Fixed Step Length Research Articles

Algorithms for solving fisk's stochastic traffic assignment model

In this, the logit-based stochastic traffic assignment model is explored. Two new improved algorithms are presented for solving this type of stochastic assignment problem. The major improvement achieved in these algorithms is that the step length in each iteration of the search process is optimized instead of using fixed step lengths as in the existing method of successive averages (MSA).

Properties of an algorithm for solving the inverse problem in radiation therapy

An iterative method for solving the inverse problem in radiation therapy is presented and the corresponding problem of minimising a functional on Rn to R1 is formulated. The appealing properties of the algorithm are that under-dosage is avoided and the physical constraints of non-negativity, which are particular to radiation therapy, are accurately incorporated. It is shown that the solution generated by the algorithm is the closest possible dose distribution to the desired dose distribution under these constraints. The linear convergence of the rather crude gradient method with fixed-step length is compensated by the fact that fast Fourier transform techniques can be used to speed up the calculations. The algorithm is applied to numerical examples and compared with a similar algorithm generating a strict minimisation according to the least-squares norm.

Efficient random walk algorithm for computing conductivity in continuum percolation systems

Random walks can be used to obtain the diffusion constant and thus the conductivity for continuum percolation problems. This paper presents an efficient algorithm that allows walkers to move very large distances in one step. The algorithm uses a first-passage time distribution for d-dimensional spherical surfaces. Results are given for overlapping nonconducting disks in two dimensions. Depending on the density of disks, it is found that the present algorithm is about 5 to 50 times faster than an equivalent algorithm using fixed step lengths.

On the improvement per iteration in Karmarkar's algorithm for linear programming

We investigate the decrease in potential at an iteration of Karmarkar's projective method for linear programming. For a fixed step length parameterα (so that we must have 0 <α ≤ 1) the best possible guaranteeδn(α) inn dimensional space is essentially ln 2 ≃ 0.69; and to achieve this we must takeα about 1. Indeed we show the precise result thatδn(α) equals ln(1 +α)-ln(1 −α/(n − 1)) forn sufficiently large. If we choose an optimal step length at each iteration then this guarantee increases only to aboutδ* ≃ 0.72. We also shed some light on the remarkable empirical observation that the number of iterations required seems scarcely to grow with the size of the problem.

A variable step method for the numerical integration of the one-dimensional Schrödinger equation

Most numerical methods which have been proposed for the approximate integration of the one-dimensional Schrödinger equation use a fixed step length of integration. Such an approach can of course result in gross inefficiency since the small step length which must normally be used in the initial part of the range of integration to obtain the desired accuracy must then be used throughout the integration. In this paper we consider the method of embedding, which is widely used with explicit Runge-Kutta methods for the solution of first order initial value problems, for use with the special formulae used to integrate the Schrödinger equation. By adopting this technique we have available at each step an estimate of the local truncation error and this estimate can be used to automatically control the step length of integration. Also considered is the problem of estimating the global truncation error at the end of the range of integration. The power of the approaches considered is illustrated by means of some numerical examples.