Non-smooth Model Research Articles

Two-Phase Image Segmentation by Nonconvex Nonsmooth Models with Convergent Alternating Minimization Algorithms

Two-Phase Image Segmentation by Nonconvex Nonsmooth Models with Convergent Alternating Minimization Algorithms

Multiscale Approach for Bounded Deformation Image Registration

Deformable image registration is a very important topic in the field of image processing. It is widely used in image fusion and shape analysis. Generally speaking, image registration models can be divided into two categories: smooth registration and non-smooth registration. During the last decades, many smooth registration models (i.e., diffeomorphic registration) were proposed. However, image with strong noise may lead to discontinuous deformation, which cannot be modelled by smooth registration. To simulate this kind of deformation, some non-smooth registration models were also proposed. However, numerical algorithms for these models are easily trapped into a local minimum because of the nonconvexity of the object functional. To overcome the local minimum of the object functional, we propose a multiscale approach for a non-smooth registration model: the bounded deformation (BD) model. The convergence of the approach is shown, and numerical tests are also performed to show the good performance of the proposed multiscale approach.

An approximate Newton-type proximal method using symmetric rank-one updating formula for minimizing the nonsmooth composite functions

Founded upon the scaled memoryless symmetric rank-one updating formula, we propose an approximation of the Newton-type proximal strategy for minimizing the nonsmooth composite functions. More exactly, to approximate the inverse Hessian of the smooth part of the objective function, a symmetric rank-one matrix is employed to straightly compute the search directions for a special category of well-known functions. Convergence of the given algorithm is argued with a nonmonotone backtracking line search adjusted for the corresponding nonsmooth model. Also, its practical advantages are computationally depicted in the two well-known real-world models.

Dynamic characteristics of series-parallel hybrid rigid-flexible coupling double-mass underactuated system on floating platform

For heavy cargo swing suppression on offshore boom cranes, a cable-driven series-parallel hybrid double-mass active anti-swing mechanism is proposed, which is a typical non-smooth coupled oscillator. To facilitate the study of dynamic characteristics of the system, the cargo swings were decomposed into two decoupled swings. Then, the kinematics is built using the influence coefficient principle. Considering the longitudinal vibration, cables were regarded as piecewise spring-damping systems. Subsequently, an 18-DOF non-smooth dynamic model with actuators was obtained. Additionally, the influence of the floating platform's excitations and the system's initial state on responses was investigated by numerical solution. Analysis tools such as the phase trajectory and Poincaré map were used to explore the dynamic characteristics, and the results exhibit the existence of out-of-plane behavior, secondary resonance, asymmetric oscillation, and chaos. Finally, experiments were performed to validate the dynamics and performance of an open-loop anti-swing method. This study will be significant for further research on anti-swing controllers.

Variational Embedding of Measured Data in Physics-Constrained Data-Driven Modeling

Abstract This paper presents a physics-constrained data-driven method that variationally embeds measured data in the modeling and analysis framework. The physics-based model is augmented with sparse but high-fidelity data through a variationally derived loss function. The structure of the loss function is analyzed in the context of variational correction to the modeled response wherein loss function penalizes the difference in the modeled response from the measured data that represents the local response of the system. The variationally embedded measured data (VEMD) method results in forward simulations that are not only driven by boundary and initial conditions but are also augmented by real measurements taken at only a small number of observation points. In the context of forward simulations, the proposed approach can be seen as inducing inductive biases that exploit the difference between the computed and measured quantities in the parametric space. With the help of a model problem, we show that the proposed method learns from the sparse high-fidelity datasets while preserving conservation properties of the balance laws. Method is applied to a non-smooth model problem and the mathematical attributes of the formulation are investigated in the context of high-fidelity sparse datasets.

