Continuous Piecewise Linear Research Articles

Canards in a minimal piecewise-linear square-wave burster.

We construct a piecewise-linear (PWL) approximation of the Hindmarsh-Rose (HR) neuron model that is minimal, in the sense that the vector field has the least number of linearity zones, in order to reproduce all the dynamics present in the original HR model with classical parameter values. This includes square-wave bursting and also special trajectories called canards, which possess long repelling segments and organise the transitions between stable bursting patterns with n and n + 1 spikes, also referred to as spike-adding canard explosions. We propose a first approximation of the smooth HR model, using a continuous PWL system, and show that its fast subsystem cannot possess a homoclinic bifurcation, which is necessary to obtain proper square-wave bursting. We then relax the assumption of continuity of the vector field across all zones, and we show that we can obtain a homoclinic bifurcation in the fast subsystem. We use the recently developed canard theory for PWL systems in order to reproduce the spike-adding canard explosion feature of the HR model as studied, e.g., in Desroches et al., Chaos 23(4), 046106 (2013).

A stabilized cut finite element method for partial differential equations on surfaces: The Laplace–Beltrami operator

We consider solving the Laplace–Beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.The resulting discrete method may be severely ill-conditioned, and the main purpose of this paper is to suggest a remedy for this problem based on adding a consistent stabilization term to the original bilinear form. We show optimal estimates for the condition number of the stabilized method independent of the location of the surface. We also prove optimal a priori error estimates for the stabilized method.

Continuous Piecewise Linear Identification with Moderate Number of Subregions

A continuous piecewise linear (CPWL) function equals an affine function in each of the subregions which tessellate the domain. By CPWL identification, one nonlinear system can be approached by a set of linear systems and then linear techniques are applicable. For the convenience of further application, the number of subregions should be moderate. However, because the relation between the subregions and parameters of a CPWL model is very complicated, existing identification methods usually lead to a lot of unexpected subregions, when training the parameters of traditional CPWL models. In this paper, domain partition based CPWL neural network, which is recently proposed, is utilized to solve the problem above. The relation between parameters and subregion structure of such model is analyzed. Then an algorithm of adjusting vertices is proposed to improve approximation precision with moderate number of subregions. Compared with some prior identification methods, the proposed algorithm uses much less subregions and achieves satisfactory precision in numerical study.

Nonlinear system identification with continuous piecewise linear neural network

This paper considers system identification using domain partition based continuous piecewise linear neural network (DP-CPLNN), which is newly proposed. DP-CPLNN has the capability of representing any continuous piecewise linear (CPWL) function, hence its identification performance can be expected. Another attractive feature of DP-CPLNN is the geometrical property of its parameters. Applying this property, this paper proposes an identification method including domain partition and parameter training. In numerical experiments, DP-CPLNN with this method outperforms hinging hyperplanes and high-level canonical piecewise linear representation, which are two widely used CPWL models, showing the flexibility of DP-CPLNN and the effectiveness of the proposed algorithm in nonlinear identification.

EXISTENCE AND APPROXIMATION OF A (REGULARIZED) OLDROYD-B MODEL

We consider the finite element approximation of the Oldroyd-B system of equations, which models a dilute polymeric fluid, in a bounded domain [Formula: see text], d = 2 or 3, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conformation tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics or a reduced version, where the tangential component on each simplicial edge (d = 2) or face (d = 3) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes satisfy a free energy bound, which involves the logarithm of the conformation tensor, without any constraint on the time step for the backward Euler-type time discretization. This extends the results of Boyaval et al. (Free-energy-dissipative schemes for the Oldroyd-B model, ESAIM: Math. Model. Numer. Anal.43 (2009) 523–561) on this free energy bound. There a piecewise constant approximation of the conformation tensor was necessary to treat the advection term in the stress equation, and a restriction on the time step, based on the initial data, was required to ensure that the approximation to the conformation tensor remained positive definite. Furthermore, for our approximation (b) in the presence of an additional dissipative term in the stress equation and a cut-off on the conformation tensor on certain terms in the system, similar to those introduced in Barrett and Süli (Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off, Math. Models Methods Appl. Sci.18 (2008) 935–971) for the microscopic–macroscopic finitely extensible nonlinear elastic model of a dilute polymeric fluid, we show (subsequence) convergence, as the spatial and temporal discretization parameters tend to zero, toward global-in-time weak solutions of this regularized Oldroyd-B system. Hence, we prove existence of global-in-time weak solutions to this regularized model. Moreover, in the case d = 2 carry out this convergence in the absence of cut-offs, but with a time step restriction dependent on the spatial discretization parameter, and hence show existence of a global-in-time weak solution to the Oldroyd-B system with an additional dissipative term in the stress equation.

