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- 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. |
- Artemov, Anton G., 1990-
(författare)
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Parallelization of dynamic algorithms for electronic structure calculations
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The aim of electronic structure calculations is to simulate behavior of complex materials by resolving interactions between electrons and nuclei in atoms at the level of quantum mechanics. Progress in the field allows to reduce the computational complexity of the solution methods to linear so that the computational time scales proportionally to the size of the physical system. To solve large scale problems one uses parallel computers and scalable codes. Often the scalability is limited by the data distribution.This thesis focuses on a number of problems arising in electronic structure calculations, such as inverse factorization of Hermitian positive definite matrices, approximate sparse matrix multiplication, and density matrix purification methods. No assumptions are made about the data distribution, instead, it is explored dynamically.The thesis consists of an introduction and five papers. Particularly, in Paper I we present a new theoretical framework for localized matrices with exponential decay of elements. We describe a new localized method for inverse factorization of Hermitian positive definite matrices. We show that it has reduced communication costs compared to other widely used parallel methods. In Paper II we present a parallel implementation of the method within the Chunks and Tasks programming model and do a scalability analysis based on critical path length estimation.We focus on the density matrix purification technique and its core operation, sparse matrix-matrix multiplication, in Papers III and IV. We analyze the sparse approximate matrix multiplication algorithm with the proposed localization framework, add a prior truncation step, and derive the asymptotic behavior of the Frobenius norm of the error. We employ the sparse approximate multiplication algorithm in the density matrix purification process and propose a method to control the error norm by choosing the right truncation threshold value. We present a new version of the Chunks and Tasks matrix library in Paper V. The library functionality and architecture are described and discussed. The efficiency of the library is demonstrated in a few computational experiments.
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| 3. |
- Axelsson, Owe, et al.
(författare)
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A black-box generalized conjugate gradient minimum residual method based on variable preconditioners and local element approximations
- 2007
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In order to control the accuracy of a preconditioner for an outer iterative process one often involves variable preconditioners. The variability may for instance be due to the use of inner iterations in the construction of the preconditioner. Both the outer and inner iterations may be based on some conjugate gradient type of method, e.g. generalized minimum residual methods.A background for such methods, including results about their computational complexity and rate of convergence, is given. It is then applied for a variable preconditioner arising for matrices partitioned in two-by-two block form. The matrices can be unsymmetric and also indefinite. The aim is to provide a black-box solver, applicable for all ranges of problem parameters such as coefficient jumps and anisotropy.When applying this approach for elliptic boundary value problems, in order to achieve the latter aim, it turns out to be efficient to use local element approximations of arising block matrices as preconditioners for the inner iterations.It is illustrated by numerical examples how the convergence rate of the inner-outer iteration method approaches that for the more expensive fixed preconditioner when the accuracies of the inner iterations increase
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| 6. |
- Axelsson, Owe, et al.
(författare)
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A general approach to analyse preconditioners for two-by-two block matrices
- 2010
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Two-by-two block matrices arise in various applications, such as in domain decomposition methods or, more generally, when solving boundary value problems discretized by finite elements from the separation of the node set of the mesh into 'fine' and 'coarse' nodes. Matrices with such a structure, in saddle point form arise also in mixed variable finite element methods and in constrained optimization problems.A general algebraic approach to construct, analyse and control the accuracy of preconditioners for matrices in two-by-two block form is presented. This includes both symmetric and nonsymmetric matrices, as well as indefinite matrices. The action of the preconditioners can involve element-by-element approximations and/or geometric or algebraic multigrid/multilevel methods.
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