Dynamics of a non-smooth model of prostate cancer with intermittent androgen deprivation therapy

Intermittent androgen deprivation therapy (IADT) is widely used in the clinic treatment for prostate cancer. We propose a Filippov system in this paper, which defines the following threshold policy: the androgen deprivation therapy (ADT) is implemented once the concentration of androgen-dependent (AD) cells exceeds the threshold level ET and it is suspended otherwise. To better understand the effect of the IADT on the evolution of prostate cancer, we perform a comprehensive qualitative analysis of global dynamics of the targeted model. As the threshold value varies, our model admits multistability of three equilibria, bistability of two equilibria, stability of the regular equilibrium or the pseudo-equilibrium, and the boundary equilibrium bifurcation. The steady-state regimes include high concentration, low concentration and no androgen-independent cells. This highlights the critical role of threshold values and the initial condition of prostate cancer cells of patients. The main results indicate that the concentration of AD cells and AI cells can be contained at a relatively satisfactory level, which can help to make treatment schedule and improve treatment outcomes for patients with prostate cancer.

A unified framework for nonconvex nonsmooth sparse and low-rank decomposition by majorization-minimization algorithm

Recovering a low-rank matrix and a sparse matrix from an observed matrix, known as sparse and low-rank decomposition (SLRD), is becoming a hot topic in recent years. The most popular model for SLRD is to use the ℓ1 norm and nuclear norm for the sparse and low-rank approximation. Since this convex model has certain limitations, various nonconvex models have been explored and found to be very promising. In this paper, we introduce a generalized nonconvex nonsmooth model for SLRD which covers a wide range of nonconvex surrogate functions that are continuous, concave and monotonically increasing on [0,∞) to approximate both the ℓ0 norm and the rank function, such as ℓp norm (0<p<1), Logarithm, Geman, SCAD and MCP functions. The choice of the nonconvex surrogates for the sparse and low-rank components can be different. Due to the nonconvexity and extensive options of the surrogates, the optimization problem is untractable. Based on the majorization-minimization (MM) algorithm, we propose a unified framework named MM-ADMM algorithm to solve this problem, which can be applied to all eligible surrogates as long as their supergradients are available. The constrained majorizing problems established under the MM framework can be easily solved by the alternating direction method of multipliers (ADMM). The theoretical convergence properties are investigated and proved, including the convergence of the sequence of objective function values generated by the designed algorithm and a weak convergence result related to the inner ADMM-iterations. Experiments on the synthetic data and real-world applications demonstrate the effectiveness of our designed MM-ADMM algorithm.

Fixed-Time Stable Neurodynamic Flow to Sparse Signal Recovery via Nonconvex L1-β2-Norm.

This letter develops a novel fixed-time stable neurodynamic flow (FTSNF) implemented in a dynamical system for solving the nonconvex, nonsmooth model L1-β2, β∈[0,1] to recover a sparse signal. FTSNF is composed of many neuron-like elements running in parallel. It is very efficient and has provable fixed-time convergence. First, a closed-form solution of the proximal operator to model L1-β2, β∈[0,1] is presented based on the classic soft thresholding of the L1-norm. Next, the proposed FTSNF is proven to have a fixed-time convergence property without additional assumptions on the convexity and strong monotonicity of the objective functions. In addition, we show that FTSNF can be transformed into other proximal neurodynamic flows that have exponential and finite-time convergence properties. The simulation results of sparse signal recovery verify the effectiveness and superiority of the proposed FTSNF.

Elastoplastic Integration Method of Mohr-Coulomb Criterion

A new method for implicit integration of the Mohr-Coulomb non-smooth multisurface plasticity models is presented, and Koiter’s requirements are incorporated exactly within the proposed algorithm. Algorithmic and numerical complexities are identified and introduced by the nonsmooth intersections of the Mohr-Coulomb surfaces; then, a projection contraction algorithm is applied to solve the classical Kuhn–Tucker complementary equations which provide the only characterization of possible active yield surfaces as a special class of variational inequalities, and the actual active yield surface is further determined by iteration. The basic idea is to calculate derivatives of the yield and potential functions with the expressions in the principal stresses and perform the return manipulations in the general stress space. Based on the principal stress characteristic equation, partial derivatives of principal stresses are calculated. The proposed algorithm eliminates the error caused by smoothing the corner of Mohr-Coulomb surfaces, avoids the numerical singularity at the intersections in the general stress space, and does not require the stress transformation needed in the principal stress space method. Lastly, several numerical examples are given to verify the validity of the proposed method.

