Path-following interior-point algorithm for monotone linear complementarity problems

This paper presents a path-following full-Newton step interior-point algorithm for solving monotone linear complementarity problems. Under new choices of the defaults of the updating barrier parameter [Formula: see text] and the threshold [Formula: see text] which defines the size of the neighborhood of the central-path, we show that the short-step algorithm has the best-known polynomial complexity, namely, [Formula: see text]. Finally, some numerical results are reported to show the efficiency of our algorithm.

Implementation of the Full-Newton Step Algorithm for Weighted Linear Complementarity Problems

We present a path-following interior-point algorithm for solving the weighted linear complementarity problem from the implementation point of view. We studied two variants, which differ only in the method of updating the parameter which characterizes the central path. The implementation was done in the C++ programming language and the obtained numerical results prove the efficiency of the proposed method.

A Full-Newton Step Interior Point Method for Fractional Programming Problem Involving Second Order Cone Constraint

Some efficient interior-point methods (IPMs) are based on using a self-concordant barrier function related to the feasibility set of the underlying problem.Here, we use IPMs for solving fractional programming problems involving second order cone constraints. We propose a logarithmic barrier function to show the self concordant property and present an algorithm to compute $\varepsilon-$solution of a fractional programming problem. Finally, we provide a numerical example to illustrate the approach.

Regularization by denoising sub-sampled Newton method for spectral CT multi-material decomposition

Spectral Computed Tomography (CT) is an emerging technology that enables us to estimate the concentration of basis materials within a scanned object by exploiting different photon energy spectra. In this work, we aim at efficiently solving a model-based maximum-a-posterior problem to reconstruct multi-materials images with application to spectral CT. In particular, we propose to solve a regularized optimization problem based on a plug-in image-denoising function using a randomized second order method. By approximating the Newton step using a sketching of the Hessian of the likelihood function, it is possible to reduce the complexity while retaining the complex prior structure given by the data-driven regularizer. We exploit a non-uniform block sub-sampling of the Hessian with inexact but efficient conjugate gradient updates that require only Jacobian-vector products for denoising term. Finally, we show numerical and experimental results for spectral CT materials decomposition. This article is part of the theme issue ‘Synergistic tomographic image reconstruction: part 1’.

PRECONDITIONED ITERATIVE METHOD FOR REACTIVE TRANSPORT WITH SORPTION IN POROUS MEDIA

This work deals with the numerical solution of a nonlinear degenerate parabolic equation arising in a model of reactive solute transport in porous media, including equilibrium sorption. The model is a simplified, yet representative, version of multicomponents reactive transport models. The numerical scheme is based on an operator splitting method, the advection and diffusion operators are solved separately using the upwind finite volume method and the mixed finite element method (MFEM) respectively. The discrete nonlinear system is solved by the Newton–Krylov method, where the linear system at each Newton step is itself solved by a Krylov type method, avoiding the storage of the full Jacobian matrix. A critical aspect of the method is an efficient matrix-free preconditioner. Our aim is, on the one hand to analyze the convergence of fixed-point algorithms. On the other hand we introduce preconditioning techniques for this system, respecting its block structure then we propose an alternative formulation based on the elimination of one of the unknowns. In both cases, we prove that the eigenvalues of the preconditioned Jacobian matrices are bounded independently of the mesh size, so that the number of outer Newton iterations, as well as the number of inner GMRES iterations, are independent of the mesh size. These results are illustrated by some numerical experiments comparing the performance of the methods.

Parallel Matrix-Free Higher-Order Finite Element Solvers for Phase-Field Fracture Problems

Phase-field fracture models lead to variational problems that can be written as a coupled variational equality and inequality system. Numerically, such problems can be treated with Galerkin finite elements and primal-dual active set methods. Specifically, low-order and high-order finite elements may be employed, where, for the latter, only few studies exist to date. The most time-consuming part in the discrete version of the primal-dual active set (semi-smooth Newton) algorithm consists in the solutions of changing linear systems arising at each semi-smooth Newton step. We propose a new parallel matrix-free monolithic multigrid preconditioner for these systems. We provide two numerical tests, and discuss the performance of the parallel solver proposed in the paper. Furthermore, we compare our new preconditioner with a block-AMG preconditioner available in the literature.