Piecewise Linear Function Approximation Research Articles

Piecewise linear approximation for MILP leveraging piecewise convexity to improve performance

To realize adaptive operation planning with MILP unit commitment, piecewise-linear approximations of the functions that describe the operating behavior of devices in the energy system have to be computed. We present an algorithm to compute a piecewise-linear approximation of a multi-variate non-linear function. The algorithm splits the domain into two regions and approximates each region with a set of hyperplanes that can be translated to a convex set of constraints in MILP. The main advantage of this “piecewise-convex approximation” (PwCA) compared to more general piecewise-linear approximation with simplices is that the MILP representation of PwCA requires only one auxiliary binary variable. For this reason, PwCA yields significantly faster solving times in large MILP problems where the MILP representation of certain functions has to be replicated many times, such as in unit commitment. To quantify the impact on solving time, we compare the performance using PwCA with the performance of simplex approximation with logarithmic formulation and show that PwCA outperforms the latter by a big margin. For this reason, we conclude that PwCA will be a useful tool to set up and solve large MILP problems such as arise in unit commitment and similar engineering optimization problems.

Piecewise reconstruction of membership function approximation errors for Takagi–Sugeno fuzzy control

We propose a conservative relaxed control method to stabilize Takagi–Sugeno (TS) fuzzy systems. First, a conventional piecewise linear function approximation method is adopted to consider the membership functions’ information in the closed-loop system’s stability analysis. Subsequently, a model reconstruction method is proposed to reformulate linear approximation error terms in the TS fuzzy form, which can effectively introduce more information into stability conditions and achieve better robust control effects. Then, based on a fuzzy Lyapunov function approach, control design conditions are derived in terms of linear matrix inequalities, with less conservatism for related results in the literature. Finally, via a benchmark experiment, it is verified that the proposed method can show better robustness with fewer decision variables in robust convex conditions.

SIMULATION OF CELLS FOR SIGNALS INTENSITY TRANSFORMATION IN MIXED IMAGE PROCESSORS AND ACTIVATION FUNCTIONS OF NEURONS IN NEURAL NETWORKS

The paper considers results of design, simulation of continuously logical pixel cells (CLPC) based on current mirrors (CM) with functions of preliminary analogue processing for image intensity transformation and coding for construction of mixed image processors (IP) and neural networks (NN). The methodology and principles of construction of such cells are based on the use of piecewise-linear approximation of functions for nonlinear transformation of analog signals. It is shown that for the realization of generalized arbitrary functions by such gamma correctors, it is possible to apply basic step functions with controlled parameters. To implement the basic step functions, it is proposed to use nodes that perform a continuous-logical operation of a limited current difference and are quite simply implemented on current reflectors (VDS). The design and modeling of continuous-logical pixel cells (CLPC) based on VDS in different modes and for different conversion functions. Such CLC has a number of advantages: high speed and reliability, simplicity, small power consumption, high integration level for linear and matrix structures. We show design of CLC variants for photocurrents transformation and their simulations. The basic element of such cells is a scheme that implements the operation of a bounded difference of continuous logic. Using a set of circuits implemented on CMOS technology, we consider generalized methods for designing cells for nonlinear conversion of the photocurrent intensity. Selection of the appropriate parameters, which can be specified as constructive constants or as parameters for external control, allows changing type of synthesized functions. Possibilities of synthesis by such cells of functions with descending sections and different types are shown: sigmoid, lambda and others. Such CLPCs consist of several dozen CMOS transistors, have low power supply voltage (1.8 ÷ 3.3V), the range of an input photocurrent is 0.1÷24μA, the transformation time is less than 1 μs, low power consumption (microwatts). The circuits and the simulation results of their design with OrCAD are shown. Examples of nonlinear image transformations are given.

Optimal Piecewise Linear Function Approximation for GPU-Based Applications

Many computer vision and human-computer interaction applications developed in recent years need evaluating complex and continuous mathematical functions as an essential step toward proper operation. However, rigorous evaluation of these kind of functions often implies a very high computational cost, unacceptable in real-time applications. To alleviate this problem, functions are commonly approximated by simpler piecewise-polynomial representations. Following this idea, we propose a novel, efficient, and practical technique to evaluate complex and continuous functions using a nearly optimal design of two types of piecewise linear approximations in the case of a large budget of evaluation subintervals. To this end, we develop a thorough error analysis that yields asymptotically tight bounds to accurately quantify the approximation performance of both representations. It provides an improvement upon previous error estimates and allows the user to control the tradeoff between the approximation error and the number of evaluation subintervals. To guarantee real-time operation, the method is suitable for, but not limited to, an efficient implementation in modern graphics processing units, where it outperforms previous alternative approaches by exploiting the fixed-function interpolation routines present in their texture units. The proposed technique is a perfect match for any application requiring the evaluation of continuous functions; we have measured in detail its quality and efficiency on several functions, and, in particular, the Gaussian function because it is extensively used in many areas of computer vision and cybernetics, and it is expensive to evaluate.

Models and Algorithms for Optimal Piecewise-Linear Function Approximation

Piecewise-linear functions can approximate nonlinear and unknown functions for which only sample points are available. This paper presents a range of piecewise-linear models and algorithms to aid engineers to find an approximation that fits best their applications. The models include piecewise-linear functions with a fixed and maximum number of linear segments, lower and upper envelopes, strategies to ensure continuity, and a generalization of these models for stochastic functions whose data points are random variables. Derived from recursive formulations, the algorithms are applied to the approximation of the production function of gas-lifted oil wells.

