Abstract: Under mild conditions, it can be induced from the Karush-Kuhn-Tucker condition that the Pareto set, in the decision space, of a continuous multiobjective optimization problem is a piecewise continuous (m - 1)-D manifold, where m is the number of objectives. Based on this regularity property, we propose a regularity model-based multiobjective estimation of distribution algorithm (RM-MEDA) for continuous multiobjective optimization problems with variable linkages. At each generation, the proposed algorithm models a promising area in the decision space by a probability distribution whose centroid is a (m - 1)-D piecewise continuous manifold. The local principal component analysis algorithm is used for building such a model. New trial solutions are sampled from the model thus built. A nondominated sorting-based selection is used for choosing solutions for the next generation. Systematic experiments have shown that, overall, RM-MEDA outperforms three other state-of-the-art algorithms, namely, GDE3, PCX-NSGA-II, and MIDEA, on a set of test instances with variable linkages. We have demonstrated that, compared with GDE3, RM-MEDA is not sensitive to algorithmic parameters, and has good scalability to the number of decision variables in the case of nonlinear variable linkages. A few shortcomings of RM-MEDA have also been identified and discussed in this paper.

Abstract: Preparing the books to read every day is enjoyable for many people. However, there are still many people who also don't like reading. This is a problem. But, when you can support others to start reading, it will be better. One of the books that can be recommended for new readers is piecewise smooth dynamical systems. This book is not kind of difficult book to read. It can be read and understand by the new readers.

Abstract: Presented is a method of smooth sliding mode control design to provide for an asymptotic second-order sliding mode on the selected sliding surface. The control law is a nonlinear dynamic feedback that in absence of unknown disturbances provides for an asymptotic second-order sliding mode. Application of the second-order disturbance observer in a combination with the proposed continuous control law practically gives the second-order sliding accuracy in presence of unknown disturbances and discrete-time control update. The piecewise constant control feedback is “smooth” in the sense that its derivative numerically taken at sampling rate does not contain high frequency components. A numerical example is presented.

Abstract: A modification of the typical minimum-structure inver-sion algorithm is presented that generates blocky, piecewise-constant earth models. Such models are often more consistent with our real or perceived knowledge of the subsurface than the fuzzy, smeared-out models produced by current minimum-structure inversions. The modified algorithm uses l1 -type measures in the measure of model structure instead of the traditional sum-of-squares, or l2 , measure. An iteratively reweighted least-squares procedure is used to deal with the nonlinearity introduced by the non- l2 measure. Also, and of note here, diagonal finite differences are included in the measure of model structure. This enables dipping interfaces to be formed. The modified algorithm retains the benefits of the minimum-structure style of inversion — namely, reliability, robustness, and minimal artifacts in the constructed model. Two examples are given: the 2D inversion of synthetic magnetotelluric data and the 3D inversion of gravity data from the Ovo...

Abstract: We consider the restoration of piecewise constant images where the number of the regions and their values are not fixed in advance, with a good difference of piecewise constant values between neighboring regions, from noisy data obtained at the output of a linear operator (e.g., a blurring kernel or a Radon transform). Thus we also address the generic problem of unsupervised segmentation in the context of linear inverse problems. The segmentation and the restoration tasks are solved jointly by minimizing an objective function (an energy) composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. The pertinence of such an energy is ensured by the analytical properties of its minimizers. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonsmooth nonconvex minimization. Indeed, the existing methods are unsatisfactory since they (implicitly or explicitly) involve a smooth approximation of the regularization term and often get stuck in shallow local minima. The goal of this paper is to design a method that efficiently handles the nonsmooth nonconvex minimization. More precisely, we propose a continuation method where one tracks the minimizers along a sequence of approximate nonsmooth energies $\{J_\eps\}$, the first of which being strictly convex and the last one the original energy to minimize. Knowing the importance of the nonsmoothness of the regularization term for the segmentation task, each $J_\eps$ is nonsmooth and is expressed as the sum of an $\ell_1$ regularization term and a smooth nonconvex function. Furthermore, the local minimization of each $J_{\eps}$ is reformulated as the minimization of a smooth function subject to a set of linear constraints. The latter problem is solved by the modified primal-dual interior point method, which guarantees the descent direction at each step. Experimental results are presented and show the effectiveness and the efficiency of the proposed method. Comparison with simulated annealing methods further shows the advantage of our method.

