arXiv subjects

Jonas Schulze

Publications and source records attributed to Jonas Schulze.

6 recordsLinked to original sources

arXiv2026

Generalizing Reduced Rank Extrapolation to Low-Rank Matrix Sequences

Reduced rank extrapolation (RRE) is an acceleration method typically used to accelerate the iterative solution of nonlinear systems of equations using a fixed-point process. In this context, the iterates are vectors generated from a fixed-point mapping function. However, when considering the iterative solution of large-scale matrix equations, the iterates are low-rank matrices generated from a fixed-point process for which, generally, the mapping function changes in each iteration. To enable acceleration of the iterative solution for these problems, we propose two novel generalizations of RRE. First, we show how to effectively compute RRE for sequences of low-rank matrices. Second, we derive a formulation of RRE that is suitable for fixed-point processes for which the mapping function changes each iteration. We demonstrate the potential of the methods on several numerical examples involving the iterative solution of large-scale Lyapunov and Riccati matrix equations.

math.NA↗

arXiv2026

Towards a mixed-precision ADI method for Lyapunov equations

We apply mixed-precision to the low-rank Lyapunov ADI (LR-ADI) by performing certain aspects of the algorithm in a lower working precision. Namely, we accumulate the overall solution, solve the linear systems comprising the ADI iteration, and store the inner low-rank factors of the residuals in various combinations of IEEE 754 single and double precision. We empirically test our implementation on Lyapunov equations arising from first- and second-order descriptor systems. For the first-order examples, accumulating the solution in single-precision yields an almost-as-small residual as for the double-precision solution. For certain applications, like computing the H2 norm of a descriptor system, low- or mixed-precision variants of the ADI can be quite competitive

math.NA↗

arXiv2025

Liftings of polynomial systems decreasing the mixed volume

The BKK theorem states that the mixed volume of the Newton polytopes of a system of polynomial equations upper bounds the number of isolated torus solutions of the system. Homotopy continuation solvers make use of this fact to pick efficient start systems. For systems where the mixed volume bound is not attained, such methods are still tracking more paths than necessary. We propose a strategy of improvement by lifting a system to an equivalent system with a strictly lower mixed volume at the expense of more variables. We illustrate this idea providing lifting constructions for arbitrary bivariate systems and certain dense-enough systems.

math.AG↗

arXiv2025

A unifying framework for ADI-like methods for linear matrix equations and beneficial consequences

We derive the alternating-directions implicit (ADI) method based on a commuting operator split and apply the results in detail to the continuous time algebraic Lyapunov equation with low-rank constant term and approximate solution, giving pointers for the Sylvester case. Previously, it has been mandatory to start the low-rank ADI for Lyapunov equations (CF-ADI, LR-ADI, G-LR-ADI) or Sylvester equations (fADI, G-fADI) with an all-zero initial value. Our approach extends the known efficient iteration schemes of low-rank increments and residuals to arbitrary low-rank initial values for all these methods. We further generalize two properties of the low-rank Lyapunov ADI to the generic ADI applied to arbitrary linear equations using a commuting operator split, namely the invariance of iterates under permutations of the shift parameters, and the efficient handling of complex shift parameters. We investigate the performance of arbitrary initial values using two outer iterations in which the low-rank Lyapunov ADI is typically called. First, we solve an algebraic Riccati equation with the Newton method. Second, we solve a differential Riccati equation with a first-order Rosenbrock method. Numerical experiments confirm that the proposed new initial value of the ADI can lead to a significant reduction in the total number of ADI steps, while also showing a 17% and 8x speed-up over the zero initial value for the two equation types, respectively.

math.NA↗

arXiv2023

Diagonally-Addressed Matrix Nicknack: How to improve SpMV performance

We suggest a technique to reduce the storage size of sparse matrices at no loss of information. We call this technique Diagonally-Adressed (DA) storage. It exploits the typically low matrix bandwidth of matrices arising in applications. For memory-bound algorithms, this traffic reduction has direct benefits for both uni-precision and multi-precision algorithms. In particular, we demonstrate how to apply DA storage to the Compressed Sparse Rows (CSR) format and compare the performance in computing the Sparse Matrix Vector (SpMV) product, which is a basic building block of many iterative algorithms. We investigate 1367 matrices from the SuiteSparse Matrix Collection fitting into the CSR format using signed 32 bit indices. More than 95% of these matrices fit into the DA-CSR format using 16 bit column indices, potentially after Reverse Cuthill-McKee (RCM) reordering. Using IEEE 754 double precision scalars, we observe a performance uplift of 11% (single-threaded) or 17.5% (multithreaded) on average when the traffic exceeds the size of the last-level CPU cache. The predicted uplift in this scenario is 20%. For traffic within the CPU's combined level 2 and level 3 caches, the multithreaded performance uplift is over 40% for a few test matrices.

math.NA↗

arXiv2021

On regular and random two-dimensional packing of crosses

Packing problems, even of objects with regular geometries, are in general non-trivial. For few special shapes, the features of crystalline as well as random, irregular two-dimensional (2D) packings are known. The packing of 2D crosses does not yet belong to the category of solved problems. We demonstrate in experiments with crosses of different aspect ratios (arm width to length) which packing fractions are actually achieved by random packing, and we compare them to densest regular packing structures. We determine local correlations of the orientations and positions after ensembles of randomly placed crosses were compacted in the plane until they jam. Short-range orientational order is found over 2 to 3 cross lengths. Similarly, correlations in the spatial distributions of neighbors extend over 2 to 3 crosses. Apparently, there is no simple relation between the geometries of the crosses and peaks in the spatial correlation functions. Some features of the orientational correlations, however, are intuitively evident.

cond-mat.stat-mech↗