Non-smooth Model Research Articles

An [formula omitted]-norm based color image deblurring model under mixed random-valued impulse and Gaussian noise

This article introduces a novel model that restores color images degraded by blurring and mixed random-valued (RV) impulse and Gaussian noise. The model includes a data-fidelity term that consists of RV impulse noise and Gaussian noise components, and total variation regularization. Specifically, the data-fidelity term uses the l0- and l2-norms to represent the RV impulse and Gaussian noise components respectively. This results in an appropriate separation of these two noise components and thus enables simultaneous elimination of the mixed noise. Moreover, total variation regularization allows sufficient denoising in homogeneous regions while maintaining edges. To solve the nonconvex and nonsmooth model, we adopt a proximal alternating minimization approach and the alternating direction method of multipliers. These techniques contribute to an iterative algorithm with a convergence analysis provided. The experimental results demonstrate the effectiveness of the proposed model compared to other existing models, as measured by visual inspection and image quality assessments.

Mathematical Modeling of the Origami Navel Shell

The well-known Nautilus shell has been modeled extensively both by mathematicians and origamists. However, there is wide disagreement on the best-fitting mathematical model — partly because there is significant variability across different Nautilus Shells found in nature, and no single model can describe all of them well. Origami structures, however, have precise repeatable folding instructions, and do not exhibit such variability. Ironically, no known mathematical models exist for these structures. In this research, we mathematically model a prominent origami design, the Navel Shell by Tomoko Fuse, believed to be based on the Nautilus.
 We use first-principles geometric and trigonometric constructs for developing a non-smooth Geometric Model of the ideal origami spiral. We then search for the best-fitting parametric smooth spiral approximation, by formulating the fitting problem as a minimization problem over four unknowns. We write a Python computer program for searching the space numerically. Our evaluations show that: (i) the Smooth spiral is an excellent fit for the Geometric Model; (ii) our models for Origami Navel Shell are different from prior mathematical models for the Nautilus shell, but they come close to a recent model for a rare species of Nautilus; (iii) the Geometric Model explains the outer edges of origami images quite well and helps identify construction errors in the inner edges; and (iv) the Smooth Model helps understand how well the ideal Navel Shell matches different variants of the Nautilus species. We hope our research lays the foundation for further mathematical modeling of origami structures.

A two-stage stochastic variational inequality model for storage and dynamic distribution of medical supplies in epidemic management

The storage and distribution of medical supplies are important parts of epidemic prevention and control. This paper first proposes a new nonsmooth two-stage stochastic equilibrium model of medical supplies in epidemic management. The first stage addresses the storage in the pre-disaster phase, and the second stage focuses on the dynamic distribution by enrolling competitions among multiple hospitals over a period of time in the post-disaster phase. The uncertainties are the numbers of infected people treated in multiple hospitals during the period of time, which are time-varying around a nominal distribution predicted by historical experience. The two-stage stochastic equilibrium model is further approximated and transformed to a monotone two-stage stochastic variational inequality (SVI) model that is computationally tractable, with the aid of a smooth approximation technique. We employ the progressive hedging method (PHM) to solve a case study in the city of Wuhan in China suffered from the COVID-19 pandemic. Numerical results are presented to demonstrate the effectiveness of the proposed model in planning the storage and dynamic distribution of medical supplies in epidemic management.

Design of nonsmooth Kalman filter for compound sandwich systems with backlash and dead zone

AbstractIn this article, a new nonsmooth Kalman filtering (KF) method is proposed for noise suppression of the compound sandwich with backlash and dead zone. Based on the characteristics of the system, a nonsmooth stochastic state‐space equation is constructed. Then, nonsmooth KF is developed to adapt the nonsmooth stochastic state space model. The filter can switch automatically among different operating zones according to the operating conditions. Moreover, the convergence of the switchable compound nonsmooth KF is discussed. Subsequently, a motor‐driving mechanical transmission system which can be described as a compound nonsmooth nonlinear sandwich system with backlash and dead zone is studied as an application case. Finally, the comparison between the proposed filtering scheme and conventional filter are presented. It is demonstrated that the nonsmooth KF has a better performance than the conventional one in terms of convergence and accuracy in state estimation under noisy environment.

Modeling and Simulation of Rocking Block Dynamics subjected to Base Motion using an Energetic Restitution Law

ABSTRACT This work uses a nonsmooth dynamic model to study the motion of rocking-block structures with base excitation. In this study, the rocking blocks are modeled with respect to two contact points that represent the bottom face of the block. Each of these contact points can assume a state of either (a) free-flight, (b) contact, or (c) impact. Two of the key differences of this work from other studies of the rocking-block problem with base motion are (a) the collision model used here strictly enforces a rigidity property on the block, and (b) impacts are characterized using global energetic coefficients of restitution, which ensure energy consistency of the system. The proposed method also yields information on the history of impulses induced on the block due to the collisions, which has not been explored previously. This work can potentially lead to further insights into the rocking-block problem, which can be applied in earthquake and structural engineering.

