Piecewise Smooth Functions Research Articles

Nonlinear approximation spaces for inverse problems

This paper is concerned with the ubiquitous inverse problem of recovering an unknown function u from finitely many measurements, possibly affected by noise. In recent years, inversion methods based on linear approximation spaces were introduced in [1, 2] with certified recovery bounds. It is however known that linear spaces become ineffective for approximating simple and relevant families of functions, such as piecewise smooth functions, that typically occur in hyperbolic PDEs (shocks) or images (edges). For such families, nonlinear spaces [3] are known to significantly improve the approximation performance. The first contribution of this paper is to provide with certified recovery bounds for inversion procedures based on nonlinear approximation spaces. The second contribution is the application of this framework to the recovery of general bidimensional shapes from cell-average data. We also discuss how the application of our results to n-term approximation relates to classical results in compressed sensing.

Stability for a Non-Smooth Filippov Ratio-Dependent Predator-Prey System through a Smooth Lyapunov Function

For nonsmooth Filippov systems, the stability of the system is assumed to be proved by nonsmooth Lyapunov functions, such as piecewise smooth Lyapunov functions. This extension was based on the Filippov solution and Clarke generalized gradient. However, it is difficult to estimate the gradient of a non-smooth Lyapunov function. In some cases, the nonsmooth system can be divided into continuous and discontinuous components. If the Lebesgue measure of the discontinuous components is zero, the smooth Lyapunov function can be utilized to prove the stability of the system owing to the inner product of the gradient of the Lyapunov function of the discontinuous components being zero. In this paper, we apply the smooth Lyapunov function to prove the stability of the nonsmooth ratio-dependent predator-prey system. In contrast to the existing literature, in this paper, although the system is divided into continuous and discontinuous components, the inner product of the gradient of the Lyapunov function of the discontinuous part is not zero but negative. In the proof of stability, the negative value condition is stricter than the zero-value condition. This proof method only needs to construct a smooth Lyapunov function, which is simpler than a non-smooth Lyapunov function or other methods.

An SPH stress correction algorithm based on the quartic piecewise smooth kernel function

<p indent="0mm">Smoothed particle hydrodynamics (SPH) is a popular adaptive Lagrangian mesh-free particle method. For the tensile instability problem in traditional SPH, according to the sufficient condition of tensile instability, a new quartic piecewise smooth kernel function with non-negative second derivative is proposed, and an SPH stress correction algorithm is obtained in this study. The numerical results show that the modified SPH algorithm can effectively describe the formation processes of 2D and 3D vdW liquid drops. Then, based on the modified SPH algorithm, the effects of different physical parameters on the particle distribution of 2D circular droplets are further discussed. The results show that, with the appropriate temperature or initial particle spacing, the boundary particles of circular droplets are neither clustered nor scattered, and as the fluid temperature or initial particle spacing increases, the boundary particles will be scattered. Conversely, the particle distribution of circular droplets becomes more uniform with the increase in the smooth length proportion coefficient.

On nonlinear metric spaces of functions of bounded variation

In the first part of the paper, the nonlinear metric space $\langle\overline{\rm G}^\infty[a,b],d\rangle$ is defined and studied. It consists of functions defined on the interval $[a,b]$ and taking the values in the extended numeric axis $\overline{\mathbb R}$. For any $x\in\overline{\rm G}^\infty[a,b]$ and $t\in(a,b)$ there are limit numbers $x(t-0),x(t+0) \in\overline{\mathbb R}$ (and numbers $x(a+0),x(b-0)\in\overline{\mathbb R}$). The completeness of the space is proved. It is the closure of the space of step functions in the metric $d$. In the second part of the work, the nonlinear space ${\rm RL}[a,b]$ is defined and studied. Every piecewise smooth function defined on $[a,b]$ is contained in ${\rm RL}[a,b]$. Every function $x\in{\rm RL}[a,b]$ has bounded variation. All one-sided derivatives (with values in the metric space $\langle\overline{\mathbb R},\varrho\rangle$) are defined for it. The function of left-hand derivatives is continuous on the left, and the function of right-hand derivatives is continuous on the right. Both functions extended to the entire interval $[a,b]$ belong to the space $\overline{\rm G}^\infty[a,b]$. In the final part of the paper, two subspaces of the space ${\rm RL}[a,b]$ are defined and studied. In subspaces, promising formulations for the simplest variational problems are stated and discussed.