Rich dynamics of a Filippov plant disease model with time delay

We propose a nonsmooth Filippov plant disease model with threshold strategy of roguing of the infected plants with a saturated roguing rate. Different from traditional Filippov models, here we incorporate time delay which represents the incubation period of the plant disease to make the model more realistic. Analytical results show that the two delayed subsystems might admit Hopf bifurcations when the delay passes through some critical values. Then we investigate some key elements to our model, including the regular/virtual equilibrium, the sliding segment, sliding mode dynamics and pseudoequilibrium. The results indicate that all solutions finally converge to either the regular equilibrium, the pseudoequilibrium or a stable standard periodic solution depending on the values of the threshold level and the time delay. Furthermore, we show that the time delay plays a significant role in discontinuity-induced bifurcations. That is, as the time delay is varied, the model exhibits complex local sliding bifurcation (boundary bifurcation) and global sliding bifurcations from a standard periodic solution to a sliding bifurcation then to a crossing bifurcation. In conclusion, a Filippov system with time delay can provide some new insights for plant disease control.

Effect of Bionic Nonsmooth Surface Vane on the Antiwear Characteristics of Double-Vane Pump.

In order to improve the antiwear characteristics of the double-vane self-priming pump, the surface structure of the Scapharca subcrenata was extracted and reconstructed according to bionic principles. Three types of nonsmooth surface models were established at the outlet end of the suction surface of the vanes, which is the most severely worn in the double-vane pump. The external characteristics, pressure field distribution, wear area distribution, and wear degree of the volute and vanes at different concentrations of nonsmooth vane structure were investigated by numerical simulation to reveal the mechanism of the nonsmooth surface structure of the wear characteristics of the vanes. The results show that the head and efficiency of pumps with four different vanes decrease and the average wear rate increases as the particle concentration increases. The different vane structures have a very small effect on the wear resistance of the volute, but a larger effect on vane wear. The circular nonsmooth surface structure, which reduces the low pressure area of the inlet section of the impeller while ensuring a smaller drop in head and efficiency, produces the best antiwear effect and improves the antiwear performance of the double-vane pump.

An approach to the modeling of honey bee colonies

Abstract. In this work, populations of adult and immature honeybees and their honey production are studied through mathematical and statistical modeling approaches. Those models are complementary and are presented in disjunct form. They were used to show different modeling methods for honey bee population dynamics. The statistical approach consisted of a generalized linear model using data from the Department of Agriculture of the United States of America (USDA), which showed that the relationship between the number of colonies and the rate of honey production is not constant in time but decrease over the years. These models showed that when a bee population is subjected to a stress factor (i.e., habitat destruction, Varroa mite, climate variability, season, neonicotinoids, among others), the abundance of individuals decreases over time as well as the honey produced by the colonies. Finally, the mathematical approach consisted of two models: (1) a smooth model, in which conditions of existence and stability of the equilibrium solutions are determined by an ecological threshold value, and (2) a non-smooth model where the mortality rate of bees is included as a function of the number of adult bees in the population.

The lower bound of nonlocal gradient for non-convex and non-smooth image patches based regularization

In the inverse problem of image processing, we have witnessed that the non-convex and non-smooth regularizers can produce clearer image edges than convex ones such as total variation (TV). This fact can be explained by the uniform lower bound theory of the local gradient in non-convex and non-smooth regularization. In recent years, although it has been numerically shown that the nonlocal regularizers of various image patches based nonlocal methods can recover image textures well, we still desire a theoretical interpretation. To this end, we propose a non-convex non-smooth and block nonlocal regularization model based on image patches. By integrating the advantages of the non-convex and non-smooth potential function in the regularization term, the uniform lower bound theory of the image patches based nonlocal gradient is given. This approach partially explains why the proposed method can produce clearer image textures and edges. Compared to some classical regularization methods, such as TV, non-convex and non-smooth regularization, nonlocal total variation and block nonlocal total variation, our experimental results show that the proposed method improves restoration quality.

