Piecewise Smooth Research Articles

Estimation and Uncertainty Quantification for Piecewise Smooth Signal Recovery

This paper presents a sparse Bayesian learning (SBL) algorithm for linear inverse problems with a high order total variation (HOTV) sparsity prior. For the problem of sparse signal recovery, SBL often produces more accurate estimates than maximum a posteriori estimates, including those that rely on l1 regularization. Moreover, rather than a single signal estimate, SBL yields a full posterior density estimate which can be used for uncertainty quantification. However, SBL is only immediately applicable to problems having a direct sparsity prior, or to those that can be formed via synthesis. This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis, and then develops a corresponding Bayesian learning method. This expands the class of problems available to Bayesian learning to include, e.g., inverse problems dealing with the recovery of piecewise smooth functions or signals from data. Numerical examples are provided to demonstrate how this new technique is effectively employed.

Toolpath generation for ultraprecision machining: Freeform surface generation and extrapolation

The rise in complexity of freeform optics drives the development of equally complex manufacturing processes needed to produce them. Complex optical designs, however, require equally complex machining toolpaths and techniques that can generate them. This paper and subsequent research develops and investigates toolpath generation techniques for explicitly defined surfaces of the form Z(X,Y). A generalized approach is developed for the toolpath generation of optical surfaces themselves as well as techniques for transitioning between surfaces in an array. All surfaces herein are continuous, well-behaved, and differentiable – constraints satisfied by most optical prescriptions today. The current paper describes the toolpath generation and extrapolation process for a tilted freeform surface defined inside a non-circular boundary. Boundaries are defined so that some non-convexity is acceptable, and it is simple to determine whether a toolpath coordinate lies inside or outside them. Using the proposed coordinate frame transformation and boundary definition, it is shown that by tilting a surface and boundary together, the tilted surface retains its original form and boundary in a rotated coordinate frame. With the proposed extrapolating surface, it is possible to modify a freeform surface outside its defined boundary so it is smooth and well-behaved inside its overall region of interest – an important consideration for both toolpath generation and opto-mechanical design. By combining freeform and extrapolating surfaces together, a smooth piecewise surface is generated whose overall continuity increases as the order of the extrapolating surface increases. As a result, it is possible to increase the overall continuity of a toolpath along its cutting direction. To verify the utility of the proposed extrapolating surface, experimental data are presented for several spiral toolpaths generated for a 6th order polynomial surface prescription defined inside a rectangular boundary. Subsequent research, using the techniques herein as a starting point, develop and investigate blending multiple freeform surfaces together.

Linearization and Perturbations of Piecewise Smooth Vector Fields with a Boundary Equilibrium

Linearization and Perturbations of Piecewise Smooth Vector Fields with a Boundary Equilibrium

Analysis of Fast-Scale Instability in Three-Level T-Type Single-Phase Inverter Feeding Diode-Bridge Rectifier With Inductive Load

In this article, the fast-scale instability in the three-level T-type single-phase inverter feeding diode-bridge rectifier with inductive load (3TSI-DR) is studied. Simulations suggest that such fast-scale instability on switching period scale can increase the harmonic content in the 3TSI-DR, which seriously affects its stable operation. To reveal the mechanism of this fast-scale instability, the state equation of the 3TSI-DR is derived, and state variables are solved based on quasi-static approximation principle. From state equation, the 3TSI-DR is periodic time-varying and piecewise smooth, belonging to Filippov system. Accordingly, the discrete-time mapping model of the 3TSI-DR is established, Filippov method is used for obtaining monodromy matrix, and Floquet theory is applied to explore instability mechanism. Theoretical results indicate that the fast-scale instability of the 3TSI-DR is caused by period-doubling bifurcation. Moreover, the Floquet multiplier sensitivities of different circuit parameters are calculated to identify key parameters; via comparing theoretical analyses with simulations, the unstable angle ranges of the fast-scale instability are given, and the stability boundaries in various parameter spaces are discussed. All these can provide design-oriented information for optimizing the 3TSI-DR to avoid instability due to period-doubling bifurcation. Finally, experimental results agreeing with simulations are presented to verify the correctness of theoretical analyses.