Vibration of Complex Euler–Bernoulli and Timoshenko–Ehrenfest Beams Through Affine GPSFs

Abstract Vibrations of thin and thick beams containing internal complexities are analyzed through generalized bases made of global piecewise-smooth functions (GPSFs). Such functional bases allow us to globally analyze multiple domains as if these latter were only one, such that a unified formulation can be used for different mechanical systems. Such bases were initially introduced to model a specific part of stress and displacement components through the thickness of multi-layered plates; subsequent extensions were introduced in the literature to allow the modeling of thin-walled beams and plates. However, in these latter cases, certain analytical difficulties were experienced when inner boundary conditions needed to be englobed into the GPSFs; in this work, such mentioned difficulties are successfully overcome through certain affine transformations which allow the analyses of vibrating complex beam systems through a straightforward analytical procedure. The complex mechanical components under investigation are Euler and/or Timoshenko models containing inner complexities (stepped beams, concentrated mass or stiffness, internal constraints, etc.). The ability of the models herein analyzed is shown through the comparison of the resulting solutions to exact counterparts, if existing, or to finite elements solutions.

Design of Saturating State Feedback With Sign-Indefinite Quadratic Forms

In this article, we propose a novel class of piecewise smooth Lyapunov functions leading to linear-matrix-inequality-based stability/performance analysis and control design for linear systems with saturating inputs. We provide conditions for global properties, and also conditions for local properties and guaranteed estimates of the basin of attraction. The backbone of our result consists in using quadratic forms with constant matrices that are not necessarily sign definite, thereby providing additional degrees of freedom. Using generalized sector conditions involving the dead-zone nonlinearity and its derivative, we formulate convex optimization conditions to verify their positivity in the region of interest. Several numerical examples with connections to existing results illustrate the potential behind our novel construction.

Smoothing approximations for piecewise smooth functions: A probabilistic approach

&lt;p style='text-indent:20px;'&gt;In this article, we present a new approach to construct smoothing approximations for piecewise smooth functions. This approach proposes to formulate any piecewise smooth function as the expectation of a random variable. Based on this formulation, we show that smoothing all elements of a defined space of piecewise smooth functions is equivalent to smooth a single probability distribution. Furthermore, we propose to use the Boltzmann distribution as a smoothing approximation for this probability distribution. Moreover, we present the theoretical results, error estimates, and some numerical examples for this new smoothing method in both one-dimensional and multiple-dimensional cases.&lt;/p&gt;

Image restoration: Structured low rank matrix framework for piecewise smooth functions and beyond

Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the total generalized variation to use sparse higher order derivatives, we derive that the Fourier samples of higher order derivatives satisfy an annihilation relation, resulting in a low rank multi-fold Hankel matrix. We further observe that the SVD of a low rank Hankel matrix corresponds to a tight wavelet frame system which can represent the image with sparse coefficients. Based on this observation, we also propose a wavelet frame analysis approach based continuous domain regularization model for the piecewise smooth image restoration. Finally, numerical results on image restoration tasks are presented as a proof-of-concept study to demonstrate that the proposed approach is compared favorably against several popular discrete regularization approaches and structured low rank matrix approaches.

