Showing papers on "Piecewise published in 2018"
15 Oct 2018
TL;DR: A general framework for privacy-preserving machine learning is designed and implemented and used to obtain new solutions for training linear regression, logistic regression and neural network models and to design variants of each building block that are secure against malicious adversaries who deviate arbitrarily.
Abstract: Machine learning is widely used to produce models for a range of applications and is increasingly offered as a service by major technology companies. However, the required massive data collection raises privacy concerns during both training and prediction stages. In this paper, we design and implement a general framework for privacy-preserving machine learning and use it to obtain new solutions for training linear regression, logistic regression and neural network models. Our protocols are in a three-server model wherein data owners secret share their data among three servers who train and evaluate models on the joint data using three-party computation (3PC). Our main contribution is a new and complete framework ($\textABY ^3$) for efficiently switching back and forth between arithmetic, binary, and Yao 3PC which is of independent interest. Many of the conversions are based on new techniques that are designed and optimized for the first time in this paper. We also propose new techniques for fixed-point multiplication of shared decimal values that extends beyond the three-party case, and customized protocols for evaluating piecewise polynomial functions. We design variants of each building block that is secure against \em malicious adversaries who deviate arbitrarily. We implement our system in C++. Our protocols are up to \em four orders of magnitude faster than the best prior work, hence significantly reducing the gap between privacy-preserving and plaintext training.
451 citations
TL;DR: It is proved that one cannot approximate a general function f∈Eβ(Rd) using neural networks that are less complex than those produced by the construction, which partly explains the benefits of depth for ReLU networks by showing that deep networks are necessary to achieve efficient approximation of (piecewise) smooth functions.
307 citations
21 May 2018
TL;DR: This paper adopts a fast marching-based path searching method to find a path on a velocity field induced by the Euclidean signed distance field (ESDF) of the map, to achieve better time allocation.
Abstract: In this paper, we propose a framework for online quadrotor motion planning for autonomous navigation in unknown environments. Based on the onboard state estimation and environment perception, we adopt a fast marching-based path searching method to find a path on a velocity field induced by the Euclidean signed distance field (ESDF) of the map, to achieve better time allocation. We generate a flight corridor for the quadrotor to travel through by inflating the path against the environment. We represent the trajectory as piecewise Bezier curves by using Bernstein polynomial basis and formulate the trajectory generation problem as typical convex programs. By using Bezier curves, we are able to bound positions and higher order dynamics of the trajectory entirely within safe regions. The proposed motion planning method is integrated into a customized light-weight quadrotor platform and is validated by presenting fully autonomous navigation in unknown cluttered indoor and outdoor environments. We also release our code for trajectory generation as an open-source package.
201 citations
TL;DR: This work presents an alternative model for multisection soft manipulator dynamics is presented based on a discrete Cosserat approach, in which the continuous COSSerat model is discretized by assuming a piecewise constant strain along the soft arm.
Abstract: Nowadays, the most adopted model for the design and control of soft robots is the piecewise constant curvature model, with its consolidated benefits and drawbacks. In this work, an alternative model for multisection soft manipulator dynamics is presented based on a discrete Cosserat approach, in which the continuous Cosserat model is discretized by assuming a piecewise constant strain along the soft arm. As a consequence, the soft manipulator state is described by a finite set of constant strains. This approach has several advantages with respect to the existing models. First, it takes into account shear and torsional deformations, which are both essential to cope with out-of-plane external loads. Furthermore, it inherits desirable geometrical and mechanical properties of the continuous Cosserat model, such as intrinsic parameterization and greater generality. Finally, this approach allows to extend to soft manipulators, the recursive composite-rigid-body and articulated-body algorithms, whose performances are compared through a cantilever beam simulation. The soundness of the model is demonstrated through extensive simulation and experimental results.
171 citations
TL;DR: An adaptive control for vehicle active suspensions with unknown nonlinearities (e.g., nonlinear springs and piecewise dampers) is proposed, such that both the transient and steady-state suspension response are guaranteed.
