arXiv subjects

Ronald Kriemann

Publications and source records attributed to Ronald Kriemann.

5 recordsLinked to original sources

arXiv2024

Floating Point Compression of Hierarchical Matrix Formats and its Impact on Matrix-Vector Multiplication

Matrix-vector multiplication forms the basis of many iterative solution algorithms and as such is an important algorithm also for hierarchical matrices which are used to represent dense data in an optimized form by applying low-rank compression. However, due to its low computational intensity, the performance of matrix-vector multiplication is typically limited by the available memory bandwidth on parallel systems. With floating point compression the memory footprint can be optimized, which reduces the stress on the memory sub system and thereby increases performance. We will look into the compression of different formats of hierachical matrices and how this can be used to speed up the corresponding matrix-vector multiplication.

cs.DC↗

arXiv2023

Hierarchical Lowrank Arithmetic with Binary Compression

With lowrank approximation the storage requirements for dense data are reduced down to linear complexity and with the addition of hierarchy this also works for data without global lowrank properties. However, the lowrank factors itself are often still stored using double precision numbers. Newer approaches exploit the different IEEE754 floating point formats available nowadays in a mixed precision approach. However, these formats show a significant gap in storage (and accuracy), e.g. between half, single and double precision. We therefore look beyond these standard formats and use adaptive compression for storing the lowrank and dense data and investigate how that affects the arithmetic of such matrices.

cs.MS↗

arXiv2022

A fast iterative algorithm for near-diagonal eigenvalue problems

We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schrödinger perturbation theory and termed Iterative Perturbative Theory (IPT). Contrary to standard eigenvalue algorithms, which are either 'direct' (to compute all eigenpairs) or 'iterative' (to compute just a few), IPT computes any number of eigenpairs with the same basic iterative procedure. Thanks to this perfect parallelism, IPT proves more efficient than classical methods (LAPACK or CUSOLVER for the full-spectrum problem, preconditioned Davidson solvers for extremal eigenvalues). We give sufficient conditions for linear convergence and demonstrate performance on dense and sparse test matrices, including one from quantum chemistry.

math.NA↗

arXiv2021

Identification of unknown parameters and prediction with hierarchical matrices

Statistical analysis of massive datasets very often implies expensive linear algebra operations with large dense matrices. Typical tasks are an estimation of unknown parameters of the underlying statistical model and prediction of missing values. We developed the H-MLE procedure, which solves these tasks. The unknown parameters can be estimated by maximizing the joint Gaussian log-likelihood function, which depends on a covariance matrix. To decrease high computational cost, we approximate the covariance matrix in the hierarchical (H-) matrix format. The H-matrix technique allows us to work with inhomogeneous covariance matrices and almost arbitrary locations. Especially, H-matrices can be applied in cases when the matrices under consideration are dense and unstructured. For validation purposes, we implemented three machine learning methods: the k-nearest neighbors (kNN), random forest, and deep neural network. The best results (for the given datasets) were obtained by the kNN method with three or seven neighbors depending on the dataset. The results computed with the H-MLE method were compared with the results obtained by the kNN method. The developed H-matrix code and all datasets are freely available online.

stat.CO↗

arXiv2019

Semi-Automatic Task Graph Construction for $\mathcal{H}$-Matrix Arithmetic

A new method to construct task graphs for \mcH-matrix arithmetic is introduced, which uses the information associated with all tasks of the standard recursive \mcH-matrix algorithms, e.g., the block index set of the matrix blocks involved in the computation. Task refinement, i.e., the replacement of tasks by sub-computations, is then used to proceed in the \mcH-matrix hierarchy until the matrix blocks containing the actual matrix data are reached. This process is a natural extension of the classical, recursive way in which \mcH-matrix arithmetic is defined and thereby simplifies the efficient usage of many-core systems. Examples for standard and accumulator based \mcH-arithmetic are shown for model problems with different block structures.

cs.MS↗