Nonsmooth Point Research Articles

Identification of piecewise linear dynamical systems using physically-interpretable neural-fuzzy networks: Methods and applications to origami structures

Self-locking origami structures are characterized by their piecewise linear constitutive relations between force and deformation, which, in practice, are always completely opaque and unmeasurable: the number of piecewise segments, the positions of non-smooth points, and the linear parameters of each segment are unknown a priori. However, acquiring this information is of fundamental importance for understanding the origami structure’s dynamic folding process and predicting its dynamic behaviors. This, therefore, arouses our interest in adopting a dynamical identification process to determine the model and to estimate the parameters. In this research, based on the piecewise linear assumption, a physically-interpretable neural-fuzzy network is built to correlate the measured input and output data. Unlike the conventional approaches, the constructed neural network possesses specific physical meaning of its components: the number of neurons relates to the number of piecewise segments, the coefficients of the local linear models relate to the parameters of the constitutive relations, and the validity functions relate to the positions of non-smooth points. By addressing several examples with different backgrounds, the network’s underlying data training methods are illustrated, including the local linear optimization for linear parameters, nested optimization for nonlinear partitions, and Local Linear Model Tree optimization for model selection. Noting that the tackled origami problem holds strong universality in terms of the unknown piecewise characteristics, the proposed approach would thus provide an effective, generic, and physically significant means for handling piecewise linear dynamical systems and meanwhile bring fresh vitality to the artificial neural network research.

The locally Chen–Harker–Kanzow–Smale smoothing functions for mixed complementarity problems

According to the structure of the projection function onto the box set $$\varPi _X$$ and the Chen–Harker–Kanzow–Smale (CHKS) smoothing function, a new class of smoothing projection functions onto the box set are proposed in this paper. The new smoothing projection functions only smooth $$\varPi _X$$ in neighborhoods of nonsmooth points of $$\varPi _X$$ , and keep unchanged with $$\varPi _X$$ at other points, hence they are referred as the locally Chen–Harker–Kanzow–Smale (LCHKS) smoothing functions. Based on the Robinson’s normal equation and the LCHKS smoothing functions, a smoothing Newton method with its convergence results is proposed for solving mixed complementarity problems. Compared with smoothing Newton methods based on various smoothing projection functions, the computations of the LCHKS smoothing function, the function value and its Jacobian matrix of the Newton equation become cheaper, and the Newton direction can be found by solving a low dimensional linear equation, hence the smoothing Newton method based on the LCHKS smoothing functions shows more efficient for large-scale mixed complementarity problems. The LCHKS smoothing functions are proved to be feasible, continuously differentiable, uniform approximations of $$\varPi _X$$ , globally Lipschitz continuous and strongly semismooth, which are important to establish the superlinear and quadratic convergence of the smoothing Newton method. The proposed smoothing Newton method is implemented in MATLAB and numerical tests are done on the MCPLIB test collection. Numerical results show that the smoothing Newton method based on the LCHKS smoothing functions is promising for mixed complementarity problems.

Stratification of free boundary points for a two-phase variational problem

In this paper we prove the local Lipschitz regularity of the minimizers of the two-phase Bernoulli type free boundary problem arising from the minimization of the functionalJ(u):=∫Ω|∇u|p+λ+pχ{u>0}+λ−pχ{u≤0},1<p<∞. Here Ω⊂RN is a bounded smooth domain and λ± are positive constants such that λ+p−λ−p>0. Furthermore, we show that for p>1 the free boundary has locally finite perimeter and the set of non-smooth points of the free boundary is of zero (N−1)-dimensional Hausdorff measure. For this, our approach is new even for the classical case p=2.

Positivity-preserving sixth-order implicit finite difference weighted essentially non-oscillatory scheme for the nonlinear heat equation

This paper presents a class of semi-implicit finite difference weighted essentially non-oscillatory (WENO) schemes for solving the nonlinear heat equation. For the discretization of second-order spatial derivatives, a sixth-order modified WENO scheme is directly implemented. This scheme preserves the positivity principle and rejects spurious oscillations close to non-smooth points. In order to admit large time steps, a class of implicit Runge–Kutta methods is used for the temporal discretization. The implicit parts of these methods are linearized in time by using the local Taylor expansion of the flux. The stability analysis of the semi-implicit WENO scheme with 3-stages form is provided. Finally, some comparative results for one-, two- and three-dimensional PDEs are included to illustrate the effectiveness of the proposed approach.