Abstract: This paper proposes an adaptive control for vehicle active suspensions with unknown nonlinearities (eg, nonlinear springs and piecewise dampers) A prescribed performance function that characterizes the convergence rate, maximum overshoot, and steady-state error is incorporated into the control design to stabilize the vertical and pitch motions, such that both the transient and steady-state suspension response are guaranteed Moreover, a novel adaptive law is used to achieve precise estimation of essential parameters (eg, mass of vehicle body and moment of inertia for pitch motion), where the parameter estimation error is obtained explicitly and then used as a new leakage term Theoretical studies prove the convergence of the estimated parameters, and compare the suggested controller with generic adaptive controllers using the gradient descent and e-modification schemes In addition to motion displacements, dynamic tire loads and suspension travel constraints are also considered Extensive comparative simulations on a dynamic simulator consisting of commercial vehicle simulation software Carsim 81 and MATLAB Simulink are provided to show the efficacy of the proposed control, and to illustrate the improved performance
162 citations
18 Jun 2018
TL;DR: A hybrid l1-l0 decomposition model is proposed that achieves visually compelling results with little halo artifacts, outperforming the state-of-the-art tone mapping algorithms in both subjective and objective evaluations.
Abstract: Tone mapping aims to reproduce a standard dynamic range image from a high dynamic range image with visual information preserved. State-of-the-art tone mapping algorithms mostly decompose an image into a base layer and a detail layer, and process them accordingly. These methods may have problems of halo artifacts and over-enhancement, due to the lack of proper priors imposed on the two layers. In this paper, we propose a hybrid l1-l0 decomposition model to address these problems. Specifically, an l1 sparsity term is imposed on the base layer to model its piecewise smoothness property. An l0 sparsity term is imposed on the detail layer as a structural prior, which leads to piecewise constant effect. We further propose a multiscale tone mapping scheme based on our layer decomposition model. Experiments show that our tone mapping algorithm achieves visually compelling results with little halo artifacts, outperforming the state-of-the-art tone mapping algorithms in both subjective and objective evaluations.
117 citations
TL;DR: A rigorous and accurate non-local (in the oversampled region) upscaling framework based on some recently developed multiscale methods is proposed based on Generalized Multiscale Finite Element Method (GMsFEM) and can provide good accuracy.
105 citations
TL;DR: This paper addresses the problem of delay-dependent robust and reliable static output feedback (SOF) control for uncertain discrete-time piecewise-affine (PWA) systems with time-delay and actuator failure in a singular system setup.
Abstract: This paper addresses the problem of delay-dependent robust and reliable $\mathscr {H}_{\infty }$ static output feedback (SOF) control for uncertain discrete-time piecewise-affine (PWA) systems with time-delay and actuator failure in a singular system setup. The Markov chain is applied to describe the actuator faults behaviors. In particular, by utilizing a system augmentation approach, the conventional closed-loop system is converted into a singular PWA system. By constructing a mode-dependent piecewise Lyapunov–Krasovskii functional, a new $\mathscr {H}_{\infty }$ performance analysis criterion is then presented, where a novel summation inequality and S-procedure are succeedingly employed. Subsequently, thanks to the special structure of the singular system formulation, the PWA SOF controller design is proposed via a convex program. Illustrative examples are finally given to show the efficacy and less conservatism of the presented approach.
101 citations
TL;DR: Using the improved unsaturated nonlinear segment activation function SignReLu, the convergence rate is faster, the gradient vanishing problem is effectively alleviated, and the accuracy of neural network identification is improved obviously.
94 citations
TL;DR: This paper addresses the problem of delay-dependent robust and reliable static output feedback (SOF) control for a class of uncertain discrete-time Takagi–Sugeno fuzzy-affine systems with time-varying delay and actuator faults in a singular system framework.
Abstract: This paper addresses the problem of delay-dependent robust and reliable $\mathscr {H}_{\infty }$ static output feedback (SOF) control for a class of uncertain discrete-time Takagi–Sugeno fuzzy-affine (FA) systems with time-varying delay and actuator faults in a singular system framework. The Markov chain is employed to describe the actuator faults behaviors. In particular, by utilizing a system augmentation approach, the conventional closed-loop system is converted into a singular FA system. By constructing a piecewise-Markovian Lyapunov–Krasovskii functional, a new $\mathscr {H}_{\infty }$ performance analysis criterion is then presented, where a novel summation inequality and S-procedure are succeedingly employed. Subsequently, thanks to the special structure of the singular system formulation, the piecewise-affine SOF controller design is proposed via a convex program. Lastly, illustrative examples are given to show the efficacy and less conservatism of the presented approach.