Applying Continuous Piecewise Linear Approximations to Affine Non-Linear Control systems

In this paper continuous piecewise linear (CPWL) approximations of non-linear affine control systems are considered. An optimal control context allows to determine asymptotic error bounds between real and approximate control systems. Then it is possible to apply the controller designed using the CPWL approximation to the real vector field with guarantee of stability. An example is presented to show the potential applications of these ideas.

Generalization of Hinging Hyperplanes

The model of hinging hyperplanes (HH) can approximate a large class of nonlinear functions to arbitrary precision, but represent only a small part of continuous piecewise-linear (CPWL) functions in two or more dimensions. In this correspondence, the influence of this drawback for black-box modeling is first illustrated by a simple example. Then it is shown that the above shortcoming can be amended by adding a sufficient number of linear functions to current hinges. It is proven that any CPWL function of n variables can be represented by a sum of hinges containing at most n+1 linear functions. Hence the model of a sum of such expanded hinges is a general representation for all CPWL functions. The structure of the novel general representation is much simpler than the existing generalized canonical representation that consists of nested absolute-value functions. This characteristic is very useful for black-box modeling. Based on the new general representation, an upper bound on the number of nestings of nested absolute-value functions of a generalized canonical representation is established, which is much smaller than the known result.

Rigorous Verification of Poincare Map Generated by a Continuous Piece-Wise Linear Vector Field and Its Application

This paper provides algorithms in order to solve an interval implicit function of the Poincare map generated by a continuous piecewise linear (CPWL) vector field, with the use of interval arithmetic. The algorithms are implemented with the use of MATLAB and INTLAB. We present an application to verification of canards in two-dimensional CPWL vector field appearing in nonlinear piecewise linear circuits frequently, and confirm that the algorithms are effective.

General Constructive Representations for Continuous Piecewise-Linear Functions

The problem of constructing a canonical representation for an arbitrary continuous piecewise-linear (PWL) function in any dimension is considered in this paper. We solve the problem based on a general lattice PWL representation, which can be determined for a given continuous PWL function using existing methods. We first transform the lattice PWL representation into the difference of two convex functions, then propose a constructive procedure to rewrite the latter as a canonical representation that consists of at most n-level nestings of absolute-value functions in n dimensions, hence give a thorough solution to the problem mentioned above. In addition, we point out that there exist notable differences between a lattice representation and the two novel general constructive representations proposed in this paper, and explain that these differences make all the three representations be of their particular interests.

Failure of the discrete maximum principle for an elliptic finite element problem

There has been a long-standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green's function is positive for triangular meshes allowing sufficiently good approximation of H 1 functions. We study this question for the Poisson problem in two dimensions discretized via the Galerkin method with continuous piecewise linears. We give examples which show that in general the answer is negative, and furthermore we extend the number of cases where it is known to be positive. Our techniques utilize some new results about discrete Green's functions that are of independent interest.

A Local Convergence Proof for the Minvar Algorithm for Computing Continuous Piecewise Linear Approximations

The class of continuous piecewise linear (PL) functions represents a useful family of approximants because invertibility can be readily imposed, and if a PL function is invertible, then it can be inverted in closed form. Many applications, arising, for example, in control systems and robotics, involve the simultaneous construction of a forward and inverse system model from data. Most approximation techniques require that separate forward and inverse models be trained, whereas an invertible continuous PL affords, simultaneously, the forward and inverse system model in a single representation. The minvar algorithm computes a continuous PL approximation to data. Local convergence of minvar is proven for the case when the data generating function is itself a PL function and available directly rather than through data.

Synthesis of nonlinear multiport resistors: a PWL approach

This paper presents a method for the approximate synthesis of nonlinear multiport resistors. According to some fundamental circuit theory results, the general problem of synthesizing a multiport resistor with given constitutive equations corresponds to that of the synthesis of nonlinear controlled sources. Following this idea, in this paper, we focus on the design of nonlinear controlled sources using a piecewise-linear (PWL) approach. The constitutive equations are first approximated by resorting to canonical expressions for continuous PWL functions, and then implemented using a set of elementary building blocks. The proposed method is applied to the synthesis of the nonlinear resistive part of an equivalent circuit of the Hodgkin-Huxley nerve membrane model.

A canonical representation of high-dimensional continuous piecewise-linear functions

A canonical representation of high-dimensional continuous piecewise-linear (PWL) functions has not been perfectly solved until now. In this brief, a constructive model of high-dimensional continuous PWL functions is proposed by using an iterative method and it is proved that the model can represent all high-dimensional continuous PWL functions.