Analytical prediction on stick-slip whirling oscillations induced by dry friction between a rotating imbalanced rotor and a flexibly supported stator

Non-smooth dry friction between the contact surfaces of rotor/stator rubbing systems can induce self-excited dry friction backward whirl, which may exhibit various discontinuity-induced sliding bifurcations corresponding to stick-slip whirling oscillations with different local dynamical characteristics around the switching manifold. The analytical method proposed in this paper aims to predict the self-excited stick-slip whirling oscillations in a general rotor/stator rubbing system, wherein both the dynamics of a rotor and a stator as well as deformation and friction at contact surfaces are taken into full consideration. The solution of the dry friction backward whirl for the four-degree-of-freedom non-smooth model with a switching manifold of a curved hypersurface is analytically derived under the hypothesis of pure rolling, which consists of a self-excited backward whirl part and a forced forward whirl part. For the validation with numerical simulation, it is found, as expected, that the analytical solution achieves high precision prediction on the response magnitudes and frequencies of the dry friction backward whirl in case of pure rolling at various rotating speeds. The analytical solution can also provide an eligible approximation on the stick-slip whirling oscillations with a relatively short slip phase in one period of stick-slip oscillation due to the negligible damping produced by slipping, such as in the case of a grazing-sliding motion, by which both the rotor and the stator undergo very large harmful amplitudes of vibration.

Dynamic tipping in the non-smooth Stommel-box model, with fast oscillatory forcing

We study the behavior at tipping points close to non-smooth fold bifurcations in non-autonomous systems. The focus is the Stommel-Box, and related climate models, which are piecewise-smooth continuous dynamical systems, modeling thermohaline circulation. We obtain explicit asymptotic expressions for the behavior at tipping points in the settings of both slowly varying freshwater forcing and rapidly oscillatory fluctuations. The results, based on combined multiple scale and local analyses, provide conditions for the sudden transitions between temperature-dominated and salinity-dominated states. In the context of high frequency oscillations, a multiple scale averaging approach can be used instead of the usual geometric approach normally required for piecewise-smooth continuous systems. The explicit parametric dependencies of advances and lags in the tipping show a competition between dynamic features of the model. We make a contrast between the behavior of tipping points close to both smooth Saddle–Node Bifurcations and the non-smooth systems studied on this paper. In particular we show that the non-smooth case has earlier and more abrupt transitions. This result has clear implications for the design of early warning signals for tipping in the case of the non-smooth dynamical systems which often arise in climate models.

The Number of Limit Cycles Bifurcating from a Degenerate Center of Piecewise Smooth Differential Systems

For two families of planar piecewise smooth polynomial differential systems, whose unperturbed system has a degenerate center at the origin, we study the biggest lower bound for the maximum number of limit cycles bifurcating from the periodic orbits of the center. These results are extensions of the known ones on unperturbed nondegenerate [Formula: see text]-center, derived from a nonsmooth harmonic oscillator model, to degenerate [Formula: see text]-center. Our study involves some new computational treatments. The main tools are the generalized polar coordinate change and the generalized Lyapunov polar coordinate change together with an averaging theory for one-dimensional piecewise smooth differential equations. Finally, we present two Maple programs for computing the averaging functions and consequently the biggest lower bound on the maximum number of limit cycles of degenerate [Formula: see text]-center under general polynomial perturbations of degree [Formula: see text].

Non-smooth models and simulation studies of the suspension system dynamics basing on piecewise linear luz(…) and tar(…) projections

Strongly nonlinear phenomena are attributes of suspension system dynamics of vehicles operated at high dynamic loads and high speeds. The causes of these phenomena are backlasches and dry friction in the mechanisms, detachments of the wheel from the roadway, impacts on the bumper elements, etc. It is well known that modelling of strong nonlinear phenomena can be based on a piecewise linear approach. To simplify the mathematical description of such phenomena, Żardecki developed special luz(…) and tar(…) projections. These piecewise-linear projections have a surprisingly simple mathematical apparatus that enables analytical operations (eg. reductions) for differential and algebraic equations and inclusions with non-smooth nonlinearities, and simplifies numerical simulations. The application of this method to model and simulation studies of the dynamics of the car steering system (including freeplay and stick-slip processes) have been reported in several authorial articles. This article presents examples related to suspension system dynamics.