Adaptive multiresolution analysis based on anisotropic triangulations

A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between the approximated function and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the Lp norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of of the approximated function in case of C2 functions).

SYNTHESIS OF CURRENT DIFFERENCING TRANSCONDUCTANCE AMPLIFIER-BASED CURRENT LIMITERS AND ITS APPLICATIONS

A synthesis of analog current limiter (CL) building blocks based on a current differencing transconductance amplifier (CDTA) is proposed. The breakpoint and the slope of the resulting transfer characteristic obtained from the proposed CDTA-based CL are electronically programmable through the external bias currents. To demonstrate versatility of the proposed electronically tunable CLs, some nonlinear applications to programmable current-mode precision full-wave rectifiers and piecewise-linear function approximation generators are also presented. PSPICE simulation and experimental results confirm the effectiveness of the proposed circuits.

Piecewise linear approximation of functions of two variables in MILP models

We consider three easy-to-implement methods for the piecewise linear approximation of functions of two variables. We experimentally evaluate their approximation quality, and give a detailed description of how the methods can be embedded in a MILP model. The advantages and drawbacks of the three methods are discussed on numerical examples.

Algorithm for building a neural network for function approximation

A technique for constructing a multilayer tree-structured neural network, which provides a continuous, piece-wise linear function approximation, is presented. The method uses a growth network with linear threshold neurons. Neurons are added to a binary tree until the approximation error at all sampling points is brought down to within a specified ±Δ. The number of neurons in the constructed network depends on the samples provided as well as the specified tolerance Δ, thus enabling a trade-off between accuracy and network size. In comparison to approaches such as backpropagation, the proposed technique requires no assumptions regarding the number of neurons, learning rate, momentum term, or initial weight values. It also does not suffer from problems of local minima. Examples are presented to illustrate the effectiveness of the technique.

Use of periodic and monotonic activation functions in multilayer feedforward neural networks trained by extended Kalman filter algorithm

The authors investigate the convergence and pruning performance of multilayer feedforward neural networks with different types of neuronal activation functions in solving various problems. Three types of activation functions are adopted in the network, namely, the traditional sigmoid function, the sinusoidal function and a periodic function that can be considered as a combination of the first two functions. To speed up the learning, as well as to reduce the network size, the extended Kalman filter (EKF) algorithm conjunct with a pruning method is used to train the network. The corresponding networks are applied to solve five typical problems, namely, 4-point XOR logic function, parity generation, handwritten digit recognition, piecewise linear function approximation and sunspot series prediction. Simulation results show that periodic activation functions perform better than monotonic ones in solving multicluster classification problems. Moreover, the combined periodic activation function is found to possess the fast convergence and multicluster classification capabilities of the sinusoidal activation function while keeping the robustness property of the sigmoid function required in the modelling of unknown systems.

Piece-wise linear approximation of functions of two variables

The goal of increasing computational efficiency is one of the fundamental challenges of both theoretical and applied research in mathematical modeling. The pursuit of this goal has lead to wide diversity of efforts to transform a specific mathematical problem into one that can be solved efficiently. Recent years have seen the emergence of highly efficient methods and software for solving Mixed Integer Programming Problems, such as those embodied in the packages CPLEX, MINTO, XPRESS-MP. The paper presents a method to develop a piece-wise linear approximation of an any desired accuracy to an arbitrary continuous function of two variables. The approximation generalizes the widely known model for approximating single variable functions, and significantly expands the set of nonlinear problems that can be efficiently solved by reducing them to Mixed Integer Programming Problems. By our development, any nonlinear programming problem, including non-convex ones, with an objective function (and/or constraints) that can be expressed as sums of component nonlinear functions of no more than two variables, can be efficiently approximated by a corresponding Mixed Integer Programming Problem.

An architecture for lossy compression of waveforms using piecewise-linear approximation

Lossy compression schemes are often desirable in many signal processing applications such as the compression of ECG data. This paper presents a relaxation of a provably good algorithm for lossy signal compression, based on the piecewise linear approximation of functions. The algorithm approximates the data within a given tolerance using a piecewise linear function. The paper also describes an architecture suitable for the single-chip implementation of the proposed algorithm. The design consists of control, two multiply/divide units, four adder/subtracter units, and an I/O interface unit. For uniformly sampled data, no division is required, and all operations can be completed in a pipelined manner in at most three cycles per sample point. The corresponding simplified architecture is also presented. >

A method for identifying nonlinear terms in parabolic initial-boundary value problems

The problem of identifying unknown nonlinear coefficients in a one dimensional diffusion equation is considered. It is shown that these coefficients can be recovered from “correct” overposed data. The main analytical tools used are the maximum principle and monotonicity, both of the underlying operator and of the overposed data. In some cases a naturally induced fixed point formulation results, while in other cases a collocation method is induced and this yields a unique piecewise linear function approximation to the undetermined coefficient. The results of some numerical simulations are given.

Two Algorithms for Piecewise-Linear Continuous Approximation of Functions of One Variable

Two simple heuristic algorithms for piecewise-linear approximation of functions of one variable are described. Both use a limit on the absolute value of error and strive to minimize the number of approximating segnents subject to the error limit. The first algorithm is faster and gives satisfactory results for sufficiently smooth functions. The second algorithm is not as fast but gives better approximations for less well-behaved functions. The two algorithms are ilustrated by several examples.