Milan Mihajlovic

Bio: Milan Mihajlovic is an academic researcher from University of Manchester. The author has contributed to research in topics: Multigrid method & Finite element method. The author has an hindex of 9, co-authored 28 publications receiving 315 citations. Previous affiliations of Milan Mihajlovic include Cardiff University & University of Niš.

HSL_MI20: An efficient AMG preconditioner for finite element problems in 3D

Abstract: Algebraic multigrid (AMG) is one of the most effective iterative methods for the solution of large, sparse linear systems obtained from the discretization of second-order scalar elliptic self-adjoint partial differential equations. It can also be used as a preconditioner for Krylov subspace methods. In this communication, we report on the design and development of a robust, effective and portable Fortran 95 implementation of the classical Ruge–Stuben AMG, which is available as package HSL_MI20 within the HSL mathematical software library. The routine can be used as a ‘black-box’ preconditioner, but it also offers the user a range of options and parameters. Proper tuning of these parameters for a particular application can significantly enhance the performance of an AMG-preconditioned Krylov solver. This is illustrated using a number of examples arising in the unstructured finite element discretization of the diffusion, the convection–diffusion, and the Stokes equations, as well as transient thermal convection problems associated with the Boussinesq approximation of the Navier–Stokes equations in 3D. Copyright © 2009 John Wiley & Sons, Ltd.

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

Abstract: We examine the convergence characteristics of a preconditioned Krylov subspace solver applied to the linear systems arising from low-order mixed finite element approximation of the biharmonic problem. The key feature of our approach is that the preconditioning can be realized using any “black-box” multigrid solver designed for the discrete Dirichlet Laplacian operator. This leads to preconditioned systems having an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. Numerical results show that the performance of the methodology is competitive with that of specialized fast iteration methods that have been developed in the context of biharmonic problems.

Efficient parallel solvers for the biharmonic equation

Abstract: We examine the convergence characteristics and performance of parallelised Krylov subspace solvers applied to the linear algebraic systems that arise from low-order mixed finite element approximation of the biharmonic problem. Our strategy results in preconditioned systems that have nearly optimal eigenvalue distribution, which consists of a tightly clustered set together with a small number of outliers. We implement the preconditioner operator in a “black-box” fashion using publicly available parallelised sparse direct solvers and multigrid solvers for the discrete Dirichlet Laplacian. We present convergence and timing results that demonstrate efficiency and scalability of our strategy when implemented on contemporary computer architectures.

Numerical solution of saddle point problems

Abstract: Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

A First Course in the Numerical Analysis of Differential Equations: Stiff equations

Abstract: Cambridge University Press. Paperback. Book Condition: New. Paperback. 480 pages. Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems. This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Paperback.

Optimal Solvers for PDE-Constrained Optimization

Abstract: Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, particularly in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large-dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems, which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other PDEs.