Спектральные свойства дифференциального оператора четного порядка с разрывной весовой функцией

This article proposes a new method for studying differential operators with a discontinuous weight function. It is assumed that the potential of the operator is a piecewise smooth function on the segment of the operator definition. The conditions of «conjugation» at the point of discontinuity of the weight function are required. The spectral properties of a differential operator defined on a finite segment with separated boundary conditions are studied. The asymptotics of the fundamental system of solutions of the corresponding differential equation for large values of the spectral parameter is obtained. With the help of this asymptotics, the «conjugation» conditions of the differential operator in question are studied. The boundary conditions of the operator under study are investigated. As a result, we obtain an equation for the eigenvalues of the operator, which is an entire function. The indicator diagram of the eigenvalue equation, which is a regular polygon, is studied. In various sectors of the indicator diagram, the asymptotics of the eigenvalues of the investigated differential operator is found. The formula for the first regularized trace of this operator by using the found asymptotics of the eigenvalues by the Lidsky–Sadovnichy method is obtained. In the case of the passage to the limit, the resulting formula leads to the trace formula for the classical operator with a smooth potential and constant weight function.

Dynamic analysis of a SIV Filippov system with media coverage and protective measures

&lt;abstract&gt;&lt;p&gt;This study aims to analyze a class of SIV systems considering the transmission rate influenced by media coverage and protective measures, in which the transmission rate is represented by a piecewise-smooth function. Firstly, for the SIV Filippov system, we take the dynamic behaviors of two subsystems into consideration, and obtain the basic reproduction number and the equilibria of the subsystems respectively. Secondly, based on the Filippov convex method, we calculate the sliding domain and the sliding mode equation, and further analyze the global dynamic behaviors of the system, through which we verify that there is no closed orbit in the system. Furthermore, we prove the global asymptotical stability of the disease-free equilibrium, two real equilibria, and the pseudo-equilibrium under certain conditions. The results demonstrate that the threshold value, the protective measures, and the media coverage could affect the number of infected individuals and the final scale of the disease. To prevent the spread of the disease, it is necessary to select an appropriate threshold and take applicable protective measures combined with media coverage. Lastly, we verify the validity of the results by numerical simulations.&lt;/p&gt;&lt;/abstract&gt;

Lowest-order equivalent nonstandard finite element methods for biharmonic plates

The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the C0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side F ∈ H−2(Ω) replaced by F ○ (JIM) and then are quasi-optimal in their respective discrete norms. The smoother JIM is defined for a piecewise smooth input function by a (generalized) Morley interpolation IM followed by a companion operator J. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj, ESAIM: M2AN 49 (2015) 977–990.] without data oscillations. This paper extends the work [Veeser and Zanotti, SIAM J. Numer. Anal. 56 (2018) 1621–1642] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.

Multi-layer Perceptron Estimator for the Total Variation Bounded Constant in Limiters for Discontinuous Galerkin Methods

The discontinuous Galerkin (DG) method is widely used in numerical solution of partial differential equations, especially for hyperbolic equations. However, for problems containing strong shocks, the DG method often needs to be supplemented by a limiter to control spurious oscillations and to ensure nonlinear stability. The total variation bounded (TVB) limiter is a popular choice and can maintain the original high order accuracy of the DG scheme in smooth regions and keep a sharp and non-oscillatory discontinuity transition, when a certain TVB constant M is chosen adequately. For scalar conservation laws, suitable choice of this constant M can be based on solid mathematical analysis. However, for nonlinear hyperbolic systems, there is no rigorous mathematical guiding principle for the determination of this constant, and numerical experiments often use ad hoc choices based on experience and through trial and error. In this paper, we develop a TVB constant artificial neural network (ANN) based estimator by constructing a multi-layer perceptron (MLP) model. We generate the training data set by constructing piecewise smooth functions containing local maxima, local minima, and discontinuities. By using the supervised learning strategy, the MLP model is trained offline. The proposed method gives the TVB constant M with robust performance to capture sharp and non-oscillatory shock transitions while maintaining the original high order accuracy in smooth regions. Numerical results using this new estimator in the TVB limiter for DG methods in one and two dimensions are given, and its performance is compared with the classical ad hoc choices of this TVB constant.