Nonsmooth Variants of Powell's BFGS Convergence Theorem

The popular BFGS quasi-Newton minimization algorithm under reasonable conditions converges globally on smooth convex functions. This result was proved by Powell in a landmark 1976 paper: we consider its implications for functions that are not smooth. In particular, an analogous convergence result holds for functions (like the Euclidean norm) whose minimizers are isolated nonsmooth points.

Optimality Conditions for Long-Run Average Rewards With Underselectivity and Nonsmooth Features

We study three existing issues associated with optimization of long-run average rewards of time-nonhomogeneous Markov processes in continuous time with continuous state spaces: 1) the underselectivity, i.e., the optimal policies do not depend on their actions in any finite time period; 2) its related issue, the bias optimality, i.e., policies that optimize both long-run average and transient total rewards, and 3) the effects of a nonsmooth point of a value function on performance optimization. These issues require considerations of the performance in the entire period with an infinite horizon, and therefore are not easily solvable by dynamic programming, which works backwards in time and takes a local view at a particular time instant. In this paper, we take a different approach called the relative optimization theory , which is based on a direct comparison of the performance measures of any two policies. We derive tight necessary and sufficient optimality conditions that take the underselectivity into consideration; we derive bias optimality conditions for both long-run average and transient rewards; and we show that the effect of a wide class of nonsmooth points, called semismooth points, of a value function on the long-run average performance is zero and can be ignored.

Single and multiple crack localization in beam-like structures using a Gaussian process regression approach

A crack or a localized damage in a structure provokes a discontinuity in the rotation. Consequently, mode-shapes are nonsmooth at the damage position and the first derivative (strictly related to the rotation) presents a jump discontinuity. Based on this simple concept, a new approach has been developed in order to predict the location of the mode-shape derivative discontinuities, and therefore the location of damage, without the need to directly differentiate. This approach applies a Gaussian process regression to the mode-shape data, using a covariance function which allows for a nonsmooth point; this point, which indicates the crack position, can be determined by a maximum likelihood algorithm. Using a finite-element model of a cracked beam, the performance of this methodology has been analyzed for the case of single crack and multiple cracks, for increasing amounts of noise. The effects of several parameters (damage position, damage severity, number of measurement points and of considered mode-shapes) on the approach accuracy have also been investigated. Finally, the method has been verified using experimental data coming from a vibrating steel beam with a cut. Results are encouraging and indicate that further developments of the technique for nondestructive testing of beam-like structures would be highly worthwhile.

Geometric design and performance analysis of a novel smooth rotor profile of claw vacuum pumps

In view of the problems that the existing rotor profiles of claw vacuum pumps contain non-smooth connection points, a novel smooth rotor profile was proposed in this paper. The proposed profile, which can achieve correct meshing, is composed of 6 circular arcs and 3 equidistant curves of epicycloid, and its design procedures as well as equations were given. The effects of geometric parameters on the shape and performances of the claw rotor were analyzed. Besides, 3D numerical simulations of the entire working process of the claw vacuum pump with the proposed rotor profile were performed, and its performances were analyzed. The study results indicate that the proposed rotor profile of claw vacuum pumps, compared with the existing rotor profiles, has remarkable advantages of better meshing performance, smaller carryover and better mechanical properties. Thus, the new rotor profile designed and the data obtained can be applied to the design and optimization of claw vacuum pumps.

High order Runge–Kutta methods for impulsive delay differential equations

The purpose of this paper is to obtain high order Runge–Kutta methods for nonlinear impulsive delay differential equations (IDDEs). In order to achieve the purpose, continuity and smoothness of the exact solutions of IDDEs are analyzed. And then the discontinuous points and non-smooth points are chosen as a part of the nodes of the Runge–Kutta methods. Furthermore, it is proved that the pth order Runge–Kutta method for IDDEs is also convergent of order p. And some simple numerical examples are given to demonstrate the theoretical results.

