a simple adaptive refinement scheme, which inserts new vertices at critical locations on the mesh such that the final piecewise linear interpolant is guaranteed to be a faithful approximation to the true geodesic distance field.

A Nonlinear Multigrid Method for Total Variation Minimization from Image Restoration

Abstract: Image restoration has been an active research topic and variational formulations are particularly effective in high quality recovery. Although there exist many modelling and theoretical results, available iterative solvers are not yet robust in solving such modeling equations. Recent attempts on developing optimisation multigrid methods have been based on first order conditions. Different from this idea, this paper proposes to use piecewise linear function spanned subspace correction to design a multilevel method for directly solving the total variation minimisation. Our method appears to be more robust than the primal-dual method (Chan et al., SIAM J. Sci. Comput. 20(6), 1964---1977, 1999) previously found reliable. Supporting numerical results are presented.

Optimal Control of Constrained Piecewise Affine Systems

Abstract: Background.- Mathematical Necessities.- Systems and Control Theory.- Receding Horizon Control.- Piecewise Affine Systems.- Optimal Control of Constrained Piecewise Affine Systems.- Constrained Finite Time Optimal Control.- Constrained Infinite Time Optimal Control.- Analysis and Post-Processing Techniques for Piecewise Affine Systems.- Linear Vector Norms as Lyapunov Functions.- Stability Analysis.- Stability Tubes.- Efficient Evaluation of Piecewise Control Laws Defined Over a Large Number of Polyhedra.

Stability and approximability of the 1 0 element for Stokes equations

Abstract: In this paper we study the stability and approximability of the 1–0 element (continuous piecewise linear for the velocity and piecewise constant for the pressure on triangles) for Stokes equations. Although this element is unstable for all meshes, it provides optimal approximations for the velocity and the pressure in many cases. We establish a relation between the stabilities of the 1–0 element (bilinear/constant on quadrilaterals) and the 1–0 element. We apply many stability results on the 1–0 element to the analysis of the 1–0 element. We prove that the element has the optimal order of approximations for the velocity and the pressure on a variety of mesh families. As a byproduct, we also obtain a basis of divergence-free piecewise linear functions on a mesh family on squares. Numerical tests are provided to support the theory and to show the efficiency of the newly discovered, truly divergence-free, 1 finite element spaces in computation. Copyright © 2006 John Wiley & Sons, Ltd.