Risk-Based Robust Statistical Learning by Stochastic Difference-of-Convex Value-Function Optimization

For the treatment of outliers, the paper “Risk-Based Robust Statistical Learning by Stochastic Difference-of-Convex Value-Function Optimization” by Junyi Liu and Jong-Shi Pang proposes a risk-based robust statistical learning model. Employing a variant of the conditional value-at-risk risk measure, called the interval conditional value-at-risk (In-CVaR), the model aims to exclude the risks associated with the left and right tails of the loss. The resulting nonsmooth and nonconvex model considers the population In-CVaR risk and distinguishes the upside and downside losses with asymmetric weights. For the solution of the model in both regression and classification, the authors show that the objective function is the difference of two convex functions each being the optimal objective value of a univariate convex stochastic program. A sampling and convex programming-based algorithm is developed with the appropriate control of incremental sample sizes, and its subsequential almost-sure convergence to a critical point is established. Numerical results illustrate the practical performance of the model and methodology.

Non-smooth dynamic modeling and simulation of an unmanned bicycle on a curved pavement

The non-smooth dynamic model of an unmanned bicycle is established to study the contact-separate and stick-slip non-smooth phenomena between wheels and the ground. According to the Carvallo-Whipple configuration, the unmanned bicycle is reduced to four rigid bodies, namely, rear wheel, rear frame, front fork, and front wheel, which are connected by perfect revolute joints. The interaction between each wheel and the ground is simplified as the normal contact force and the friction force at the contact point, and these forces are described by the Hunt-Crossley contact force model and the LuGre friction force model, respectively. According to the characteristics of flat and curved pavements, calculation methods for contact forces and their generalized forces are presented. The dynamics of the system is modeled by the Lagrange equations of the first kind, a numerical solution algorithm of the dynamic equations is presented, and the Baumgarte stabilization method is used to restrict the drift of the constraints. The correctness of the dynamic model and the numerical algorithm is verified in comparison with the previous studies. The feasibility of the proposed model is demonstrated by simulations under different motion states.

Two-modal resonance control with an encapsulated nonlinear energy sink

Nonlinear energy sink (NES) has received extensive attention as a broadband vibration control strategy. Generally, the vibration of the NES vibrator is unconstrained. Moreover, the vibration suppression is for a single modal resonance. In this paper, an encapsulated NES (E-NES) is proposed to control two resonance modes. Meanwhile, a strategy for optimizing two-resonance mode is also proposed. In practical applications, the vibration of the NES vibrator needs to be limited within a certain range. The vibration of the vibrator of the proposed E-NES is limited by a spring and a damping capacity with a gap. A dynamic model with non-smooth nonlinearity of a two-degree-of-freedom (two-DOF) system coupled with an E-NES is established. The non-smooth model is fitted by a hyperbolic tangent function. Then the continuous nonlinear model is analyzed approximately analytically and simulated numerically. The experimental platform is designed to verify the effectiveness of theoretical research. Moreover, the E-NES parameters are optimized by the differential evolution algorithm. The results show that the optimized design of the E-NES can achieve almost 90% suppression efficiency for both resonance peaks. On the whole, the proposed NES is a promising vibration control strategy due to its lower cost and high efficiency. Moreover, this NES can be accommodated in a box, which presents strong convenience in engineering.

