On piecewise-linear basis functions and piecewise-linear signal expansions

The electric current volume integral equation formulation is discretized with piecewise linear basis and testing functions. The resulting weak formulation is more complicated, and contains additional terms compared with the previous discretization utilizing piecewise constant approximations. However, the use of fully linear functions provides superior solution accuracy while keeping good convergence properties of the original integral equation.

Recently Gray and Lynch (1977, 1978) introduced a promising finite element method for solving the shallow water equations. Rather than working with the governing equations in conservation form, their ‘wave equation’ method (WEM) involves transforming the continuity equation to a second order partial differential equation. The revised system of equations is then solved with a Galerkin FEM, piecewise linear basis functions, and centred time stepping.

The Galerkin finite elements method (GAFEM) is used to find the numerical solution (NS) of the couple linear elliptic boundary value problem (CLEBVP). The solution of the CLEBVP is approximated by a linear combination of piecewise linear and continuous basis functions, upon substituting this approximated solution into the CLEBVP the problem then reduces to solve a “Galerkin” linear algebraic system (GALAS). This system is solved by using the classical Cholesky method (CChM). Some illustrative examples are given; to show the high efficiency of the GAFEM to solve such for the CLEBVPs. Figures provide the NS with “results”.

A novel f-f model is developed constructively which can express any n-dimensional piecewise linear (PWL) function by a superposition of basis functions if it is defined over an nth-order degenerate intersection formed by (n-1)th-order minimal degenerate intersections. We also propose the concrete functional forms of nth-order basis functions. Being the simplest type of the minimal degenerate intersection, the basis function is the most elementary "building block" of a PWL function defined in an arbitrary-dimensional space. In addition, the model constitutes a natural continuation to Julian's canonical formulation and can bridge the lattice PWL model and the well-established canonical representation.

The moments produced by the aerodynamic control effectors on aircraft are typically non-linear functions of displacement with a region of linearity about zero displacement. In this paper we briefly review a novel approach that was developed at AFRL for solving the control allocation problem where the control moments are piecewise linear functions of the control displacement. The advantages of this approach are that it avoids cumbersome, high-order polynomial curve fits of the aerodynamic data and thus avoids solution of a non-linear programming problem. Stability and control derivatives are almost always stored in multi-dimensional look-up tables where it is assumed that the data is piecewise linear. The approach that is presented utilizes this piecewise linear assumption for the control effector moment data and accounts for non-linearities in the moment-effector relationships and position constraints on the effectors as well. The control allocation problem using piecewise linear functions is posed as a mixed-integer linear program (MILP) and solved using a freely available GNU-licensed mixed-integer linear programming code. We demonstrate two applications of the piecewise linear control allocation (PLCA) approach. The first application to be considered is to use the PLCA approach in the inner-loop control law of a re-usable launch vehicle on approach and landing. The second application is constraint estimation for trajectory reshaping and re-targeting. In order to successfully re-target a trajectory when an aircraft has experienced degraded performance due to a failure or damage to the vehicle, the effects of the failure or damage on the lift, drag, and trimmability of the vehicle must be known a priori over the entire flight envelope

The infinite relaxations in integer programming were introduced by Gomory and Johnson to provide a general framework for the theory of cutting planes: the so-called valid functions, and in particular the minimal and extreme functions, can be seen as automatic rules for the generation of cuts. However, while many extreme functions are piecewise linear and therefore easy to describe, the set of extreme functions turns out to have a very complicated mathematical structure, as several extreme functions are known that exhibit a somewhat pathological behavior. In this paper we show that if some smoothness assumption is imposed on an extreme function $$\pi $$, then $$\pi $$ is necessarily piecewise linear. More precisely, we show that if a continuous extreme function for the Gomory–Johnson one-dimensional infinite group relaxation is a piecewise $${\mathcal {C}}^2$$ function, then it is a piecewise linear function.

This paper investigates the performance of current-mode chaotic neural networks that are optimized for travelling salesman problem (TSP). Piece-wise linear (PWL) functions used as the output functions in chaotic neurons are found to ease the design of neural network circuits. Three different PWL output functions are compared by solving 10-city TSP. Simulation results showed excellent performance of the chaotic neural networks with PWL output function. The design parameters to implement the PWL output function suitable for TSP chaotic neural networks are obtained.

Nonlinear effects of a power amplifier (PA) in communications systems include spectral spreading, which causes adjacent channel interference. For predistorter design, it is crucial to employ a model that can sufficiently model the intermodulation distortion (IMD) caused by the PA. The order of IMD generated by a polynomial is determined by its highest order basis. For a piecewise linear function, the order of nonlinearity cannot be directly deduced from its basis function. This paper investigates how IMD is introduced by the simplicial canonical piecewise linear (SCPWL) function. We examine the operation imposed by the SCPWL basis function on sinusoid and multi-tone signal in the time domain and frequency domain. Our analysis results in expressions that can be used to predict the output spectrum of the SCPWL basis. The SCPWL basis operation involves spreading of the input signal spectrum. The output spectrum of each SCPWL basis includes infinite IMD components. The IMD order of the SCPWL model is not determined by the highest order basis. Instead, the number of SCPWL parameters, i.e., the number of linear affine segments that defines the function, determines how accurately it can model the IMD components.

The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546]. Both formulations are similar in that they use binary variables and logical implications modelled by big-[Formula: see text] constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing. Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two formulations, including a series of comparative experimental results across 10 data sets that appeared across both papers. We hope that our results will allow readers to take an objective view as to which implementation they should use.

