| 1. |
- Weiss, Michael, 1990-, et al.
(författare)
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Schur complement-based preconditioner applied to 3D controlled-source electromagnetic forward problems of geophysical applications
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper, we propose an iterative algorithm for 3D controlled-source electromagnetic simulations preconditioned by a block lower-triangular preconditioning scheme based on the Schur complement approach. The complex-valued linear system of equations stemming from the spectral element discretisation of the underlying problem is transformed into an equivalent real-valued two-by-two block system. This system is then solved using a generalised conjugate residual method. The algorithm is preconditioned with a block lower-triangular preconditioner which is either based on the Schur complement or an element-by-element approximation thereof. Application of these preconditioners requires computing the inverses of the two diagonal blocks in the block lower-triangular preconditioner as well as one sparse matrix-vector multiplication with the non-zero off-diagonal block. Solving these systems with the two diagonal blocks is either done using a generalised conjugate residual method preconditioned with the multigrid-based auxiliary-space Maxwell preconditioner AMS or using a direct solver. Numerical tests show that simulations based on the preconditioner using the Schur complement converges in accordance to theory meaning in a couple of iterations for most tested cases if the Schur complement is solved directly. In contrast, the number of iterations increases if one solves system with the Schur complement iteratively. Numerical experiments further show that using the element-by-element approximation instead of the Schur complement leads to an increase in the number of iteration required which is expected. Our results also disclose an increase of the iteration count with decreasing conductivity and frequency values. We draw upon eigenvalue spectra to investigate the observed trends and find that the main problem lies in the ill-conditioning of the Schur complement or its approximation.
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| 2. |
- Axelsson, Owe, et al.
(författare)
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An Implementation of a Coarse-Fine Mesh Stabilized Schwarz Method for a Three-Space Dimensional PDE-Problem
- 2024
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Ingår i: Large-Scale Scientific Computations, LSSC 2023. - : Springer. - 9783031562075 - 9783031562082 ; , s. 3-18
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Konferensbidrag (refereegranskat)abstract
- When solving very large scale problems on parallel computer platforms, we consider the advantages of domain decomposition in strips or layers, compared to general domain decomposition splitting techniques. The layer sub-domains are grouped in pairs, ordered as odd-even respectively even-odd and solved by a Schwarz alternating iteration method, where the solution at the middle interfaces of the odd-even groups is used as Dirichlet boundary conditions for the even-odd ordered groups and vice versa. To stabilize the method the commonly used coarse mesh method can be replaced by a coarse-fine mesh method. A component analysis of the arising eigenvectors demonstrates that this solution framework leads to very few Schwarz iterations. The resulting coarse-fine mesh method entails a coarse mesh of a somewhat large size. In this study it is solved by two methods, a modified Cholesky factorization of the whole coarse mesh matrix and a block-diagonal preconditioner, based on the coarse mesh points and the inner node points. Extensive numerical tests show that the latter method, being also computationally cheaper, needs very few iterations, in particular when the domain has been divided in many layers and the coarse to fine mesh size ratio is not too large.
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| 3. |
- Axelsson, Owe, et al.
(författare)
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Numerical and computational efficiency of solvers for two-phase problems
- 2012
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We consider two-phase flow problems, modelled by the Cahn-Hilliard equation. In our work, the nonlinear fourth-order equation is decomposed into a system of two second-order equations for the concentration and the chemical potential.We analyse solution methods based on an approximate two-by-two block factorization of the Jacobian of the nonlinear discrete problem. We propose a preconditioning technique that reduces the problem of solving the non-symmetric discrete Cahn-Hilliard system to the problem of solving systems with symmetric positive definite matrices where off-the-shelf multilevel and multigrid algorithms are directly applicable. The resulting solution methods exhibit optimal convergence and computational complexity properties and are suitable for parallel implementation.We illustrate the efficiency of the proposed methods by various numerical experiments, including parallel results for large scale three dimensional problems.
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| 4. |
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| 5. |
- Axelsson, Owe, et al.
(författare)
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Operator splittings for solving nonlinear, coupled multiphysics problems with an application to the numerical solution of an interface problem
- 2011
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- This work summarizes operator splitting methods to solve various kinds of coupled multiphysics problems. Such coupled problems are usually stiff. Furthermore, one is often interested in obtaining stationary solutions, which require integration over long time intervals. Therefore, an implicit and stable time-stepping method of at least second order of accuracy must be used, to allow for larger timesteps. To control the operator splitting errors for nonlinear problems, an approximate Newton solution method is proposed for each separate equation. After completion of some (normally few) Newton steps, the equations are updated with the current solution, thereby preparing for the next sequence of Newton steps.An application for a nonlinear model of interface tracking problem arising in a multiphase flow is described. Hereby an inner-outer iterative solution method with a proper preconditioning for solving the arising linearized algebraic equations, which results in few iterations, is analyzed. There is no need to update the preconditioner during the iterations.
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| 6. |
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| 7. |
- Axelsson, Owe, et al.
(författare)
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Parallel solution methods and preconditioners for evolution equations
- 2018
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Ingår i: Mathematical Modelling and Analysis. - : Vilnius Gediminas Technical University. - 1392-6292 .- 1648-3510. ; 23, s. 287-308
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Tidskriftsartikel (refereegranskat)abstract
- The recent development of the high performance computer platforms shows a clear trend towards heterogeneity and hierarchy. In order to utilize the computational power, particular attention must be paid to finding new algorithms or adjust existing ones so that they better match the HPC computer architecture. In this work we consider an alternative to classical time-stepping methods based on use of time-harmonic properties and discuss solution approaches that allow efficient utilization of modern HPC resources. The method in focus is based on a truncated Fourier expansion of the solution of an evolutionary problem. The analysis is done for linear equations and it is remarked on the possibility to use two- or multilevel mesh methods for nonlinear problems, which can enable further, even higher degree of parallelization. The arising block matrix system to be solved admits a two-by-two block form with square blocks, for which a very efficient preconditioner exists. It leads to tight eigenvalue bounds for the preconditioned matrix and, hence, to a very fast convergence of a preconditioned Krylov subspace or iterative refinement method. The analytical background is shown as well as some illustrating numerical examples.
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| 8. |
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| 9. |
- Axelsson, Owe, et al.
(författare)
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Preconditioning of boundary value problems using elementwise Schur complements
- 2006
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Based on a particular node ordering and corresponding block decomposition of the matrix we analyse an efficient, algebraic multilevel preconditioner for the iterative solution of finite element discretizations of elliptic boundary value problems. Thereby an analysis of a new version of block-factorization preconditioning methods is presented. The approximate factorization requires an approximation of the arising Schur complement matrix. In this paper we consider such approximations derived by the assembly of the local macro-element Schur complements.The method can be applied also for non-selfadjoint problems but for the derivation of condition number bounds we assume that the corresponding differential operator is selfadjoint and positive definite.
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