Abstract: The cesam code is a consistent set of programs and routines which perform calculations of 1D quasi-hydrostatic stellar evolution including microscopic diffusion of chemical species and diffusion of angular momentum. The solution of the quasi-static equilibrium is performed by a collocation method based on piecewise polynomials approximations projected on a B-spline basis; that allows stable and robust calculations, and the exact restitution of the solution, not only at grid points, even for the discontinuous variables. Other advantages are the monitoring by only one parameter of the accuracy and its improvement by super-convergence. An automatic mesh refinement has been designed for adjusting the localisations of grid points according to the changes of unknowns. For standard models, the evolution of the chemical composition is solved by stiffly stable schemes of orders up to four; in the convection zones mixing and evolution of chemical are simultaneous. The solution of the diffusion equation employs the Galerkin finite elements scheme; the mixing of chemicals is then performed by a strong turbulent diffusion. A precise restoration of the atmosphere is allowed for.

Abstract: Due to dynamic changes of supply voltage, envelope-tracking (ET) power amplifiers (PAs) exhibit very distinct characteristics in different power regions. It is very difficult to compensate the distortion induced by these amplifiers by employing conventional digital predistortion techniques. In this paper, by introducing a new piecewise Volterra model based on a vector threshold decomposition technique, we first set several thresholds in the input power level according to the PA characteristics, and decompose the input complex envelope signal into several sub-signals by using these thresholds. We then process each sub-signal separately by employing the dynamic deviation reduction-based Volterra series, and finally recombine them together to produce the predistorted output. Experimental results show that by using this new decomposed piecewise digital predistorter model, the distinct characteristics of the ET system at different signal power levels can be accurately modeled, and thus, the distortion, including both static nonlinearities and memory effects, caused by the amplifier nonlinear behavior can be effectively compensated.

Abstract: We relax the monotonicity requirement of Lyapunov?s theorem to enlarge the class of functions that can provide certificates of stability. To this end, we propose two new sufficient conditions for global asymptotic stability that allow the Lyapunov functions to increase locally, but guarantee an average decrease every few steps. Our first condition is non-convex, but allows an intuitive interpretation. The second condition, which includes the first one as a special case, is convex and can be cast as a semidefinite program. We show that when non-monotonic Lyapunov functions exist, one can construct a more complicated function that decreases monotonically. We demonstrate the strength of our methodology over standard Lyapunov theory through examples from three different classes of dynamical systems. First, we consider polynomial dynamics where we utilize techniques from sum-of-squares programming. Second, analysis of piecewise affine systems is performed. Here, connections to the method of piecewise quadratic Lyapunov functions are made. Finally, we examine systems with arbitrary switching between a finite set of matrices. It will be shown that tighter bounds on the joint spectral radius can be obtained using our technique.

Abstract: This article describes how to use smooth solvers for simulation of a class of piecewise smooth systems of ordinary differential equations, called Filippov systems, with discontinuous vector fields. In these systems constrained motion along a discontinuity surface (so-called sliding) is possible and requires special treatment numerically. The introduced algorithms are based on an extension to Filippov's method to stabilise the sliding flow together with accurate detection of the entrance and exit of sliding regions. The methods are implemented in a general way in MATLAB and sufficient details are given to enable users to modify the code to run on arbitrary examples. Here, the method is used to compute the dynamics of three example systems, a dry-friction oscillator, a relay feedback system and a model of an oil well drill-string.

Abstract: We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of two-dimensional maps with uniformly bounded second derivative, but we are confident that the present approach can be successful in much greater generality (we hope including higher dimensional billiards). For the class of systems at hand, we obtain a complete description of the SRB measures, their statistical properties and their stability with respect to many types of perturbations, including deterministic and random perturbations and holes.

Abstract: We prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hattractors.

Abstract: We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations.