89 citations
TL;DR: In this article, a system of fractional differential equations involving non-singular Mittag-Leffler kernel is considered and the regularity and existence of its solution is studied.
Abstract: A system of fractional differential equations involving non-singular Mittag-Leffler kernel is considered. This system is transformed to a type of weakly singular integral equations in which the weak singular kernel is involved with both the unknown and known functions. The regularity and existence of its solution is studied. The collocation methods on discontinuous piecewise polynomial space are considered. The convergence and superconvergence properties of the introduced methods are derived on graded meshes. Numerical results provided to show that our theoretical convergence bounds are often sharp and the introduced methods are efficient. Some comparisons and applications are discussed.
TL;DR: In this paper, the authors proposed the first comprehensive treatment of high-dimensional time series factor models with multiple change-points in their second-order structure, using wavelets to estimate the number and locations of changepoints consistently as well as identifying whether they originate in the common or idiosyncratic components.
TL;DR: This paper formulates a simplified traffic smoothing model for guiding movements of connected automated vehicles on a general one-lane highway segment and discovers a set of elegant theoretical properties for the general objective function and the associated constraints in the proposed simplified model.
Abstract: This paper formulates a simplified traffic smoothing model for guiding movements of connected automated vehicles on a general one-lane highway segment. Adapted from the shooting heuristic proposed by Zhou et al. (2017) and Ma et al. (2017), this model confines each vehicle’s trajectory as a piecewise quadratic function with no more than five pieces and lets all trajectories in the same platoon share identical acceleration and deceleration rates. Similar to the shooting heuristic, the proposed simplified model is able to control the overall smoothness of a platoon of connected automated vehicles and approximately optimize traffic performance in terms of fuel efficiency and driving comfort. While the shooting heuristic relies on numerical meta-heuristic algorithms that cannot ensure solution optimality, we discover a set of elegant theoretical properties for the general objective function and the associated constraints in the proposed simplified model, and consequentially propose an efficient analytical algorithm for solving this problem to the exact optimum. Interestingly, this exact algorithm has intuitive physical interpretations, i.e., stretching the transitional parts of the trajectories (i.e., parts with acceleration and deceleration adjustments) as far as they reach the upstream end of the investigated segment, and then balancing the acceleration and deceleration magnitudes as close as possible. This analytical exact model can be considered as a core module to a range of general trajectory optimization problems at various infrastructure settings. Numerical examples reveal that this exact algorithm has much more efficient computational performance and the same or better solution quality compared with the previously proposed shooting heuristic. These examples also illustrate how to apply this model to CAV control problems on signalized segments and at non-stop intersections. Further, we study a homogeneous special case of this model and analytically formulate the relationship between queue propagation and trajectory smoothing. One counter-intuitive finding is that trajectory smoothing may not always cause longer queue propagation but instead may mitigate queue propagation with appropriate settings. This theoretical finding has valuable implications to joint optimization of queuing management and traffic smoothing in complex transportation networks.
TL;DR: It is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of $\Vert \cdot \Vert _\infty$, if sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled.
Abstract: This paper employs a Takagi–Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in the sense of spatial $L^\infty$ norm $\Vert \cdot \Vert _\infty$ for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensors are discretely distributed in space. Initially, a T-S fuzzy PDE model is assumed to be derived by the sector nonlinearity method to accurately describe complex spatiotemporal dynamics of the nonlinear DPSs. Subsequently, a static sampled-data fuzzy local state feedback controller is constructed based on the T-S fuzzy PDE model. By constructing an appropriate Lyapunov–Krasovskii functional candidate and employing vector-valued Wirtinger's inequalities, a variation of vector-valued Poincare–Wirtinger inequality in one-dimensional spatial domain, as well as a vector-valued Agmon's inequality, it is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of $\Vert \cdot \Vert _\infty$ , if sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled. Moreover, an LMI relaxation technique is utilized to enhance exponential stabilization ability of the suggested sampled-data fuzzy controller. Finally, the satisfactory and better performance of the suggested sampled-data fuzzy controller are demonstrated by numerical simulation results of two examples.