Numerical investigation of Crouzeix's conjecture

Crouzeix's conjecture states that for all polynomials p and matrices A, the inequality ‖p(A)‖≤2‖p‖W(A) holds, where the quantity on the left is the 2-norm of the matrix p(A) and the norm on the right is the maximum modulus of the polynomial p on W(A), the field of values of A. We report on some extensive numerical experiments investigating the conjecture via nonsmooth minimization of the Crouzeix ratio f≡‖p‖W(A)/‖p(A)‖, using Chebfun to evaluate this quantity accurately and efficiently and the BFGS method to search for its minimal value, which is 0.5 if Crouzeix's conjecture is true. Almost all of our optimization searches deliver final polynomial-matrix pairs that are very close to nonsmooth stationary points of f with stationary value 0.5 (for which W(A) is a disk) or smooth stationary points of f with stationary value 1 (for which W(A) has a corner). Our observations have led us to some additional conjectures as well as some new theorems. We hope that these give insight into Crouzeix's conjecture, which is strongly supported by our results.

Grazing-induced bifurcations in impact oscillators with elastic and rigid constraints

This paper investigates differences between the grazing-induced bifurcations in impact oscillators with one-sided elastic and rigid constraints by a path-following (continuation) method. The grazing bifurcations are computed and classified for both oscillators. Two-parameter smooth (period-doubling, saddle-node) and non-smooth (grazing) bifurcations are analyzed. Frequency response curves including bifurcation points are determined for different values of stiffness ratio and restitution of energy coefficient. As the stiffness ratio increases, the constraint changes from elastic to rigid and the bifurcation structure varies correspondingly. For the first time our numerical results presented in [17] and in the current work show that for the impact oscillators with one-sided elastic constraint, the smooth (period-doubling, saddle-node) bifurcations approach the non-smooth (grazing) bifurcations as the stiffness ratio increases. However, for the impact oscillators with one-sided rigid constraint, there is no smooth bifurcations near the non-smooth (grazing) bifurcation points. Basins of attraction, computed by our newly developed Matlab-based computational suite ABESPOL [5], complement our study.

Trajectory Optimization Based on Multi-Interval Mesh Refinement Method

In order to improve the optimization accuracy and convergence rate for trajectory optimization of the air-to-air missile, a multi-interval mesh refinement Radau pseudospectral method was introduced. This method made the mesh endpoints converge to the practical nonsmooth points and decreased the overall collocation points to improve convergence rate and computational efficiency. The trajectory was divided into four phases according to the working time of engine and handover of midcourse and terminal guidance, and then the optimization model was built. The multi-interval mesh refinement Radau pseudospectral method with different collocation points in each mesh interval was used to solve the trajectory optimization model. Moreover, this method was compared with traditional h method. Simulation results show that this method can decrease the dimensionality of nonlinear programming (NLP) problem and therefore improve the efficiency of pseudospectral methods for solving trajectory optimization problems.

Billiards in convex bodies with acute angles

In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body K ⊂ Rd has the property that the tangent cone of every non-smooth point q ∉ ∂K is acute (in a certain sense), then there is a closed billiard trajectory in K.

Non-smooth predictive control for mechanical transmission systems with backlash-like hysteresis

This paper proposes a non-smooth predictive control approach for mechanical transmission systems described by dynamic models with preceded backlash-like hysteresis. In this type of system, the work platform is driven by a DC motor through a gearbox. The work platform is represented by a linear dynamic sub-model connected in series with a backlash-like hysteresis inherent in gearbox. Here, backlash-like hysteresis is modeled as a non-smooth function with multi-valued mapping. In this case, the conventional model predictive control for such system cannot be implemented directly since the gradients of the control objective function with respect to control variables do not exist at non-smooth points. In order to solve this problem, a non-smooth receding horizon strategy is proposed. Moreover, the stability of predictive control of such non-smooth dynamic systems is analyzed. Finally, a numerical example and a simulation study on a mechanical transmission system are presented for validating the proposed method.