Abstract: We solve a steady Darcy–Forchheimer flow in a bounded region by means of piecewise constant velocities and nonconforming piecewise $${\mathbb{P}_1}$$ pressures For the computation, we solve the nonlinearity by an alternating-directions algorithm and we decouple the computation of the velocity from that of the pressure by a gradient algorithm We prove a priori error estimates of the scheme and convergence of the alternating-directions algorithm

Abstract: Empirical mode decomposition (EMD) is a relatively new, data-driven adaptive technique for analyzing multicomponent signals. Although it has many interesting features and often exhibits an ability to decompose nonlinear and nonstationary signals, it lacks a strong theoretical basis which would allow a performance analysis and hence the enhancement and optimization of the method in a systematic way. In this paper, the optimization of EMD is attempted in an alternative manner. Using specially defined multicomponent signals, the optimum outputs can be known in advance and used in the optimization of the EMD-free parameters within a genetic algorithm framework. The contributions of this paper are two-fold. First, the optimization of both the interpolation points and the piecewise interpolating polynomials for the formation of the upper and lower envelopes of the signal reveal important characteristics of the method which where previously hidden. Second, basic directions for the estimates of the optimized parameters are developed, leading to significant performance improvements.

Abstract: This paper studies multiperiodicity and attractivity for a class of recurrent neural networks (RNNs) with unsaturating piecewise linear transfer functions and variable delays. Using local inhibition, conditions for boundedness and global attractivity are established. These conditions allow coexistence of stable and unstable trajectories. Moreover, multiperiodicity of the network is investigated by using local invariant sets. It shows that under some interesting conditions, there exists one periodic trajectory in each invariant set which exponentially attracts all trajectories in that region correspondingly. Simulations are carried out to illustrate the theories.

Abstract: We introduce a family of box splines for efficient, accurate, and smooth reconstruction of volumetric data sampled on the body-centered cubic (BCC) lattice, which is the favorable volumetric sampling pattern due to its optimal spectral sphere packing property. First, we construct a box spline based on the four principal directions of the BCC lattice that allows for a linear C0 reconstruction. Then, the design is extended for higher degrees of continuity. We derive the explicit piecewise polynomial representations of the C0 and C2 box splines that are useful for practical reconstruction applications. We further demonstrate that approximation in the shift-invariant space - generated by BCC-lattice shifts of these box splines - is twice as efficient as using the tensor-product B-spline solutions on the Cartesian lattice (with comparable smoothness and approximation order and with the same sampling density). Practical evidence is provided demonstrating that the BCC lattice not only is generally a more accurate sampling pattern, but also allows for extremely efficient reconstructions that outperform tensor-product Cartesian reconstructions.

Abstract: In this paper we continue to consider differential equations with piecewise constant argument of generalized type (EPCAG) [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA 66 (2007) 367–383]. A deviating function of a new form is introduced. The linear and quasilinear systems are under discussion. The structure of the sets of solutions is specified. Necessary and sufficient conditions for stability of the zero solution are obtained. Our approach can be fruitfully applied to the investigation of stability, oscillations, controllability and many other problems of EPCAG. Some of the results were announced at The International Conference on Hybrid Systems and Applications, University of Louisiana, Lafayette, 2006.

Abstract: This paper is concerned with stability analysis and H infin decentralized control of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions. The fuzzy large-scale systems consist of J interconnected discrete-time Takagi-Sugeno (T-S) fuzzy subsystems, and the stability analysis is based on Lyapunov functions that are piecewise quadratic. It is shown that the stability of the discrete-time fuzzy large-scale systems can be established if a piecewise quadratic Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. The H infin controllers are also designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions. It is demonstrated via numerical examples that the stability and controller synthesis results based on the piecewise quadratic Lyapunov functions are less conservative than those based on the common quadratic Lyapunov functions.

