General Splitting Method Research Articles

Modified general splitting method for the split feasibility problem

Modified general splitting method for the split feasibility problem

An alternated inertial general splitting method with linearization for the split feasibility problem

In this paper, we propose an alternated inertial general splitting method with linearization for a split feasibility problem. Four rules of inertial parameters and relaxation parameters are discussed, where the adaptive inertial parameters are firstly investigated. The convergence of the proposed method is established under standard conditions. The numerical examples are presented to test four choices of the parameters and illustrate the advantage of our methods. An application in signal processing is presented to illustrate the recovery of the proposed algorithm for the sparse signal.

General splitting methods with linearization for the split feasibility problem

In this article, we introduce a general splitting method with linearization to solve the split feasibility problem and propose a way of selecting the stepsizes such that the implementation of the method does not need any prior information about the operator norm. We present the constant and adaptive relaxation parameters, and the latter is “optimal” in theory. These ways of selecting stepsizes and relaxation parameters are also practised to the relaxed splitting method with linearization where the two closed convex sets are both level sets of convex functions. The weak convergence of two proposed methods is established under standard conditions and the linear convergence of the general splitting method with linearization is analyzed. The numerical examples are presented to illustrate the advantage of our methods by comparing with other methods.

Sampling Conditionally on a Rare Event via Generalized Splitting

We propose and analyze a generalized splitting method to sample approximately from a distribution conditional on the occurrence of a rare event. This has important applications in a variety of cont...

Efficient Monte Carlo simulation via the generalized splitting method

We describe a new Monte Carlo algorithm for the consistent and unbiased estimation of multidimensional integrals and the efficient sampling from multidimensional densities. The algorithm is inspired by the classical splitting method and can be applied to general static simulation models. We provide examples from rare-event probability estimation, counting, and sampling, demonstrating that the proposed method can outperform existing Markov chain sampling methods in terms of convergence speed and accuracy.

On a new class of splitting type iterative methods

This paper deals with the stability analysis of a new class of iterative methods for elliptic problems. These schemes are based on a general splitting method, which decomposes a multidimensional parabolic problem into a system of 1D implicit problems. The solution of the given linear system of equations is approximated by p-component vector of approximations. Each splitted operator is applied to only one specific component of this vector. We use energy and spectral stability analysis and investigate the convergence rate of two iterative schemes. Finally, some results of numerical experiments are presented. The dependence of the convergence rate on the smoothness of the solution is investigated.

On a new class of splitting type iterative methods

This paper deals with the stability analysis of a new class of iterative methods for elliptic problems. These schemes are based on a general splitting method, which decomposes a multidimensional parabolic problem into a system of 1D implicit problems. The solution of the given linear system of equations is approximated by p-component vector of approximations. Each splitted operator is applied to only one specific component of this vector. We use energy and spectral stability analysis and investigate the convergence rate of two iterative schemes. Finally, some results of numerical experiments are presented. The dependence of the convergence rate on the smoothness of the solution is investigated.

STABILITY ANALYSIS OF SEIDEL TYPE MULTICOMPONENT ITERATIVE METHOD

This paper deals with the stability analysis of multicomponent iterative methods for solving elliptic problems. They are based on a general splitting method, which decomposes a multidimensional parabolic problem into a system of one dimensional implicit problems. Error estimates in the L 2 norm are proved for the first method. For the stability analysis of Seidel type iterative method we use a spectral method. Two dimensional and three dimensional problems are investigated. Finally, we present results of numerical experiments. Our goal is to investigate the dependence of convergence rates of multicomponent iterative methods on the smoothness of the solution. Hence we solve a discrete problem, which approximates the 3D Poisson's problem. It is proved that the number of iterations depends weakly on the number of grid points if the exact solution and the initial approximation are smooth functions, both. The same problem is also solved by the Stability Correction iterative method. The obtained results indicate a similar behavior.

Matrix stretching for sparse least squares problems

For linear least squares problems min(x) parallel to Ax - b parallel to(2) where A is sparse except for a few dense rows, a straightforward application of Cholesky or QR factorization will lead to catastrophic fill in the factor R. We consider handling such problems by a matrix stretching technique, where the dense rows are split into several more sparse rows. We develop both a recursive binary splitting algorithm and a more general splitting method. We show that for both schemes the stretched problem has the same set of solutions as the original least squares problem. Further. the condition number of the stretched problem differs from that of the original by only a modest factor, and hence the approach is numerically stable. Experimental results from applying the recursive binary scheme to a set of modified matrices from the Harwell-Boeing collection are given. We conclude that when A has a small number of dense rows relative to its dimension, there is a significant gain in sparsity of the factor R. A crude estimate of the optimal number of splits is obtained by analysing a simple model problem. Copyright (C) 2000 John Wiley & Sons, Ltd.