Recovery of Piecewise Smooth Density and Lamé Parameters from High Frequency Exterior Cauchy Data

Recovery of Piecewise Smooth Density and Lamé Parameters from High Frequency Exterior Cauchy Data

A new prescribed-time stability theorem for impulsive piecewise-smooth systems and its application to synchronization in networks

In this paper, the prescribed-time synchronization (PTs) issue in networks of nonlinear piecewise-smooth (PWS) systems with impulse effects is discussed. Firstly, a time-varying piecewise function is constructed to develop a novel theorem with respect to the prescribed-time stability for nonlinear impulsive PWS systems, in which the convergence time is pre-set by users. Secondly, under the designed feedback controller with the discontinuous factor, by utilizing generalized Lyapunov functional method, non-smooth analysis theory, and inequality analysis technique, the global prescribed-time stability of synchronization error system is proved analytically. Meanwhile, the PTs conditions are addressed in terms of linear matrix inequalities (LMIs). Finally, an application to Chua’s chaotic circuit system is given to verify the correctness of theoretical results and the feasibility of the designed control strategy.

Tchebycheffian B-splines in isogeometric Galerkin methods

Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most of the properties known in the polynomial spline case. In particular, under suitable assumptions, Tchebycheffian splines admit a representation in terms of basis functions, called Tchebycheffian B-splines (TB-splines), completely analogous to polynomial B-splines. A particularly interesting subclass consists of Tchebycheffian splines with pieces belonging to null-spaces of constant-coefficient linear differential operators. They grant the freedom of combining polynomials with exponential and trigonometric functions with any number of individual shape parameters. Moreover, they have been recently equipped with efficient evaluation and manipulation procedures. In this paper we consider the use of TB-splines with pieces belonging to null-spaces of constant-coefficient linear differential operators as an attractive substitute for standard polynomial B-splines and rational NURBS in isogeometric Galerkin methods. We discuss how to exploit the large flexibility of the geometrical and analytical features of the underlying Tchebycheff spaces according to problem-driven selection strategies. TB-splines offer a wide and robust environment for the isogeometric paradigm beyond the limits of the rational NURBS model.

Nontrivial Limit Cycles in a Kind of Piecewise Smooth Generalized Abel Equation

In this paper, we focus on the number of nontrivial limit cycles in a kind of piecewise smooth generalized Abel equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Under the condition [Formula: see text], employing Melnikov functions of any order and using properties of Chebyshev systems, we prove that if [Formula: see text] is odd, then the maximum number of nontrivial limit cycles bifurcating from the periodic annulus of the unperturbed system is 6 and it is attainable, and if [Formula: see text] is even, then the maximum number is 3, and it can be attained too.

Bifurcation of Limit Cycles of a Piecewise Smooth Hamiltonian System

Bifurcation of Limit Cycles of a Piecewise Smooth Hamiltonian System

Bifurcation Analysis of a Piecewise Smooth Map with Two Asymptotes

In this paper, we consider a discontinuous piecewise smooth system involving four parameters and two asymptotes, recently introduced as a model in engineering sciences. We classify and investigate its bifurcation behaviour. A local bifurcation analysis of the system in the range of parameters which has not been studied so far is undertaken and then supported by numerical computations. This reveals the existence of a flip bifurcation depends on the power singularity. Moreover, we state that a set of positive measure of points with divergent dynamic behaviour exists.