Proceedings Article•
29 Apr 2018
TL;DR: This work proposes the first trainable probabilistic deep architecture for hybrid domains that features tractable queries and relieves the user from deciding a-priori the parametric form of the random variables but is still expressive enough to effectively approximate any distribution and permits efficient learning and inference.
Abstract: While all kinds of mixed data---from personal data, over panel and scientific data, to public and commercial data---are collected and stored, building probabilistic graphical models for these hybrid domains becomes more difficult. Users spend significant amounts of time in identifying the parametric form of the random variables (Gaussian, Poisson, Logit, etc.) involved and learning the mixed models. To make this difficult task easier, we propose the first trainable probabilistic deep architecture for hybrid domains that features tractable queries. It is based on Sum-Product Networks (SPNs) with piecewise polynomial leaf distributions together with novel nonparametric decomposition and conditioning steps using the Hirschfeld-Gebelein-Renyi Maximum Correlation Coefficient. This relieves the user from deciding a-priori the parametric form of the random variables but is still expressive enough to effectively approximate any distribution and permits efficient learning and inference.Our experiments show that the architecture, called Mixed SPNs, can indeed capture complex distributions across a wide range of hybrid domains.
TL;DR: Experimental and simulation results show that PLS method has significant improvement in image quality compared with the TLS reconstruction, and is among the first ones using experimental EIT data.
Abstract: This paper presents an image reconstruction method based on parametric level set (PLS) method using electrical impedance tomography. The conductivity to be reconstructed was assumed to be piecewise constant and the geometry of the anomaly was represented by a shape-based PLS function, which we represent using Gaussian radial basis functions (GRBF). The representation of the PLS function significantly reduces the number of unknowns, and circumvents many difficulties that are associated with traditional level set (TLS) methods, such as regularization, re-initialization and use of signed distance function. PLS reconstruction results shown in this article are some of the first ones using experimental EIT data. The performance of the PLS method was tested with water tank data for two-phase visualization and with simulations which demonstrate the most popular biomedical application of EIT: lung imaging. In addition, robustness studies of the PLS method w.r.t width of the Gaussian function and GRBF centers were performed on simulated lung imaging data. The experimental and simulation results show that PLS method has significant improvement in image quality compared with the TLS reconstruction.
TL;DR: The proposed stabilizing controller can be solved directly and effectively, which is applicable to more general situations than those previously covered, and given to illustrate the effectiveness of the proposed method.
TL;DR: The existence and the exponential stability of piecewise differentiable pseudo-almost periodic solutions for a class of impulsive neutral high-order Hopfield neural networks with mixed time-varying delays and leakage delays are established by employing the fixed point theorem, Lyapunov functional method and differential inequality.
Abstract: In this paper, the existence and the exponential stability of piecewise differentiable pseudo-almost periodic solutions for a class of impulsive neutral high-order Hopfield neural networks with mixed time-varying delays and leakage delays are established by employing the fixed point theorem, Lyapunov functional method and differential inequality. Numerical example with graphical illustration is given to illuminate our main results.
TL;DR: A family of piecewise functions is proposed, based on which the fractional order integration of the Müntz-Legendre wavelets are easy to calculate, and this operational matrix with the collocation points is used to reduce the under study problem to a system of algebraic equations.
Abstract: This paper presents a new computational technique for solving fractional pantograph differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use Muntz-Legendre wavelet and its operational matrix of fractional-order integration. First, the Muntz-Legendre wavelet is presented. Then a family of piecewise functions is proposed, based on which the fractional order integration of the Muntz-Legendre wavelets are easy to calculate. The proposed approach is used this operational matrix with the collocation points to reduce the under study problem to a system of algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.
TL;DR: The spectral/hp element method as mentioned in this paper combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes.
Abstract: The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a C0 - continuous expansion. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.
TL;DR: This paper investigates the problem of pinning cluster synchronization for colored community networks via adaptive aperiodically intermittent control by introducing a novel piecewise continuous auxiliary function and derived criteria are rigorously derived according to Lyapunov stability theory and piecewise analysis approach.
