Showing papers on "Piecewise linear function published in 1990"

Improving the learning speed of 2-layer neural networks by choosing initial values of the adaptive weights

Abstract: The authors describe how a two-layer neural network can approximate any nonlinear function by forming a union of piecewise linear segments. A method is given for picking initial weights for the network to decrease training time. The authors have used the method to initialize adaptive weights over a large number of different training problems and have achieved major improvements in learning speed in every case. The improvement is best when a large number of hidden units is used with a complicated desired response. The authors have used the method to train the truck-backer-upper and were able to decrease the training time from about two days to four hours

Improved versions of learning vector quantization

Abstract: The author introduces a variant of (supervised) learning vector quantization (LVQ) and discusses practical problems associated with the application of the algorithms. The LVQ algorithms work explicitly in the input domain of the primary observation vectors, and their purpose is to approximate the theoretical Bayes decision borders using piecewise linear decision surfaces. This is done by purported optimal placement of the class codebook vectors in signal space. As the classification decision is based on the nearest-neighbor selection among the codebook vectors, its computation is very fast. It has turned out that the differences between the presented algorithms in regard to the remaining discretization error are not significant, and thus the choice of the algorithm may be based on secondary arguments, such as stability in learning, in which respect the variant introduced (LVQ2.1) seems to be superior to the others. A comparative study of several methods applied to speech recognition is included

Piecewise linear risk function and portfolio optimization

Abstract: A new portfolio optimization model using a piecewise linear risk function is proposed. This model is similar to, but has several advantages over the classical Markowitz's quadratic risk model. First, it is much easier to generate an optimal portfolio since the problem to be solved is a linear program instead of a quadratic program. Second, integer constraints associated with real transaction can be incorporated without making the problem intractable. Third, it enables us to distinguish two distributions with the same first and second moment but with different third moment. Fourth, we can generate the capital-market line and derive CAPM type equilibrium relations. We compared the piecewise linear risk model with the quadratic risk model using historical data of Tokyo Stock Market, whose results partly support the claims stated above.

A generalized canonical piecewise-linear representation

Abstract: An extension of the well-known canonical representation for continuous piecewise-linear functions is introduced. This form is capable of describing all piecewise-linear functions in two dimensions, being no longer subject to any restrictions. Moreover, it is shown that just one nesting of absolute-value functions is sufficient to cover the whole class. >

Deterministic minimal time vessel routing

Abstract: We develop general methodologies for the minimal time routing problem of a vessel moving in stationary or time dependent environments, respectively. Local optimality considerations, combined with global boundary conditions, result in piecewise continuous optimal policies. In the stationary case, the velocity of the traveling vessel within each subregion depends only on the direction of motion. Variational calculus is used to derive the geometry of piecewise linear extremals. For the time dependent problem, the speed of the vessel within each subregion is assumed to be a known function of time and the direction of motion. Optimal control theory is used to reveal the nature of piecewise continuous optimal policies.

Fredholm determinant for piecewise linear transformations

Abstract: We call the number ξ the lower Lyapunov number. We will study Spec^) , the spectrum of P \\BV> the restriction of P to the subspace BV of functions with bounded variation. The generating function of P is determined by the orbits of the division points of the partition, and the orbits are characterized by a finite dimensional matrix Φ(z) which is defined by a renewal equation (§ 3). Hence, we can show that D(z)=det(I— Φ(#))> which we call a Fredholm determinant, is the determinant of /— #P=ΣίΓ-o zP in the following sense: Theorem A. Let λ G C and assume that \\\\\\>e~. Then λ belongs to Sρec(F) if and only if z—\\~ is a zero of D(z):

Optimal control of the vacation scheme in a M/G/1 queue

Abstract: In this article the M/G/1 queue with server vacations is considered with the assumption that the decision whether or not to take a new vacation, when the system is empty, depends on the number of vacations already taken through a random outcome. Both descriptive and optimization issues are considered, where the latter is done under the expected long-run average cost criterion with linear holding costs, fixed setup costs and a concave piecewise linear reward function for being on vacation. The optimization problem results in an infinite dimensional fractional program of which the solution yields a (deterministic) policy of the control limit type.

