Numerical integration of ordinary differential equations with some invariants is considered. For such a purpose, certain projection methods have proved its high accuracy and efficiency, but sometimes they can exhibit instability. In this paper, a new, highly efficient projection method is proposed based on explicit Runge–Kutta methods. The key there is to employ the idea of the perturbed colloc...

We show how the collocation framework that is prevalent in the radial basis function literature can be modified so that the methods can be interpreted in the framework of standard pseudospectral methods. This implies that many of the standard algorithms and strategies used for solving time-dependent as well as time-independent partial differential equations with (polynomial) pseudospectral meth...

A method of estimating a variety of curves by a sequence of piecewise polynomials is proposed, motivated by a Bayesian model and an appropriate summary of the resulting posterior distribution. A joint distribution is set up over both the number and the position of the knots de®ning the piecewise polynomials. Throughout we use reversible jump Markov chain Monte Carlo methods to compute the poste...

We consider the problem of fitting a continuous piecewise linear function to a finite set of data points, modeled as a mathematical program with convex objective. We review some fitting problems that can be modeled as convex programs, and then introduce mixed-binary generalizations that allow variability in the regions defining the best-fit function’s domain. We also study the additional constr...

Most stochastic gradient descent algorithms can optimize neural networks that are sub-differentiable in their parameters; however, this implies the network’s activation function must exhibit a degree of continuity which limits network model’s uniform approximation capacity to continuous functions. This paper focuses on case where discontinuities arise from distinct sub-patterns, each defined di...

In this paper, a self starting two step continuous block hybrid formulae (CBHF) with four Off-step points is developed using collocation and interpolation procedures. The CBHF is then used to produce multiple numerical integrators which are of uniform order and are assembled into a single block matrix equation. These equations are simultaneously applied to provide the approximate solution for t...