Abstract: This paper investigates the problem of pinning cluster synchronization for colored community networks via adaptive aperiodically intermittent control. Firstly, a general colored community network model is proposed, where the isolated nodes can interact through different kinds of connections in different communities and the interactions between different pair of communities can also be different, and moreover, the nodes in different communities can have different state dimensions. Then, an adaptive aperiodically intermittent control strategy combined with pinning scheme is developed to realize cluster synchronization of such colored community network. By introducing a novel piecewise continuous auxiliary function, some globally exponential cluster synchronization criteria are rigorously derived according to Lyapunov stability theory and piecewise analysis approach. Based on the derived criteria, a guideline to illustrate which nodes in each community should be preferentially pinned is given. It is noted that the adaptive intermittent pinning control is aperiodic, in which both control width and control period are allowed to be variable. Finally, a numerical example is provided to show the effectiveness of the theoretical results obtained.
11 Jan 2018
TL;DR: This paper studies the problem of simulating the time evolution of a lattice Hamiltonian, and proves a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires (nT) gates in the worst case.
Abstract: We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and encompasses all physically reasonable Hamiltonians. Our algorithm simulates the time evolution of such a Hamiltonian on n qubits for time T up to error e using O(T polylog(nT/e)) gates with depth O(T polylog(nT/e)). Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in nT and polylogarithmic in 1/e. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires (nT) gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb-Robinson bounds. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
TL;DR: In this article, a novel jerk system with a smooth piecewise quadratic nonlinearity is introduced, which provides a similar smoothness as the cubic polynomial function, but a faster response and a simpler circuitry.
Abstract: In this contribution, a novel Jerk system with a smooth piecewise quadratic nonlinearity is introduced. The new nonlinearity provides a similar smoothness as the cubic polynomial function, but a faster response and a simpler circuitry. The basic dynamical properties of the model are discussed in terms of its parameters by using standard nonlinear analysis tools including phase space trajectory plots, frequency spectra, bifurcation diagrams and Lyapunov exponent plots. The bifurcation analysis yields very rich and interesting scenarios such as period-doubling bifurcations, antimonotonicity (i.e. the concurrent creation and annihilation of periodic orbits), periodic windows, and symmetry recovering crises. One of the main findings of this work is the presence of a window in the parameter space in which the novel jerk system experiences the unusual and striking feature of multiple coexisting attractors (i.e. coexistence of four or six disconnected periodic and chaotic attractors) for the same parameters’ setting. Correspondingly, basins of attraction of various competing attractors display extremely complex basin boundaries. Compared to some lower dimensional systems (e.g. Leipnik–Newton system, modified Sprott B system) capable of displaying such type of behavior reported to date, the jerk system introduced in this work represents the simplest and the most ‘elegant’ paradigm. An electronic circuit for allowing an illustration of the theoretical model is proposed and implemented in PSpice. The results obtained in this work let us conjecture that there exist some regions in its parameter space (that need to be uncovered) in which the universal Chua’s circuit experiences six disconnected non static attractors similar to those presented in this work.
TL;DR: A new monolithic equation is proposed based on the best fitting model, which can be used as an efficient alternative to the existing piecewise analytic equations, which is proposed and tested for modeling the classic 1D dam-break flow problem.
TL;DR: This work advocates a gradient-based, full space solver where the mesh and conservation law solution converge to their optimal values simultaneously and therefore never require the solution of the discrete conservation law on a non-aligned mesh.
TL;DR: In this paper, the Lyapunov constants at infinity of piecewise polynomial systems with no singular points at infinity were derived and an example of 11 limit cycles bifurcating from infinity was presented.
Abstract: In this paper, we study bifurcation of limit cycles from the equator of piecewise polynomial systems with no singular points at infinity. We develop a method for computing the Lyapunov constants at infinity of piecewise polynomial systems. In particular, we consider cubic piecewise polynomial systems and study limit cycle bifurcations in the neighborhood of the origin and infinity. Moreover, an example is presented to show 11 limit cycles bifurcating from infinity.