A minimax method for finding saddle points of upper semi-differentiable locally Lipschitz continuous functional in Banach space and its convergence

In 2005, a minimax characterization for nonsmooth saddle points in Banach spaces was presented by Yao and Zhou. According to this characterization, a descent-max method was devised. But, there was no numerical experiment and convergence result for the method. In this paper, to a class of locally Lipschitz continuous functionals, a new minimax method is designed. Numerical experiment is carried out and convergence results are established. This method extends two minimax methods given by Yao and Zhou in 2005 and Yao in 2013.

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On the premise of appropriate learning accuracy, the number of the neurons of a neural network should be as less as possible (constructional sparsification), so as to reduce the cost, and to improve the robustness and the generalization accuracy. We study the constructional sparsification of feedforward neural networks by using regularization methods. Apart from the traditional L 1 regularization for sparsification, we mainly use the L 1/2 regularization. To remove the oscillation in the iteration process due to the nonsmoothness of the L 1/2 regularizer, we propose to smooth it in a neighborhood of the nonsmooth point to get a smoothing L 1/2 regularizer. By doing so, we expect to improve the efficiency of the L 1/2 regularizer so as to surpass the L 1 regularizer. Some of our recent works in this respect are summarized in this paper, including the works on BP feedforward neural networks, higher order neural networks, double parallel neural networks and Takagi-Sugeno fuzzy models.

Smoothing Approximations for Some Piecewise Smooth Functions

In this paper, we study smoothing approximations for some piecewise smooth functions. We first present two approaches for one-dimensional case: a global approach is to construct smoothing approximations over the whole domain and a local approach is to construct smoothing approximations within appropriate neighborhoods of the nonsmooth points. We obtain some error estimate results for both approaches and discuss whether the smoothing approximations can inherit the convexity of the original functions. Furthermore, we extend the global approach to some multiple dimensional cases.

Bifurcations in piecewise-smooth steady-state problems: abstract study and application to plane contact problems with friction

The paper presents a local study of bifurcations in a class of piecewise-smooth steady-state problems for which the regions of smooth behaviour permit analytical expressions. A system of piecewise-linear equations capturing the essential features of branching scenarios around points of non-smoothness is derived under the assumptions that (i) the points lie in the intersection of the boundaries of the regions where the gradients of the respective smooth selections have the full rank, (ii) there is no solution branch whose tangential direction is tangent to the boundary of any of the regions. The simplest cases of this system are studied in detail and the most probable branching scenarios are described. A criterion for detecting bifurcation points is proposed and a procedure for its realisation in the course of numerical continuation of solution curves is designed for large problems. Application of the general frame to discretised plane contact problems with Coulomb friction is explained. Simple as well as more realistic model examples of bifurcations are shown.

Non-smooth Predictive Control for Wiener Systems with Backlash-Like Hysteresis

In this paper, a nonsmooth predictive control method for Wiener systems with backlash-like hysteresis is proposed. In this type of system, the backlash-like hysteresis is connected in series with a preceded linear dynamic subsystem. A backlash-like hysteresis is modeled as a nonsmooth function with multivalued mapping; the corresponding objective function of the control system is also defined as being nonsmooth. In this case, the implementation of conventional model predictive control for such systems will encounter a challenge, since the gradients of the control criterion function with respect to control variables do not exist at nonsmooth points. In order to solve this problem, a nonsmooth receding horizon strategy based on Clarke subgradients is proposed. Moreover, the robust stability of the predictive control of such nonsmooth Wiener systems is analyzed. Finally, a numerical example and a simulation study on a mechanical transmission system are implemented to validate the proposed method.

AN ADAPTIVE FINITE DIFFERENCE METHOD USING FAR-FIELD BOUNDARY CONDITIONS FOR THE BLACK-SCHOLES EQUATION

We present an accurate and efficient numerical method for solving the Black-Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid gen- eration: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demon- strate the accuracy and efficiency of the proposed method. Theresults show that the computational time is reduced substantially with the ac- curacy being maintained.