Research on restoration algorithm of discrete tone image based on noise model

The ℓ0 “norm” regularized variational model is widely used in signal/image restoration. It is suitable for sparsity based regularization model design. This paper proposes a wavelet frame based Poisson noise removal model is proposed and the alternating direction method of multipliers (ADMM) is adopted to solve the corresponding optimization problem. Due to the nonconvex and noncontinuous property of the objective function with ℓ0 term, plenty of results in the literature cannot be directly applied to verifying the convergence of this ADMM based algorithm. We established the convergence property of the proposed algorithm for Poisson noise image restoration and reconstruction. This theoretical foundation is valuable to guarantee the stability performance of the algorithm in numerical simulation. Numerical experiments on different image processing tasks, such as gray image deblurring, CT image reconstruction and graph data denoising, also verified the effectiveness of this newly proposed nonconvex and nonsmooth model. Moreover all the results are possible to be extended to more general nonconvex noncontinuous optimization problem.

Noise-induced transitions in a non-smooth SIS epidemic model with media alert.

We investigate a non-smooth stochastic epidemic model with consideration of the alerts from media and social network. Environmental uncertainty and political bias are the stochastic drivers in our mathematical model. We aim at the interfere measures assuming that a disease has already invaded into a population. Fundamental findings include that the media alert and social network alert are able to mitigate an infection. It is also shown that interfere measures and environmental noise can drive the stochastic trajectories frequently to switch between lower and higher level of infections. By constructing the confidence ellipse for each endemic equilibrium, we can estimate the tipping value of the noise intensity that causes the state switching.

Muscular Strength Spurts in Adolescent Male Basketball Players: The INEX Study

Although successful performance in basketball requires high levels of muscular strength during adolescence, its development is confounded by the effects of normal growth. We examine the timing, intensity and sequence of muscular strength according to biological age (years from peak height velocity (PHV)) and hypothesize that young basketball players attain their peak muscular strength spurts around PHV. A total of 160 adolescent male basketballers, aged 11–15 years, were followed bi-annually over 3 consecutive years. The years from attainment of PHV and peak weight velocity (PWV) were estimated and five muscular strength measures (sit-ups, handgrip, seated medicine ball throw, squat jump and countermovement jump) were aligned to years from PHV in 3-month intervals. Strength velocities were estimated using a non-smooth mathematical model. The mean ages at-PHV and at-PWV were 13.90 ± 1.40 years and 13.90 ± 1.79 years, respectively. Maximal velocity in sit-ups was attained 6 months prior to attainment of PHV (intensity = 10.69 repetitions·year−1), whereas maximal velocity in squat jump occurred 6 months after-PHV (intensity = 3.93 cm·year−1). Handgrip strength, seated medicine ball throw and countermovement jump maximal velocity peaked at-PHV (intensity = 8.47 kgf·year−1, intensity = 0.75 m·year−1, intensity = 5.59 cm·year−1, respectively). In general, maximal velocity spurts did not differ in their timing, with the velocities reaching a peak concurrent with PHV and PWV or within 6 months of its attainment. Basketball coaches, as well as strength and conditioning trainers, should consider individual differences in strength development and be aware of rapid periods of growth in stature when planning and designing muscular strength training regimes.

A Penalty Relaxation Method for Image Processing Using Euler's Elastica Model

Euler's elastica model has been widely used in image processing. Since it is a challenging nonconvex and nonsmooth optimization model, most existing algorithms do not have convergence theory for it...

Non-smooth classification model based on new smoothing technique

This work describes a framework for solving support vector machine with kernel (SVMK). Recently, it has been proved that the use of non-smooth loss function for supervised learning problem gives more efficient results [1]. This gives the idea of solving the SVMK problem based on hinge loss function. However, the hinge loss function is non-differentiable (we can’t use the standard optimization methods to minimize the empirical risk). To overcome this difficulty, a special smoothing technique for the hinge loss is proposed. Thus, the obtained smooth problem combined with Tikhonov regularization is solved using a stochastic gradient descent method. Finally, some numerical experiments on academic and real-life datasets are presented to show the efficiency of the proposed approach.

Movement characteristics of a non-smooth model with a closed curve equilibrium

The main aim of this paper is to analyze the dynamical properties of a model with a closed curve equilibrium. The corresponding three-variable model is given as a set of nonlinear ordinary differential equations containing non-smooth functions. The dynamics of the model are studied depending on three parameters. For this purpose, new methods, as the 0-1 test for chaos and approximate entropy, are applied. Using these tools, the dynamics are quantified and qualified. It is shown that depending on the system’s parameters, the system exhibits both irregular (chaotic) and regular (periodic) character.