TL;DR: In this article, the authors propose an end-to-end trainable deep network which is inspired by the state-of-the-art fine-grained recognition model and is tailored for the FSFG task, which consists of a bilinear feature learning module and a classifier mapping module.
Abstract: Humans are capable of learning a new fine-grained concept with very little supervision, \emph{e.g.}, few exemplary images for a species of bird, yet our best deep learning systems need hundreds or thousands of labeled examples. In this paper, we try to reduce this gap by studying the fine-grained image recognition problem in a challenging few-shot learning setting, termed few-shot fine-grained recognition (FSFG). The task of FSFG requires the learning systems to build classifiers for novel fine-grained categories from few examples (only one or less than five). To solve this problem, we propose an end-to-end trainable deep network which is inspired by the state-of-the-art fine-grained recognition model and is tailored for the FSFG task.
Specifically, our network consists of a bilinear feature learning module and a classifier mapping module: while the former encodes the discriminative information of an exemplar image into a feature vector, the latter maps the intermediate feature into the decision boundary of the novel category. The key novelty of our model is a "piecewise mappings" function in the classifier mapping module, which generates the decision boundary via learning a set of more attainable sub-classifiers in a more parameter-economic way. We learn the exemplar-to-classifier mapping based on an auxiliary dataset in a meta-learning fashion, which is expected to be able to generalize to novel categories. By conducting comprehensive experiments on three fine-grained datasets, we demonstrate that the proposed method achieves superior performance over the competing baselines.
TL;DR: An adaptive-scale active contour model (ASACM) based on image entropy and semi-naive Bayesian classifier is proposed, which achieves simultaneous segmentation and bias field estimation for images with severe intensity inhomogeneity.
Posted Content•
TL;DR: In this article, the authors consider two topological transforms based on Euler calculus: the persistent homology transform (PHT) and the Euler characteristic transform (ECT).
Abstract: In this paper we consider two topological transforms based on Euler calculus: the Persistent Homology Transform (PHT) and the Euler Characteristic Transform (ECT). Both of these transforms are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. Both transforms take a shape, viewed as a tame subset $M$ of $\mathbb{R}^d$, and associates to each direction $v\in S^{d-1}$ a shape summary obtained by scanning $M$ in the direction $v$. These shape summaries are either persistence diagrams or piecewise constant integer valued functions called Euler curves. By using an inversion theorem of Schapira, we show that both transforms are injective on the space of shapes---each shape has a unique transform. We also introduce a notion of a "generic shape", which we prove can be uniquely identified up to an element of $O(d)$ by using the pushforward of the Lebesgue measure from the sphere to the space of Euler curves. Finally, our main result proves that any shape in a certain uncountable set of non-axis aligned shapes can be specified using only finitely many Euler curves.
TL;DR: In this paper, a shell finite element with a variable kinematic field based on a new zig-zag power function is proposed for the analysis of laminated shell structures, which is written by using an arbitrary number of continuous piecewise polynomial functions.
Abstract: In the present work, a shell finite element with a variable kinematic field based on a new zig-zag power function is proposed for the analysis of laminated shell structures . The kinematic field is written by using an arbitrary number of continuous piecewise polynomial functions. The polynomial expansion order of a generic subdomain is a combination of zig-zag power functions depending on the shell thickness coordinate. As in the classical layer-wise approach, the shell thickness can be divided into a variable number of mathematical subdomains. The expansion order of each subdomain is an input parameter of the analysis. This feature enables the solution to be locally refined over generic regions of the shell thickness by enriching the kinematic field. The advanced finite shell elements with variable kinematics are formulated in the framework of the Carrera Unified Formulation. The finite element arrays are formulated in terms of fundamental nuclei, which are invariants of the theory approximation order and the modelling technique (Equivalent-Single-Layer, Layer-Wise). In this work, the attention is focused on linear static stress analyses of composite laminated shell structures. The governing equations are obtained by applying the Principle of Virtual Displacements, and they are solved using the Finite Element method . Furthermore, the Mixed Interpolated Tensorial Components (MITC) method is employed to contrast the shear locking phenomenon . Several numerical investigations are carried out to validate and demonstrate the accuracy and efficiency of the present shell element.