Power flow analysis of integrated energy microgrid considering non‐smooth characteristics

With the development of electric energy substitution technologies such as electric auxiliary heating, the relationship between electric energy and thermal energy is getting closer, and it is of great significance to in-depth study of integrated energy micro-grids containing electricity and heat. However, the integrated energy microgrid power flow (PF) model has non-smooth characteristics such as distributed generations (DGs) limits, converter stations limits and the unknown direction of pipeline mass flow, which can easily lead to the failure of PF convergence. This paper establishes an integrated energy microgrid model including three-phase unbalanced AC/DC hybrid microgrid and heating system. Firstly, the correctness of the converter station model is verified by electromagnetic transient simulation, and then the non-smooth constraint models considering DG and converter station's reactive power limits and pipeline mass flow direction unknown are established based on complementary constraints, and the non-smooth constraints are smoothed by Fischer-Burmeister (FB) function. Finally, the integrated energy microgrid is used for PF analysis, and the results verify the rationality of the model and the effectiveness of the proposed non-smooth characteristics treatment method.

Bionic Nonsmooth Drag Reduction Mathematical Model Construction and Subsoiling Verification.

In this study, a bionic nonsmooth drag-reducing surface design method was proposed; a mathematical model was developed to obtain the relationship between the altitude of the nonsmooth drag-reducing surface bulges and the spacing of two bulges, as well as the speed of movement, based on which two subsoiler shovel tips were designed and verified on field experiments. The mechanism of nonsmooth surface drag reduction in soil was analyzed, inspired by the efficient digging patterns of antlions. The nonsmooth surface morphology of the antlion was acquired by scanning electron microscopy, and a movement model of the nonsmooth surface in soil was developed, deriving that the altitude of the nonsmooth drag-reducing surface bulge is proportional to the square of the distance between two bulges and inversely proportional to the square of the movement speed. A flat subsoiler shovel tip and a curved tip were designed by applying this model, and the smooth subsoiler shovel tips and the pangolin scale bionic tips were used as controls, respectively. The effect of the model-designed subsoilers on drag reduction was verified by subsoiling experiments in the field. The results showed that the resistance of the model-designed curved subsoiler was the lowest, the resistance of the pangolin scale bionic subsoiler was moderate, and the resistance of the smooth surface subsoiler was the highest; the resistance of the curved subsoiler was less than the flat subsoilers; the resistance reduction rate of the model-designed curved subsoiler was 24.6% to 33.7% at different depths. The nonsmooth drag reduction model established in this study can be applied not only to the design of subsoilers but also to the design of nonsmooth drag reduction surfaces of other soil contacting parts.

Non-smooth Modeling and Variable Structure Control of a Class of Hybrid Dynamical Systems

In this work, we propose a modeling formulation and controller design for a class of hybrid dynamical systems. In this formulation, a switching dynamical system is modeled as a dynamical system with discontinuous right hand side. More specifically, the system is transformed to a nonlinear system with discontinuous nonlinearities. Then, a synthesis of feedback linearization and sliding mode control is employed for output tracking control problem. Application and implementation of this approach is illustrated via a chemical process example.

An Optimization-Based Meta-Learning Model for MRI Reconstruction with Diverse Dataset

This work aims at developing a generalizable Magnetic Resonance Imaging (MRI) reconstruction method in the meta-learning framework. Specifically, we develop a deep reconstruction network induced by a learnable optimization algorithm (LOA) to solve the nonconvex nonsmooth variational model of MRI image reconstruction. In this model, the nonconvex nonsmooth regularization term is parameterized as a structured deep network where the network parameters can be learned from data. We partition these network parameters into two parts: a task-invariant part for the common feature encoder component of the regularization, and a task-specific part to account for the variations in the heterogeneous training and testing data. We train the regularization parameters in a bilevel optimization framework which significantly improves the robustness of the training process and the generalization ability of the network. We conduct a series of numerical experiments using heterogeneous MRI data sets with various undersampling patterns, ratios, and acquisition settings. The experimental results show that our network yields greatly improved reconstruction quality over existing methods and can generalize well to new reconstruction problems whose undersampling patterns/trajectories are not present during training.