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Item type:Publication, Access status: with files , Multilinearization of polynomial models by linear state transformation(Elsevier, 2026-09-05); ; ; Many problems in engineering involve the analysis and model-based control of nonlinear systems. Often a tradeoff exists between the fidelity of the model and the computational complexity of analysis and design algorithms. Multilinear models offer significant advantages in this regard since they can be represented as tensors which enables efficient representation and computation, even in high-dimensional settings. This paper deals with the multilinearization of polynomial models through linear transformation. We are able to provide a comprehensive analysis for the case of two variables. For the higher dimensional case, we propose partial solutions which exploit our results for the two-dimensional case. We perform theoretical runtime analyses of the resulting algorithms which we also illustrate through numerical results.Publicationtype: Journal Article17 7 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Coupled clustering in hierarchical matrices for the oseen problemFluid flow problems can be modelled by the Navier-Stokes or, after linearization, by the Oseen equations. Their discretization results in discrete saddle point problems. These systems of equations are typically very large and need to be solved iteratively. Standard (block-) preconditioning techniques for saddle point problems rely on an approximation of the Schur complement. Such an approximation can be obtained by a hierarchical ((Formula presented.) -) matrix LU-decomposition which first approximates the Schur complement explicitly. The computational complexity of this computation depends, among other things, on the hierarchical block structure of the involved hierarchical matrices. However, widely used techniques do not consider the connection between the discretization grids for the velocity field and the pressure, respectively. Here we present a hierarchical block structure for the finite element discretization of the gradient operator which is improved by considering the connection between the two involved grids. We prove a logarithmic depth estimate for cluster trees generated with the coupled clustering. Numerical results will show that the improved block structure allows for a faster computation of the Schur complement which is the bottleneck for the set-up of the (Formula presented.) -matrix LU-decomposition. The presented coupled clustering is also applicable to other types of mixed (finite element) problems.Publicationtype: Journal Article34 39 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , RBF-FD discretization of the Oseen equationsThe radial basis function - finite difference (RBF-FD) method is a (meshless) technique for the discretization of differential operators on scattered node sets. In recent years, it has been successfully applied mostly to scalar partial differential equations (PDEs). The extension to the application to the steady state Oseen equations on (several) scattered node sets is not straightforward but requires novel components which are the subject of this paper. We consider the steady-state Oseen equations in three spatial dimensions, and as a radial basis function, we restrict ourselves to the polyharmonic spline (PHS) with polynomial augmentation. However, the following contributions of our paper may also be applied to other model problems and RBFs. In particular, we will consider the selection of two node sets for the two types of unknowns, velocity and pressure, and subsequent (flexible order) RBF-FD discretization of the various differential operators in the coupled system. We discuss variants for the discretization of the pressure constraint as well as the influence of the viscosity parameter on the convergence of the RBF-FD discretization. Finally, we provide numerical tests for the Oseen equations in three dimensions on complex domains using several node arrangements, convection directions and parameters inherent to the PHS RBF-FD method. The tests demonstrate that the proposed method is stable for discretization step widths between ℎ𝑢 = 0.01 and ℎ𝑢 = 0.5 and viscosities in the range of 10−3 to 1 not just on the unit cube but also on a more complicated three-dimensional bunny-shaped domain. In particular, for even degrees of polynomial augmentation of the Laplacian (and lower degrees for involved first order differential operators), we can reach convergence of the same (even) order.Publicationtype: Journal Article34 22 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Smaller stencil preconditioners for linear systems in RBF-FD discretizationsRadial basis function finite difference (RBF-FD) discretization has recently emerged as an alternative to classical finite difference or finite element discretization of (systems) of partial differential equations. In this paper, we focus on the construction of preconditioners for the iterative solution of the resulting linear systems of equations. In RBF-FD, a higher discretization accuracy may be obtained by increasing the stencil size. This, however, leads to a less sparse and often also worse conditioned stiffness matrix which are both challenges for subsequent iterative solvers. We propose to construct preconditioners based on stiffness matrices resulting from RBF-FD discretization with smaller stencil sizes compared to the one for the actual system to be solved. In our numerical results, we focus on RBF-FD discretizations based on polyharmonic splines (PHS) with polynomial augmentation. We illustrate the performance of smaller stencil preconditioners in the solution of the three-dimensional convection-diffusion equation.Publicationtype: Journal Article25 59 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Using surrogate models to accelerate load step methods for nonlinear finite element problems in hyperelasticityFinite element methods for displacement problems in hyperelasticity lead to systems of nonlinear equations. These equations are usually solved with Newton's method or a related method. The convergence of Newton's method depends heavily on the proximity of the initial guess to the numerical solution. Load step methods overcome problems with divergence by applying the load in increments, leading to a sequence of sub‐problems with initial guesses closer to the numerical solution of each sub‐problem, supporting the convergence. The downside of this approach is the high computational effort