Numerical Algorithms 

About: Numerical Algorithms is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Numerical analysis & Nonlinear system. It has an ISSN identifier of 1017-1398. Over the lifetime, 3431 publications have been published receiving 58958 citations.

REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems

Abstract: The package REGULARIZATION TOOLS consists of 54 Matlab routines for analysis and solution of discrete ill-posed problems, i.e., systems of linear equations whose coefficient matrix has the properties that its condition number is very large, and its singular values decay gradually to zero. Such problems typically arise in connection with discretization of Fredholm integral equations of the first kind, and similar ill-posed problems. Some form of regularization is always required in order to compute a stabilized solution to discrete ill-posed problems. The purpose of REGULARIZATION TOOLS is to provide the user with easy-to-use routines, based on numerical robust and efficient algorithms, for doing experiments with regularization of discrete ill-posed problems. By means of this package, the user can experiment with different regularization strategies, compare them, and draw conclusions from these experiments that would otherwise require a major programming effert. For discrete ill-posed problems, which are indeed difficult to treat numerically, such an approach is certainly superior to a single black-box routine. This paper describes the underlying theory gives an overview of the package; a complete manual is also available.

A multiprojection algorithm using Bregman projections in a product space

Abstract: Generalized distances give rise to generalized projections into convex sets. An important question is whether or not one can use within the same projection algorithm different types of such generalized projections. This question has practical consequences in the area of signal detection and image recovery in situations that can be formulated mathematically as a convex feasibility problem. Using an extension of Pierra's product space formalism, we show here that a multiprojection algorithm converges. Our algorithm is fully simultaneous, i.e., it uses in each iterative stepall sets of the convex feasibility problem. Different multiprojection algorithms can be derived from our algorithmic scheme by a judicious choice of the Bregman functions which govern the process. As a by-product of our investigation we also obtain blockiterative schemes for certain kinds of linearly constraned optimization problems.

Numerical integration using sparse grids

Abstract: We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49] In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules in several numerical experiments and applications For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction

Detailed error analysis for a fractional Adams method

Abstract: We investigate a method for the numerical solution of the nonlinear fractional differential equation D * α y(t)=f(t,y(t)), equipped with initial conditions y (k)(0)=y 0 (k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.

Regularization Tools Version 4.0 for Matlab 7.3

Abstract: This communication describes version 4.0 of Regularization Tools, a Matlab package for analysis and solution of discrete ill-posed problems. The new version allows for under-determined problems, and it is expanded with several new iterative methods, as well as new test problems and new parameter-choice methods.