Abstract: We present algorithms for automated macromodeling of nonlinear mixed-signal system blocks. A key feature of our methods is that they automate the generation of general-purpose macromodels that are suitable for a wide range of time- and frequency-domain analyses important in mixed-signal design flows. In our approach, a nonlinear circuit or system is approximated using piecewise-polynomial (PWP) representations. Each polynomial system is reduced to a smaller one via weakly nonlinear polynomial model-reduction methods. Our approach, dubbed PWP, generalizes recent trajectory-based piecewise-linear approaches and ties them with polynomial-based model-order reduction, which inherently captures stronger nonlinearities within each region. PWP-generated macromodels not only reproduce small-signal distortion and intermodulation properties well but also retain fidelity in large-signal transient analyses. The reduced models can be used as drop-in replacements for large subsystems to achieve fast system-level simulation using a variety of time- and frequency-domain analyses (such as dc, ac, transient, harmonic balance, etc.). For the polynomial reduction step within PWP, we also present a novel technique [dubbed multiple pseudoinput (MPI)] that combines concepts from proper orthogonal decomposition with Krylov-subspace projection. We illustrate the use of PWP and MPI with several examples (including op-amps and I/O buffers) and provide important implementation details. Our experiments indicate that it is easy to obtain speedups of about an order of magnitude with push-button nonlinear macromodel-generation algorithms.

Abstract: Fundamentals of Conventional and Piecewise Constant Systems Preliminary Theorems and Techniques for Analysis of Nonlinear Piecewise Constant Systems Piecewise Constant Dynamical Systems and Their Behavior Analytical and Semi-Analytical Solution Development with Piecewise Constant Arguments Numerical and Improved Semi-Analytical Approaches Implementing Piecewise Constant Arguments Application of P-T Method on Multi-Degree-of-Freedom Nonlinear Dynamic Systems Periodicity-Ratio and Its Application in Diagnosing Irregularities of Nonlinear Systems.

Abstract: In this paper we consider a multi-threshold compound Poisson risk model. A piecewise integro-differential equation is derived for the Gerber–Shiu discounted penalty function. We then provide a recursive approach to obtain general solutions to the integro-differential equation and its generalizations. Finally, we use the probability of ruin to illustrate the applicability of the approach.

Abstract: airfoil types: NACA 4415 and WTEA-TE1, as well as for 17 modified WTEA-TE1 airfoil shapes, obtained by displacing the flexible wing upper surface using a single point control mechanism. The second derivative of the pressure distribution is calculated, using two interpolation schemes: piecewise cubic Hermite interpolating polynomial and Spline, from which it is determined that transition may be identified as the location of maximum curvature in the pressure distribution. The results of this method are validated using the well-known XFoil code, which is used to theoretically calculate the transition point position. Advantages of this new method in the real-time control of the location of the transition point are presented.

Abstract: We consider a newsvendor problem with partially observed Markovian demand. Demand is observed if it is less than the inventory. Otherwise, only the event that it is larger than or equal to the inventory is observed. These observations are used to update the demand distribution from one period to the next. The state of the resulting dynamic programming equation is the current demand distribution, which is generally infinite dimensional. We use unnormalized probabilities to convert the nonlinear state transition equation to a linear one. This helps in proving the existence of an optimal feedback ordering policy. So as to learn more about the demand, the optimal order is set to exceed the myopic optimal order. The optimal cost decreases as the demand distribution decreases in the hazard rate order. In a special case with finitely many demand values, we characterize a near-optimal solution by establishing that the value function is piecewise linear.

Abstract: It is well known that high-order finite-difference methods may become unstable due to the presence of boundaries and the imposition of boundary conditions. For uniform grids, Gustafsson, Kreiss, and Sundstrom theory and the summation-by-parts method provide sufficient conditions for stability. For non-uniform grids, clustering of nodes close to the boundaries improves the stability of the resulting finite-difference operator. Several heuristic explanations exist for the goodness of the clustering, and attempts have been made to link it to the Runge phenomenon present in polynomial interpolations of high degree. By following the philosophy behind the Chebyshev polynomials, a non-uniform grid for piecewise polynomial interpolations of degree q⩽N is introduced in this paper, where N + 1 is the total number of grid nodes. It is shown that when q=N, this polynomial interpolation coincides with the Chebyshev interpolation, and the resulting finite-difference schemes are equivalent to Chebyshev collocation methods. Finally, test cases are run showing how stability and correct transient behaviours are achieved for any degree q