Adaptive estimation of multivariate piecewise polynomials and bounded variation functions by optimal decision trees

Proposed by Donoho (Ann. Statist. 25 (1997) 1870–1911), Dyadic CART is a nonparametric regression method which computes a globally optimal dyadic decision tree and fits piecewise constant functions in two dimensions. In this article, we define and study Dyadic CART and a closely related estimator, namely Optimal Regression Tree (ORT), in the context of estimating piecewise smooth functions in general dimensions in the fixed design setup. More precisely, these optimal decision tree estimators fit piecewise polynomials of any given degree. Like Dyadic CART in two dimensions, we reason that these estimators can also be computed in polynomial time in the sample size N via dynamic programming. We prove oracle inequalities for the finite sample risk of Dyadic CART and ORT, which imply tight risk bounds for several function classes of interest. First, they imply that the finite sample risk of ORT of order r≥0 is always bounded by CklogN N whenever the regression function is piecewise polynomial of degree r on some reasonably regular axis aligned rectangular partition of the domain with at most k rectangles. Beyond the univariate case, such guarantees are scarcely available in the literature for computationally efficient estimators. Second, our oracle inequalities uncover minimax rate optimality and adaptivity of the Dyadic CART estimator for function spaces with bounded variation. We consider two function spaces of recent interest where multivariate total variation denoising and univariate trend filtering are the state of the art methods. We show that Dyadic CART enjoys certain advantages over these estimators while still maintaining all their known guarantees.

A two-stage approximation strategy for piecewise smooth functions in two and three dimensions

Abstract Given values of a piecewise smooth function $f$ on a square grid within a domain $[0,1]^d$, $d=2,3$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. The insight used here is that the behavior near the boundaries, or near a singularity curve, is fully characterized and identified by the values of certain differences of the data across the boundary and across the singularity curve. We refer to these values as the signature of $f$. In this paper, we aim at using these values in order to define the approximation. That is, we look for an approximation whose signature is matched to the signature of $f$. Given function data on a grid, assuming the function is piecewise smooth, first, the singularity structure of the function is identified. For example, in the two-dimensional case, we find an approximation to the curves separating between smooth segments of $f$. Secondly, simultaneously, we find the approximations to the different segments of $f$. A system of equations derived from the principle of matching the signature of the approximation and the function with respect to the given grid defines a first stage approximation. A second stage improved approximation is constructed using a global approximation to the error obtained in the first stage approximation.

Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs

In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required.

SPECIAL SPLINE APPROXIMATION FOR THE SOLUTION OF THE NON-STATIONARY 3-D MASS TRANSFER PROBLEM

In this paper we consider the conservative averaging method (CAM) with special spline approximation for solving the non-stationary 3-D mass transfer problem. The special hyperbolic type spline, which interpolates the middle integral values of piece-wise smooth function is used. With the help of these splines the initial-boundary value problem (IBVP) of mathematical physics in 3-D domain with respect to one coordinate is reduced to problems for system of equations in 2-D domain. This procedure allows reduce also the 2-D problem to a 1-D problem and thus the solution of the approximated problem can be obtained analytically. The accuracy of the approximated solution for the special 1-D IBVP is compared with the exact solution of the studied problem obtained with the Fourier series method. The numerical solution is compared with the spline solution. The above-mentioned method has extensive physical applications, related to mass and heat transfer problems in 3-D domains.

Generalized A-GPSFS and modeling of vibrating elements with internal complexities