needed to solve the load steps. Based on a benchmark problem in high‐order FEM, we extend traditional load step methods to a new approach exploiting the hierarchical basis used for the spatial discretization of the problem and saving up to 50% of computation time (vs. benchmark).Publicationtype: Journal Article38 33 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , A block householder-based algorithm for the QR decomposition of hierarchical matricesHierarchical matrices are dense but data-sparse matrices that use low-rank factorizations of suitable submatrices to reduce the storage and computational cost to linear-polylogarithmic complexity. In this paper, we propose a new approach to efficiently compute QR factorizations in the hierarchical matrix format based on block Householder transformations. To prevent unnecessarily high ranks in the resulting factors and to increase speed and accuracy, the algorithm meticulously tracks for which intermediate results low-rank factorizations are available. We also use a special storage scheme for the block Householder reflector to further reduce computational and storage costs. Numerical tests for two- and three-dimensional Laplacian boundary element matrices, different radial basis function kernel matrices, and matrices of typical hierarchical matrix structures but filled with random entries illustrate the performance of the new algorithm in comparison to some other QR algorithms for hierarchical matrices from the literature.Publicationtype: Journal Article64 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , An Error-Based Low-Rank Correction for Pressure Schur Complement PreconditionersWe describe a multiplicative low-rank correction scheme for pressure Schur complement preconditioners to accelerate the iterative solution of the linearized Navier-Stokes equations. The application of interest is a model for buoyancydriven fluid flows described by the Boussinesq approximation which combines the Navier-Stokes equations enhanced with a Coriolis term and a temperature advection-diffusion equation. The update method is based on a low-rank approximation to the error between the identity and the preconditioned Schur complement. Numerical results on a cube and a shell geometry illustrate the action of the lowrank correction on spectra of preconditioned Schur complements using known preconditioning techniques, the least-squares commutator and the SIMPLE method. The computational costs of the update method are also investigated. The goal is to analyze whether such an update method can lead to accelerated solvers. Numerical experiments show that the update technique can reduce iteration counts in some cases but (counter-intutively) may increase iteration counts in other settings.Publicationtype: Book Part38 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , A low-rank update for relaxed Schur complement preconditioners in fluid flow problemsThe simulation of fluid dynamic problems often involves solving large-scale saddle-point systems. Their numerical solution with iterative solvers requires efficient preconditioners. Low-rank updates can adapt standard preconditioners to accelerate their convergence. We consider a multiplicative low-rank correction for pressure Schur complement preconditioners that is based on a (randomized) low-rank approximation of the error between the identity and the preconditioned Schur complement. We further introduce a relaxation parameter that scales the initial preconditioner. This parameter can improve the initial preconditioner as well as the update scheme. We provide an error analysis for the described update method. Numerical results for the linearized Navier–Stokes equations in a model for atmospheric dynamics on two different geometries illustrate the action of the update scheme. We numerically analyze various parameters of the low-rank update with respect to their influence on convergence and computational time.Publicationtype: Journal Article32 83 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , A block Cholesky-LU-based QR factorization for rectangular matricesThe Householder method provides a stable algorithm to compute the full QR factorization of a general matrix. The standard version of the algorithm uses a sequence of orthogonal reflections to transform the matrix into upper triangular form column by column. In order to exploit (level 3 BLAS or structured matrix) computational advantages for block-partitioned algorithms, we develop a block algorithm for the QR factorization. It is based on a well-known block version of the Householder method which recursively divides a matrix columnwise into two smaller matrices. However, instead of continuing the recursion down to single matrix columns, we introduce a novel way to compute the QR factors in implicit Householder representation for a larger block of several matrix columns, that is, we start the recursion at a block level instead of a single column. Numerical experiments illustrate to what extent the novel approach trades some of the stability of Householder's method for the computational efficiency of block methods.Publicationtype: Journal ArticleScopus© Citations 1 81 607 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Coupled clustering strategies for hierarchical matrix preconditioners in saddle point problemsFluid flow problems can be modeled by the Navier–Stokes or, after linearization, by the Oseen equations. Their discretization results in discrete saddle point problems. These systems of equations are typically very large and need to be solved iteratively. Standard (block-) preconditioning techniques for saddle point problems rely on an approximation of the Schur complement. Such an approximation can be obtained by a hierarchical (H-) matrix LU-decomposition, which first approximates the Schur complement explicitly. The computational complexity of this computation depends, among other things, on the hierarchical block structure of the involved matrices. However, widely used techniques do not consider the connection between the discretization grids for the velocity field and the pressure, respectively. Here, we present a hierarchical block structure for the finite element discretization of the gradient operator that is improved by considering the connection between the two involved grids. Numerical results imply that the improved block structure allows for a faster computation of the Schur complement, which is the bottleneck for the set-up of the H-matrix LU-decomposition.Publicationtype: Journal Article44 48