A high order PDE-constrained optimization for the image denoising problem

In the present work, we investigate the inverse problem of identifying simultaneously the denoised image and the weighting parameter that controls the balance between two diffusion operators for an evolutionary partial differential equation (PDE). The problem is formulated as a non-smooth PDE-constrained optimization model. This PDE is constructed by second- and fourth-order diffusive tensors that combines the benefits from the diffusion model of Perona–Malik in the homogeneous regions, the Weickert model near sharp edges and the fourth-order term in reducing staircasing. The existence and uniqueness of solutions for the proposed PDE-constrained optimization system are provided in a suitable Sobolev space. Also, an optimization problem for the determination of the weighting parameter is introduced based on the Primal–Dual algorithm. Finally, simulation results show that the obtained parameter usually coincides with the better choice related to the best restoration quality of the image.

Incremental DC optimization algorithm for large-scale clusterwise linear regression

The objective function in the nonsmooth optimization model of the clusterwise linear regression (CLR) problem with the squared regression error is represented as a difference of two convex functions. Then using the difference of convex algorithm (DCA) approach the CLR problem is replaced by the sequence of smooth unconstrained optimization subproblems. A new algorithm based on the DCA and the incremental approach is designed to solve the CLR problem. We apply the Quasi-Newton method to solve the subproblems. The proposed algorithm is evaluated using several synthetic and real-world data sets for regression and compared with other algorithms for CLR. Results demonstrate that the DCA based algorithm is efficient for solving CLR problems with the large number of data points and in particular, outperforms other algorithms when the number of input variables is small.

Lexicographic derivatives of nonsmooth glucose-insulin kinetics under normal and artificial pancreatic responses

Parametric sensitivities of nonsmooth glucose-insulin kinetics models are investigated in this article. In particular, we study several variations on a model of the so-called Intravenous Glucose Tolerance Test (IVGTT), which measures a subject’s insulin response to glucose over time and yields data useful for characterizing and diagnosing diabetes. Nonsmoothness refers here to continuous behavior that is punctuated by “switches” (physiological events) arising due to biochemical and control thresholds being crossed. Insulin secretion from a typical pancreatic response and insulin infusion from an external device (for diabetic patients) are both considered here. The presence of nonsmoothness means that classical sensitivity theory and associated numerical methods are ill-equipped to handle these models. Motivated by this, lexicographic directional differentiation and lexicographic “sensitivity functions” are used, which are recently developed tools in nonsmooth analysis. Using this new approach, we derive nonsmooth sensitivity systems associated with the considered glucose kinetics models, whose unique solutions provide local sensitivity information analogous to classical forward parametric sensitivity functions. Moreover, the theory is practically implementable and the resulting sensitivity functions are computationally relevant, as they can be supplied to dedicated nonsmooth numerical methods (e.g., optimization). The nonsmooth sensitivity systems are solved numerically, giving new insights into the model, and thus into the dynamics of glucose kinetics under the simulated conditions. It is hoped that the results here may be used to improve the design of assessments for diabetes (e.g., the IVGTT) and its treatment.

Finite-Time Command-Filtered Composite Adaptive Neural Control of Uncertain Nonlinear Systems.

This article presents a new command-filtered composite adaptive neural control scheme for uncertain nonlinear systems. Compared with existing works, this approach focuses on achieving finite-time convergent composite adaptive control for the higher-order nonlinear system with unknown nonlinearities, parameter uncertainties, and external disturbances. First, radial basis function neural networks (NNs) are utilized to approximate the unknown functions of the considered uncertain nonlinear system. By constructing the prediction errors from the serial-parallel nonsmooth estimation models, the prediction errors and the tracking errors are fused to update the weights of the NNs. Afterward, the composite adaptive neural backstepping control scheme is proposed via nonsmooth command filter and adaptive disturbance estimation techniques. The proposed control scheme ensures that high-precision tracking performances and NN approximation performances can be achieved simultaneously. Meanwhile, it can avoid the singularity problem in the finite-time backstepping framework. Moreover, it is proved that all signals in the closed-loop control system can be convergent in finite time. Finally, simulation results are given to illustrate the effectiveness of the proposed control scheme.

A novel truncated nonconvex nonsmooth variational method for SAR image despeckling

ABSTRACT Speckle reduction is a fundamental problem in coherent imaging systems. In this paper, to suppress the speckle in SAR images, we propose a novel truncated nonconvex nonsmooth model. It incorporates a truncated nonconvex regularization term and an I-divergence fidelity term. The truncated () regularization can better recover neat edges and simultaneously prevent contrast reduction artefact. The I-divergence fidelity term is used to suppress the multiplicative noise effectively. We also propose an efficient algorithm based on variable-splitting and alternating direction method of multipliers (ADMM) method to solve the model. Compared to state-of-the-art speckle suppression methods, intensive experimental results on a variety of SAR images show the superiority of the proposed method qualitatively and quantitatively.