A new algorithm for learning in piecewise-linear neural networks

Piecewise-linear (PWL) neural networks are widely known for their amenability to digital implementation. This paper presents a new algorithm for learning in PWL networks consisting of a single hidden layer. The approach adopted is based upon constructing a continuous PWL error function and developing an efficient algorithm to minimize it. The algorithm consists of two basic stages in searching the weight space. The first stage of the optimization algorithm is used to locate a point in the weight space representing the intersection of N linearly independent hyperplanes, with N being the number of weights in the network. The second stage is then called to use this point as a starting point in order to continue searching by moving along the single-dimension boundaries between the different linear regions of the error function, hopping from one point (representing the intersection of N hyperplanes) to another. The proposed algorithm exhibits significantly accelerated convergence, as compared to standard algorithms such as back-propagation and improved versions of it, such as the conjugate gradient algorithm. In addition, it has the distinct advantage that there are no parameters to adjust, and therefore there is no time-consuming parameters tuning step. The new algorithm is expected to find applications in function approximation, time series prediction and binary classification problems.

Numerical Analysis of Time-Depending PrimalElastoplasticity with Hardening

The quasi-static elastoplastic evolution problem with combined isotropic and kinematic hardening is considered with emphasis on optimal convergence of the lowest order scheme. In each time-step of a generalized midpoint scheme such as the implicit Euler or the Crank--Nicolson scheme, the spatial discretization consists of minimizing a convex but nonsmooth function on a subspace of continuous piecewise linear, resp., piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which, for smooth data and a smooth exact solution, proves linear convergence as the mesh-size tends to zero. Strong convergence of the time-derivatives is established under mild conditions on the mesh- and time-step sizes. Numerical experiments confirm our theoretical predictions on the improved spatial convergence and indicate that the Crank--Nicolson scheme is not always superior over the implicit Euler scheme in practice.

A parametrization of piecewise linear Lyapunov functions via linear programming

In this paper, we present a parametrization of piecewise linear (PWL) Lyapunov functions. To this end, we consider the class of all continuous PWL functions defined over a simplicial partition. We take advantage of a recently developed high level canonical PWL (HL CPWL) representation, which expresses the PWL function in a compact and closed form. Once the parametrization problem is properly stated, we focus on its application to the stabiilty analysis of dynamic systems. We consider uncertain non-linear systems and extend the sector condition obtained by Ohta et al. In addition, we propose a method of selecting an optimal candidate. One of the main advantages of this approach is that the parametrization and choice of the Lyapunov candidate, as well as the stability analysis, result in linear programming problems.

An Improved Error Bound for a Lagrange--Galerkin Method for Contaminant Transport with Non-Lipschitzian Adsorption Kinetics

In this paper we consider the Lagrange--Galerkin finite element approximation by continuous piecewise linears in space of the following problem: We obtain error bounds which improve on those in the literature.

Explicit piecewise-linear models

A proper explicit model of piecewise-linear (PWL) functions is a problem under current investigation in the research area of nonlinear systems. In this correspondence this problem is considered from a fundamental scheme. Based on this scheme, two explicit models are derived for all continuous PWL functions. The proposed models are suitable for a network implementation and can be used in applications, e.g., neural networks or nonlinear signal processing. >

Canonical representation: from piecewise-linear function to piecewise-smooth functions

The canonical representation of piecewise-linear (PWL) functions provides a global compact formulation of continuous PWL functions, which has significant advantages in the research and applications concerning nonlinear systems. This work studies the generalization of the canonical representation from PWL functions to piecewise-smooth (PWS) functions. First a class of PWS functions, called the regular PWS functions, is defined as a generalization of the continuous PWL functions. An important example of the regular PWS functions is the continuous piecewise-polynomial function. The continuous PWL function with a PWL partition is also covered by the regular PWS function. Then the canonical representation of the PWS function is defined and the existence conditions are discussed. The PWS generalization of the canonical representation is significant in applications where a PWS scheme can improve the performance of a PWL scheme in the approximation of a nonlinear function, i.e., in approximating the input/output (I/O) relation of a nonlinear system or a mapping neural network or in nonlinear signal processing. >

Steady-state analysis of piecewise-linear dynamic systems

Piecewise-linear (PWL) systems form a subclass of the general class of nonlinear systems. However, because PWL functions are not differentiable everywhere, results derived for general differentiable systems do not generally apply to PWL systems. In this paper, the properties of the solutions of continuous PWL dynamic systems are investigated in detail and theorems on the continuity and differentiability of the solutions with respect to initial conditions are derived. The results obtained are applied in the study of the convergence properties of steadystate algorithms when applied to continuous PWL dynamic systems. It is found that under fairly mild conditions the convergence is of order two, which is the same order of convergence of the algorithms when applied to differentiable dynamic systems. It is also found, however, that there are special cases when the order of convergence may be reduced to one.