Mechanical components containing internal complexities in free vibration mode are analyzed through generalized bases made of global piecewise smooth functions. Such bases introduced in 2002, as GPSFs, were made adaptive in 2015 to reduce the computational efforts (A-GPSFs) while, a recent study in 2018 used supplementary analytical conditions along with A-GPSFs to study free vibrations of thin-walled beams and plates. In this paper, generalized bases of A-GPSFs are introduced (i) in a self-contained manner, id est, without needing any supplementary analytical conditions and even (ii) extending the possibility of modeling mechanical components containing internal jumps. It is shown how such generalized A-GPSFs can be used to model continuous functions which have different degrees of smoothness; even discontinuous functions (i.e. containing jumps) can be modeled and always without involving significant approximations or letting rise to any Gibbs phenomenon. Thus, a free vibration analysis of mechanical components including internal complexities of engineering interest becomes part of a simple unified procedure based on a global approximation where unknown eigenvalues and eigenfunctions can be singled out only through generalized bases of A-GPSFs; the results of previous formulations are then obtained as particular cases. The efficiency and ability of the proposed models result from the comparison between calculated eigen-parameters and those of other models presented in the open literature.

Single image restoration through [formula omitted]-relaxed truncated [formula omitted] analysis-based sparse optimization in tight frames

Image restoration problems, i.e., recovery of an original high-quality image from the degraded observation, arise in various science and engineer areas. Over the past decades, the framelet-based methods are particularly investigated and adopted, owing to the excellent ability of sparse approximating the piecewise-smooth functions such as natural images. In this paper, we propose a novel tight frame-based ℓ2-relaxed truncated ℓ0 analysis-sparsity model that simultaneously exploiting the sparsity and support priors. The resulting nonconvex nonsmooth optimization problem is addressed by using the proposed proximal alternating adaptive hard thresholding (PAAHT) method. We also proved that the sequence generated by the proposed algorithm sublinearly converges. Numerical experiments on several typical image restoration problems demonstrate that the proposed method is more effective than the standard sparsity-inducing algorithms and outperforms several state-of-the-art methods in both objective and perceptual quality.

On Inf-Convolution-Based Robust Practical Stabilization Under Computational Uncertainty

This work is concerned with practical stabilization of nonlinear systems by means of inf-convolution-based sample-and-hold control. It is a fairly general stabilization technique based on a generic non-smooth control Lyapunov function (CLF) and robust to actuator uncertainty, measurement noise, etc. The stabilization technique itself involves computation of descent directions of the CLF. It turns out that non-exact realization of this computation leads not just to a quantitative, but also qualitative obstruction in the sense that the result of the computation might fail to be a descent direction altogether and there is also no straightforward way to relate it to a descent direction. Disturbance, primarily measurement noise, complicate the described issue even more. This work suggests a modified inf-convolution-based control that is robust w. r. t. system and measurement noise, as well as computational uncertainty. The assumptions on the CLF are mild, as, e. g., any piece-wise smooth function, which often results from a numerical LF/CLF construction, satisfies them. A computational study with a three-wheel robot with dynamical steering and throttle under various tolerances w. r. t. computational uncertainty demonstrates the relevance of the addressed issue and the necessity of modifying the used stabilization technique. Similar analyses may be extended to other methods which involve optimization, such as Dini aiming or steepest descent.

Determining the theoretical trajectory of the centre of tractor’s mass when turning in a combined way

One of the ways to increase the stability of movement and improve the manoeuvrability and controllability of a wheeled universal row-crop tractor in curvilinear motion is the choice of a rational turning method. To solve the tasks, the authors proposed a combined method of turning the tractor, which consists in crabbing in the entry into the rotation, after which the rear wheels automatically return to the neutral position and then the direction of movement in the established rotation and exit from the rotation uses front steering wheels. The article developed a methodology for determining the theoretical trajectory of the centre of mass of a wheeled tractor with a combined method of rotation for the three distinct sections of the movement included in it. Based on the conditions of smoothness and continuity, we glued together three characteristic sections of the curvilinear trajectory of the centre of mass and obtained a continuous non-linear piecewise smooth function of the investigated combined rotation method. It is possible to apply the research results for theoretical calculations of the parameters for the curvilinear motion of tractors, as well as other wheeled vehicles, that allows choosing rational laws to change the speed and control modes of turning the wheels, to identify the patterns of change for ensuring the necessary stability, manoeuvrability and controllability and confirming effectiveness in comparison with known and previously investigated laws.