Blue-Sky Catastrophic Bifurcations Behind Emergence and Disappearance of Quasiperiodic Rubbing Oscillations in a Piecewise Smooth Rotor–Stator System

In this paper, an approach combining the methods of multigrid Point Mapping under Cell Reference and Stagger-and-step is proposed, which can efficiently determine both attractors and unstable saddle-type invariant sets in higher-dimensional dynamical systems. Then, the bifurcations around quasiperiodic partial rub oscillations of a piecewise smooth rotor–stator system are investigated, which are categorized into two classes depending on whether Nonlinear Normal Modes (NNMs) exist or not. It is found that the quasiperiodic rubbing oscillations, within the parameter region with an NNM, emerge through a Hopf bifurcation from a periodic rubbing oscillation and disappear through a collision with a saddle torus with an accompanying variation of system parameters. Within the parameter region without NNM, the bifurcation scenarios of two sequences of quasiperiodic rubbing oscillations, which emerge subsequently with the increasing of the rotating speed, are similar but somewhat complicated. That is, the quasiperiodic rubbing oscillation emerges through an inverse doubling of torus from a chaotic attractor, which is produced from a chaotic saddle after chaotic crisis. The quasiperiodic rubbing oscillation disappears through a collision between the stable and the unstable quasiperiodic tori. The insight into blue-sky catastrophic bifurcations in the quasiperiodic rubbing oscillations of the piecewise smooth rotor–stator rubbing system is thus obtained through this study.

On the structure of regularization paths for piecewise differentiable regularization terms

Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression and machine learning. Since the choice of the regularization parameter is crucial but often difficult, path-following methods are used to approximate the entire regularization path, i.e., the set of all possible solutions for all regularization parameters. Due to their nature, the development of these methods requires structural results about the regularization path. The goal of this article is to derive these results for the case of a smooth objective function which is penalized by a piecewise differentiable regularization term. We do this by treating regularization as a multiobjective optimization problem. Our results suggest that even in this general case, the regularization path is piecewise smooth. Moreover, our theory allows for a classification of the nonsmooth features that occur in between smooth parts. This is demonstrated in two applications, namely support-vector machines and exact penalty methods.

Boundary Hölder Estimates for a Class of Degenerate Elliptic Equations in Piecewise Smooth Domains

Boundary Hölder Estimates for a Class of Degenerate Elliptic Equations in Piecewise Smooth Domains

Stability conditions for refractive partially integrable piecewise smooth vector fields

In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our main goal is to study stability problems inside the class of 3−dimensional refractive piecewise smooth vector fields. Our concern is to study refractive vector fields that admit a first integral that leaves invariant any sphere centered at the origin. Global stability conditions on generic one-parameter families of refractive piecewise smooth vector fields on a two-dimensional sphere are presented and used to prove our main result, which establishes necessary conditions for the structural stability inside that class.

Activity Patterns of a Two-Timescale Neuronal Ring Model with Voltage-Dependent, Piecewise Smooth Inhibitory Coupling

Activity Patterns of a Two-Timescale Neuronal Ring Model with Voltage-Dependent, Piecewise Smooth Inhibitory Coupling

Quantitatively defining species boundaries with more efficiency and more biological realism

We introduce a widely applicable species delimitation method based on the multispecies coalescent model that is more efficient and more biologically realistic than existing methods. We extend a threshold-based method to allow the ancestral speciation rate to vary through time as a smooth piecewise function. Furthermore, we introduce the cutting-edge proposal kernels of StarBeast3 to this model, thus enabling rapid species delimitation on large molecular datasets and allowing the use of relaxed molecular clock models. We validate these methods with genomic sequence data and SNP data, and show they are more efficient than existing methods at achieving parameter convergence during Bayesian MCMC. Lastly, we apply these methods to two datasets (Hemidactylus and Galagidae) and find inconsistencies with the published literature. Our methods are powerful for rapid quantitative testing of species boundaries in large multilocus datasets and are implemented as an open source BEAST 2 package called SPEEDEMON.

Non-Negative Matrix Factorization Based on Smoothing and Sparse Constraints for Hyperspectral Unmixing.