In this paper we apply one of the main results from the theory of noninvertible maps to predict the oscillations amplitude of a chaotic attractor, using only the lines where the piece-wise linear (PWL) map is not differentiable, their first iterates (called critical lines or curves), and their finite-rank iterates. This approach is also valid for smooth differentiable noninvertible maps, where the critical lines are defined as the first iterates of the set of points for which the Jacobian determinant of the map cancels. Moreover, a novel bifurcation encountered in the case of PWL map is presented: chaotic orbit arising from the collision of a typical bifurcation for smooth maps (the Neimark-Hopf, or center bifurcation) with the bifurcation typical only for piece-wise smooth and non differentiable maps (the border-collision bifurcation). Finally, it is shown that this novel bifurcation can give birth to two (or more) chaotic attractors, and the conditions allowing to predict the number of chaotic attractors which could coexist are proposed and discussed in the context of possible applications.

The paper studies a second order nonlinear differential equation whose right hand side is a piecewise linear function. It proves the coexistence of a countably set of periodic solutions and an uncountable set of bounded non-periodic solutions. The Weierstrass approximation theorem is applied so that the piecewise linear function can be approximated by a polynomial on the finite interval. The result can be also used to explain the chaos on some smooth nonlinear dynamical systems.

PurposeA wide range of micro‐electro‐mechanical‐systems are based on the electrostatic principle, and for their design the computation of the electric capacities is of great importance. The purpose of this paper is to efficiently compute the capacities as a function of all possible positions of the two electrode structures within the transducer by an enhanced boundary element method (BEM).Design/methodology/approachA Galerkin BEM is developed and the arising algebraic system of equations is efficiently solved by a CG‐method with a multilevel preconditioner and an appropriate fast multipole algorithm for the matrix‐vector operations within the CG‐iterations.FindingsIt can be demonstrated that the piecewise linear and discontinuous trial functions give an approximation, which is almost as good as the one of the piecewise constant trial functions on the refined mesh, at lower computational costs and at about the same memory requirements.Originality/valueThe paper can proof mathematically and demonstrate in practice, that a higher order of convergence is achieved by using piecewise linear, globally discontinuous basis functions instead of piecewise constant basis functions. In addition, an appropriate preconditioner (artificial multilevel boundary element preconditioner, which is based on the Bramble Pasciak Xu like preconditioner) has been developed for the fast iterative solution of the algebraic system of equations.

Hybrid systems built up with CMOS devices and Single-Electron Transistors (SET), have been foreseen as a potential alternative in the long term when CMOS downscaling reaches its physical fundamental limits. Although there are still many problems in the fabrication of these nanoelectronic devices, the design flow for hybrid systems requires further development not only in those aspects directly linked to the design and synthesis, but also in aspects regarding the verification of the design, such as electrical simulation. This paper introduces an appropriate model for the SET that can be easily implemented for co-simulation of hybrid systems. The model constitutes a functional model for the SET in the form of an explicit piecewise linear (PWL) formulation that can be easily coded into a high level language.

In this paper, mapping networks will be considered from the viewpoint of the piecewise-linear (PWL) approximation. The so-called canonical representation plays a kernel role in the PWL representation theory. While this theory has been researched intensively in the contents of mathematics and circuit simulations, little has been seen in the research area about the theoretical aspect of neural networks. This paper modifies this theory and applies it as a mathematical support for mapping networks. The main modification is a "higher-level" generalization of the canonical representation with proofs of its availability in the set of PWL functions. The modified theory will first be used to study the canonical PWL feature of the popular multilayer perceptron-like (MLPL) networks. Second, it will be seen that the generalized canonical representation is itself suitable for a network implementation, which is called the standard canonical PWL network. More generally, the family of (generalized) canonical PWL networks is defined as those which may take the canonical PWL representation as a mathematical model. This family is large and practically meaningful. The standard canonical PWL networks may be taken as representatives in the family. The modification of the PWL representation theory as well as the introduction of this theory in the theoretical study of mapping networks, which provide a new concept of mapping networks, i.e., the canonical PWL network family, may be regarded as the main contributions of the paper.

A space deformation is a mapping from a source region to a target region within Euclidean space, which best satisfies some userspecified constraints. It can be used to deform shapes embedded in the ambient space and represented in various forms -- polygon meshes, point clouds or volumetric data. For a space deformation method to be useful, it should possess some natural properties: e.g. detail preservation, smoothness and intuitive control. A harmonic map from a domain ω ⊂ R d to R d is a mapping whose d components are harmonic functions. Harmonic mappings are smooth and regular, and if their components are coupled in some special way, the mapping can be detail-preserving, making it a natural choice for space deformation applications. The challenge is to find a harmonic mapping of the domain, which will satisfy constraints specified by the user, yet also be detail-preserving, and intuitive to control. We generate harmonic mappings as a linear combination of a set of harmonic basis functions, which have a closed-form expression when the source region boundary is piecewise linear. This is done by defining an energy functional of the mapping, and minimizing it within the linear span of these basis functions. The resulting mapping is harmonic, and a natural "As-Rigid-As-Possible" deformation of the source region. Unlike other space deformation methods, our approach does not require an explicit discretization of the domain. It is shown to be much more efficient, yet generate comparable deformations to state-of-the-art methods. We describe an optimization algorithm to minimize the deformation energy, which is robust, provably convergent, and easy to implement.