Ritz method for dynamic analysis of large discrete linear systems with non-proportional damping

Abstract: Real and complex Ritz vector bases for dynamic analysis of large linear systems with non-proportional damping are presented and compared. Both vector bases are generated utilizing load dependent vector algorithms that employ recurrence equations analogous to the Lanczos algorithm. The choice of static response to fixed spatial loading distribution, as a starting vector in recurrence equations, is motivated by the static correction concept. Different phases of dynamic response analysis are compared with respect to computational efficiency and accuracy. It is concluded that the real vector basis approach is approximately eight times more efficient than the complex vector basis approach. The complex vector basis has some advantages with respect to accuracy, if the excitation is of piecewise linear form, since the exact solution can be utilized. In addition, it is demonstrated that both Ritz vector bases, real and complex, possess superior accuracy over the adequate eigenvector bases.

Adaptive-clustering optical neural net.

Abstract: Pattern recognition techniques (for clustering and linear discriminant function selection) are combined with neural net methods (that provide an automated method to combine linear discriminant functions into piecewise linear discriminant surfaces). The resulting adaptive-clustering neural net is suitable for optical implementation and has certain desirable properties in comparison with other neural nets. Simulation results are provided.

A projection method for the uncapacitated facility location problem

Abstract: Several algorithms already exist for solving the uncapacitated facility location problem. The most efficient are based upon the solution of the strong linear programming relaxation. The dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function. This paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections. The amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints, wherem is the number of clients. For comparison, the underlying linear programming dual hasmn + m + n variables andmn +n constraints, wheren is the number of potential locations for the facilities. The method is flexible as it can handle side constraints. In particular, when there is a duality gap, the linear programming formulation can be strengthened by adding cuts. Numerical results for some classical test problems are included.

Univariate Multiquadric Approximation: Reproduction of Linear Polynomials

Abstract: It is known that multiquadric radial basis function approximations can reproduce low order polynomials when the centres form an infinite regular lattice. We make a start on the interesting question of extending this result in a way that allows the centres to be in less restrictive positions. Specifically, univariate multiquadric approximations are studied when the only conditions on the centres are that they are not bounded above or below. We find that all linear polynomials can be reproduced on IR, which is a simple conclusion if the multiquadrics degenerate to piecewise linear functions. Our method of analysis depends on a Peano kernel formulation of linear combinations of second divided differences, a crucial point being that it is necessary to employ differences in order that certain infinite sums are absolutely convergent. It seems that standard methods cannot be used to identify the linear space that is spanned by the multiquadric functions, partly because it is shown that this space provides uniform convergence to any continuous function on any finite interval of the real line.

A new coordinate transform for the finite element solution of axisymmetric problems in magnetostatics

Abstract: It is pointed out that, although field computation in two-dimensional (2-D) axial symmetric problems has a longstanding history and its software has acquired the status of robustness and reliability, problems with singularities in the neighborhood of r=0 still remain a point of concern. A simple example is shown which exhibits a disturbing lack of accuracy and seems to mock all ideas of reasonability of finite-element solutions. A simple solution to the problem is presented, consisting of transforming the vector potential into a form which is better adapted to the bilinear interpolation used in first-order finite elements. For one-dimensional problems with piecewise linear materials this approach yields exact solutions; an increase in accuracy was obtained for other problems also. >

A subdomain Galerkin/Least squares method for first-order elliptic systems in the plane

Abstract: A finite element method for the approximation of solutions of linear, first-order elliptic systems in two-dimensional domains is considered. The method differs from previous techniques in that the discretization is effected prior to the application of a least squares method. Optimal error estimates in ${\text{\bf H}}^1 (\Omega )$ and ${\text{\bf L}}^2 (\Omega )$-norms are derived for piecewise linear finite element trial spaces. Numerical examples are provided that illustrate the theoretical convergence rates and that indicate that the method is easy to implement and is computationally efficient.