Hyperspectral unmixing (HU) is a technique for estimating a set of pure source signals (end members) and their proportions (abundances) from each pixel of the hyperspectral image. Non-negative matrix factorization (NMF) can decompose the observation matrix into the product of two non-negative matrices simultaneously and can be used in HU. Unfortunately, a limitation of many traditional NMF-based methods, i.e., the non-convexity of the objective function, may lead to a sub-optimal solution. Thus, we put forward a new unmixing method based on NMF under smoothing and sparse constraints to obtain a better solution. First, considering the sparseness of the abundance matrix, a weight sparse regularization is introduced into the NMF model to ensure the sparseness of the abundance matrix. Second, according to the similarity prior of the same feature in the adjacent pixels, a Total Variation regularization is further added to the NMF model to improve the smoothness of the abundance map. Finally, the signatures of each end member are modified smoothly in spectral space. Moreover, it is noticed that discontinuities may emerge due to the removal of noisy bands. Therefore, the spectral data are piecewise smooth in spectral space. Then, in this paper, a piecewise smoothness constraint is further applied to each column of the end-member matrix. Experiments are conducted to evaluate the effectiveness of the proposed method based on two different datasets, including a synthetic dataset and the real-life Cuprite dataset, respectively. Experimental results show that the proposed method outperforms several state-of-the-art HU methods. In the Cuprite hyperspectral dataset, the proposed method’s Spectral Angle Distance is 0.1694, compared to the TV-RSNMF method’s 0.1703, L1/2NMF method’s 0.1925, and VCA-FCLS method’s 0.1872.

A Hybrid RBF Collocation Method and Its Application in the Elastostatic Symmetric Problems

In this paper, a new hybrid radial basis function collocation method (HRBF-CM) is proposed to help resolve two-dimensional elastostatic symmetric problems. In the new approach, the hybrid radial basis function (HRBF) combines the infinitely smooth RBF and piecewise smooth RBF, containing two parameters (the shape parameter and the weight parameter). Discretization schemes are presented in detail. We use MATLAB to implement the HRBF-CM and produce numerical results which demonstrate the potential of this method. The new method’s accuracy is higher than that of the traditional methods, especially in the case of a more significant number of nodes. We discuss the new method’s effectiveness compared to the widely used traditional RBF and also investigate the effect of parameters on the method’s performance under the new method.

Prescribed-time synchronization in networks of piecewise smooth systems via a nonlinear dynamic event-triggered control strategy

This paper discusses the prescribed-time synchronization (PTs) issue in networks of piecewise smooth (PWS) systems. Firstly, two new lemmas with respect to the prescribed-time stability are developed for nonlinear PWS systems, where the stable time can be preset arbitrarily, and is irrelevant to any control parameters and initial values. Secondly, a feedback controller and a novel event-triggered controller are designed to realize PTs objective, respectively. Different from the existing linear dynamic event-triggered condition, the adjustable internal dynamic variable is governed by a nonlinear system, which is related to the prescribed time. Thirdly, by adopting set-valued analysis technique and generalized Lyapunov functional approach, the conditions of PTs are achieved in terms of linear matrix inequalities (LMIs). Finally, two application examples to chaotic Sprott circuit are provided to certificate the correctness of the obtained results and the effectiveness of the designed control schemes.

There Exist Transitive Piecewise Smooth Vector Fields on $$\mathbb {S}^2$$ but Not Robustly Transitive

It is well known that smooth (or continuous) vector fields cannot be topologically transitive on the sphere $\S^2$. Piecewise-smooth vector fields, on the other hand, may present non-trivial recurrence even on $\S^2$. Accordingly, in this paper the existence of topologically transitive piecewise-smooth vector fields on $\S^2$ is proved, see Theorem \ref{teorema-principal}. We also prove that transitivity occurs alongside the presence of some particular portions of the phase portrait known as {\it sliding region} and {\it escaping region}. More precisely, Theorem \ref{main:transitivity} states that, under the presence of transitivity, trajectories must interchange between sliding and escaping regions through tangency points. In addition, we prove that every transitive piecewise-smooth vector field is neither robustly transitive nor structural stable on $\S^2$, see Theorem \ref{main:no-transitive}. We finish the paper proving Theorem \ref{main:general} addressing non-robustness on general compact two-dimensional manifolds.