TUHH Open Research (TORE)https://tore.tuhh.deTORE captures, stores, indexes, preserves, and distributes digital research material.Fri, 27 Jan 2023 14:01:48 GMT2023-01-27T14:01:48Z5061- Domain-decomposition Based H-LU Preconditionershttp://hdl.handle.net/11420/10614Title: Domain-decomposition Based H-LU Preconditioners
Authors: Le Borne, Sabine; Grasedyck, Lars; Kriemann, Ronald
Abstract: Hierarchical matrices (in short: Hilbert space -matrices) have first been introduced in 1998 [7] and since then have entered into a wide range of applications. They provide a format for the data-sparse representation of fully populated matrices.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/11420/106142007-01-01T00:00:00Z
- Domain decomposition based H-LU preconditioninghttp://hdl.handle.net/11420/10611Title: Domain decomposition based H-LU preconditioning
Authors: Grasedyck, Lars; Kriemann, Ronald; Le Borne, Sabine
Abstract: Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an H-matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In this paper, we develop a new approach to construct the necessary partition based on domain decomposition. Compared to standard geometric bisection based H-matrices, this new approach yields H-LU factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements. These rigorously proven and numerically verified improvements result from an H-matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an LU factorization. We provide numerical results in which a domain decomposition based H-LU factorization is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation. © 2009 Springer-Verlag.
Tue, 03 Mar 2009 00:00:00 GMThttp://hdl.handle.net/11420/106112009-03-03T00:00:00Z
- Parallel black box ℋ-LU preconditioning for elliptic boundary value problemshttp://hdl.handle.net/11420/10606Title: Parallel black box ℋ-LU preconditioning for elliptic boundary value problems
Authors: Grasedyck, Lars; Kriemann, Ronald; Le Borne, Sabine
Abstract: Hierarchical ( ℋ-) matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an ℋ-matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In the context of finite element discretisations of elliptic boundary value problems, ℋ-matrices can be used for the construction of preconditioners such as approximate ℋ-LU factors. In this paper, we develop a new black box approach to construct the necessary partition. This new approach is based on the matrix graph of the sparse stiffness matrix and no longer requires geometric data associated with the indices like the standard clustering algorithms. The black box clustering and a subsequent ℋ-LU factorisation have been implemented in parallel, and we provide numerical results in which the resulting black box ℋ-LU factorisation is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.
Tue, 01 Apr 2008 00:00:00 GMThttp://hdl.handle.net/11420/106062008-04-01T00:00:00Z
- Adaptive geometrically balanced clustering of ℋ-matriceshttp://hdl.handle.net/11420/10637Title: Adaptive geometrically balanced clustering of ℋ-matrices
Authors: Grasedyck, Lars; Hackbusch, Wolfgang; Le Borne, Sabine
Abstract: In [8], a class of (data-sparse) hierarchical (ℋ-) matrices is introduced that can be used to efficiently assemble and store stiffness matrices arising in boundary element applications. In this paper, we develop and analyse modifications in the construction of an ℋ-matrix that will allow an efficient application to problems involving adaptive mesh refinement. In particular, we present a new clustering algorithm such that, when an ℋ-matrix has to be updated due to some adaptive grid refinement, the majority of the previously assembled matrix entries can be kept whereas only a few new entries resulting from the refinement have to be computed. We provide an efficient implementation of the necessary updates and prove for the resulting ℋ-matrix that the storage requirements as well as the complexity of the matrix-vector multiplication are almost linear, i.e., O(nlog(n)).
Fri, 21 May 2004 00:00:00 GMThttp://hdl.handle.net/11420/106372004-05-21T00:00:00Z
- Challenges of order reduction techniques for problems involving polymorphic uncertaintyhttp://hdl.handle.net/11420/2963Title: Challenges of order reduction techniques for problems involving polymorphic uncertainty
Authors: Pivovarov, Dmytro; Willner, Kai; Steinmann, Paul; Brumme, Stephan; Müller, Michael; Srisupattarawanit, Tarin; Ostermeyer, Georg Peter; Henning, Carla; Ricken, Tim; Kastian, Steffen; Reese, Stefanie; Moser, Dieter; Grasedyck, Lars; Biehler, Jonas; Pfaller, Martin; Wall, Wolfgang; Kohlsche, Thomas; Estorff, Otto von; Gruhlke, Robert; Eigel, Martin; Ehre, Max; Papaioannou, Iason; Straub, Daniel; Leyendecker, Sigrid
Abstract: Modeling of mechanical systems with uncertainties is extremely challenging and requires a careful analysis of a huge amount of data. Both, probabilistic modeling and nonprobabilistic modeling require either an extremely large ensemble of samples or the introduction of additional dimensions to the problem, thus, resulting also in an enormous computational cost growth. No matter whether the Monte-Carlo sampling or Smolyak's sparse grids are used, which may theoretically overcome the curse of dimensionality, the system evaluation must be performed at least hundreds of times. This becomes possible only by using reduced order modeling and surrogate modeling. Moreover, special approximation techniques are needed to analyze the input data and to produce a parametric model of the system's uncertainties. In this paper, we describe the main challenges of approximation of uncertain data, order reduction, and surrogate modeling specifically for problems involving polymorphic uncertainty. Thereby some examples are presented to illustrate the challenges and solution methods.
Wed, 01 May 2019 00:00:00 GMThttp://hdl.handle.net/11420/29632019-05-01T00:00:00Z
- ℋ-Matrix preconditioners in convection-dominated problemshttp://hdl.handle.net/11420/10607Title: ℋ-Matrix preconditioners in convection-dominated problems
Authors: Le Borne, Sabine; Grasedyck, Lars
Abstract: Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. In this paper we exploit ℋ-matrix techniques to approximate the LU-decompositions of stiffness matrices as they appear in (finite element or finite difference) discretizations of convectiondominated elliptic partial differential equations. These sparse ℋ-matrix approximations may then be used as preconditioners in iterative methods. Whereas the approximation of the matrix inverse by an ℋ-matrix requires some modification in the underlying index clustering when applied to convectiondominant problems, the ℋ-LU-decomposition works well in the standard ℋ-matrix setting even in the convection dominant case. We will complement our theoretical analysis with some numerical examples. © 2006 Society for Industrial and Applied Mathematics.
Mon, 31 Jul 2006 00:00:00 GMThttp://hdl.handle.net/11420/106072006-07-31T00:00:00Z