An internal variable formulation for perfectly plastic and linear hardering relations in plasticity

Abstract: We consider a simple internal vairable model of plasticity, with particular emphasis on the case of isotropic hardening. We show that, for this case, the formulation involving sign constrained internal variables appropriate for uniaxial behaviour and piecewise linear yield surfaces can be modified by a simple transformation to provide a formulation which is applicable to continuously differentiable yield surfaces

On the optimal plastic synthesis of frames

Abstract: This paper concerns the first-order optimal plastic synthesis (design) of frames under combined axial force and bending caused by a single load condition. A two phase solution scheme is presented whereby both the restrictions of an initially assumed linear variation in element plastic capacities and of fixed, possibly inaccurate, piecewise linear yield polygons are removed. The first phase implements the so-called nonlinear growth laws while the second phase automatically refines the yield curve linearizations. A simple frame example is used to illustrate these concepts.

A Yield Surface Linearization Procedure in Limit Analysis

Abstract: Within the framework of discretized structural models and piecewise linear yield polyhedra, a heuristic is presented to iteratively improve the limit load estimation by approaching locally the plastic admissibility domain. For the two-dimensional active stress situation, the scheme requires the generation of at the most three vertices, in addition to the initial four, to describe the improved inscribed yield polygon. Two examples are given to illustrate the procedure and to clarify various aspects of the algorithm presented.

Sliding mode-a new way to control series resonant converters

Abstract: The application of the theory of variable structure control systems for the control of resonant DC-DC converters is illustrated. The full-bridge configuration of series-resonant converters (SRCs) is used. The converter is operated at resonant frequency to obtain both output-to-input voltage characteristics similar to those of the buck converter and high efficiency. To arrive at an exact mathematical model, the SRC is viewed as a piecewise linear system. However, under some approximations and a suitably selected set of state variables, the SRC is characterized by a linear continuous-time model which allows determination of the equivalent control via the sliding mode control (SMC) theory. To obtain a sliding regime, the appropriate sliding surface is selected by assigning poles. This control law constitutes a new approach to the control of resonant DC-DC converters. >

A boundary element Galerkin method for a hypersingular integral equation on open surfaces

Abstract: A hypersingular boundary integral equation of the first kind on an open surface piece Γ is solved approximately using the Galerkin method. As boundary elements on rectangles we use continuous, piecewise bilinear functions which vanish on the boundary of Γ. We show how to compensate for the effect of the edge and corner singularities of the true solution of the integral equation by using an appropriately graded mesh and obtain the same convergence rate as for the case of a smooth solution. We also derive asymptotic error estimates in lower-order Sobolev norms via the Aubin–Nitsche trick. Numerical experiments for the Galerkin method with piecewise linear functions on triangles demonstrate the effect of graded meshes and show experimental rates of convergence which underline the theoretical results.

Event-driven behavioural simulation of analogue transfer functions

Abstract: A method is proposed for simulating analogue circuits, represented by their transfer functions, in an event-driven simulator. Analogue waveforms may be voltage or current, and are represented by a piecewise linear approximate. A novel timestep control technique is presented, whereby individual models maintain this approximation within tolerances, permitting different timesteps in different parts of the circuit. The method is demonstrated by the simulation of an FM receiver architecture. >

A local refinement finite-element method for one-dimensional parabolic systems

Abstract: Vector systems of parabolic partial differential equations in one space dimension are solved by an adaptive local mesh refinement Galerkin finite-element procedure. Piecewise linear polynomials are used for the spatial representation of the solution and the backward Euler method is used for temporal integration. A local error indicator based on an estimate of the local discretization error is used to control an adaptive-feedback mesh refinement strategy, where finer space-time meshes are recursively added to coarser ones in regions where greater solution resolution is needed. A posteriori estimates of the local discretization error are obtained by a p-refinement procedure that uses piecewise quadratic hierarchic finite-element approximations in space and trapezoidal rule integration in time. Superconvergence properties of the finite-element method are used to neglect errors at nodes and thereby increase the computational efficiency of the error estimation procedure. Further improvements in computational e...