GAMM2025

Session

S17: Applied and numerical linear algebra

S17

8 Apr 2025, 08:30

Lecture and Conference Centre

Conveners

S17: Applied and numerical linear algebra: S17.01

• Volker Mehrmann

S17: Applied and numerical linear algebra: S17.02

• Luka Grubišić

S17: Applied and numerical linear algebra: S17.03

• Michal Outrata

Description

The aim of this section is to bring together experts in the field of applied and numerical linear algebra, discussing recent theoretical and algorithmic developments.

599. Accelerating operator Sinkhorn iteration with overrelaxation

André Uschmajew

08/04/2025, 08:30

The operator scaling problem is a generalization of the famous matrix scaling problem and admits several important applications. The operator Sinkhorn iteration (OSI) is an iterative algorithm for finding a solution. In the talk we will introduce the basic convergence theory of this method based on the Hilbert metric on positive definite matrices. In addition, we propose accelerated versions...

600. Mixed-Precision Parallel Tensor Train Operations

Eda Oktay

08/04/2025, 09:10

Tensor Train (TT) decomposition is a widely used low-rank tensor factorization technique known for its memory efficiency and scalability in high-dimensional data. Its advantages have motivated the development of various TT-based methods and applications across fields, such as chemistry, quantum physics, and machine learning. However, constructing low-rank tensors and performing operations in...

601. Mixed-precision techniques for the low-rank Lyapunov ADI

Jonas Schulze

08/04/2025, 09:30

Continuous-time algebraic Lyapunov equations arise in the fields of, e.g., optimal control and model order reduction. For many applications, the coefficient matrices are large and sparse, while the solution matrix has a low numerical rank. In this setting, the alternating-directions implicit (ADI) method, which directly operates on the low-rank factors of the solution matrix, is one of the...

602. A hybrid Chebyshev-Tucker tensor format with applications to multi-particle modelling

Bonan Sun

08/04/2025, 09:50

In this talk, we introduce a mesh-free two-level hybrid Tucker tensor format for the approximation of multivariate functions. This new format combines the product Chebyshev interpolation with the ALS-based Tucker decomposition of the coefficients tensor. The benefits of this tensor approximation are two-fold. On the one hand, it allows to avoid the rank-structured approximation of functional...

603. Mixed-precision iterative refinement for low-rank Lyapunov equations

Xiaobo Liu

08/04/2025, 10:10

We investigate the use of mixed precisions in an iterative refinement framework for solving low-rank Lyapunov matrix equations, where the sign function method with Newton iteration is used as the solver. We also discuss the possibility of using mixed precisions within the Newton iteration itself and propose a mixed-precision truncation scheme based on the QR factorization with column pivoting....

604. Structured rational matrices and their linearizations

Froilán Dopico

08/04/2025, 16:30

Rational matrices, that is, matrices whose entries are univariate rational functions appear in control problems and also in the numerical solution of non-linear eigenvalue problems as approximations of other matrices whose entries are more general univariate functions. Very often the rational matrices arising in applications have particular structures that should be preserved/used in the...

605. System representation of rational functions with poles outside an annulus

Michał Wojtylak

08/04/2025, 17:10

A well known realisation theorem says that a rational matrix function, contractive on the disc, has a representationF(z) = D + C(I−zA)⁻¹B, with appropriately chosen matrices A,B,C,D such that the block matrix [A,B;C,D] is a contraction. A multivariate varsion of this result was obtained in [Grinshpan et al. 2016]. We will use that result to obtain a one-variable result for F(z) rational...

606. The closure of the bundle of a matrix pencil

Patryk Pagacz

08/04/2025, 17:30

Bundles of matrix pencils (under strict equivalence) are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are an infinite union of orbits under strict equivalence). The notion of a bundle for matrix pencils was introduced in the 1990's, following the same notion for matrices under similarity, introduced by Arnold in 1971, and it has been extensively...

607. Regularization and stabilization of port-Hamiltonian descriptor systems via output feedback

Volker Mehrmann

08/04/2025, 17:50

The structure preserving regularization, stabilization, and passivation of (possibly non-regular) linear port-Hamiltonian descriptor (pHDAE) systems by output feedback is discussed. For general descriptor systems the characterization when there exist output feedbacks that lead to an asymptotically stable closed loop system is a very hard and partially an open problem. For systems in pHDAE...

608. On port-Hamiltonian partial differential algebraic equations

Till Preuster

08/04/2025, 18:10

We explore a class of port-Hamiltonian partial differential algebraic equations (pH-pDAEs), characterized by a self-adjoint linear relation and a skew-adjoint operator. We parametrize all their self-adjoint and skew-adjoint realizations making use of extension theory of symmetric relations. To this end, we explicitly construct a boundary triplet for a class of linear relations in range...

610. Towards understanding Krylov subspace methods through examples

Jörg Liesen

09/04/2025, 16:30

Taking the 1952 landmark paper of Hestenes and Stiefel on the conjugate gradient method as their historical starting point, Krylov subspace methods for solving linear algebraic systems have been around for more than 70 years. Tens of thousands of research articles on Krylov subspace methods and their applications have been published by authors coming from the most diverse scientific...

611. Spectral properties of certain nonsymmetric saddle point matrices

Justus Ramme

09/04/2025, 16:50

Linear algebraic systems in saddle point form arise in numerous applications in science and engineering. In general saddle point matrices are highly indefinite which slows down the convergence of many iterative methods, including Krylov subspace methods. By negating the second block row, symmetry of the saddle point matrix is traded for a spectrum that is contained in the right half plane....

612. Spectral analysis of preconditioners for fully implicit Runge-Kutta methods

Michal Outrata

09/04/2025, 17:10

When using implicit Runge-Kutta methods (IRK) for solving linear parabolic or hyperbolic PDEs, solving the system of stage equations M * k = bis usually the computational bottleneck as the dimension of this problem for an s-stage IRK method becomes O(sn) where the spatial discretization dimension n can be very large. Hence the solution process requires the use of iterative solvers, whose...

613. On a matrix-Newton-type framework for solving NEPv

Tom Werner

09/04/2025, 17:30

In this talk, we want to introduce a novel framework for solving nonlinear matrix equations by inexact matrix-valued Newton-Krylov methods. In this framework, we consider global Krylov subspace methods for solving the correction equation and discuss the benefits of this approach. Finally, we show how to use the proposed algorithm to solve nonlinear eigenvector dependent eigenvalue problems...

614. Generic eigenvalue algorithms and singular value algorithms for matrices of quaternions, reduced biquaternions, and dual numbers

Ivan Slapničar

09/04/2025, 17:50

Quaternions are a four-dimensional non-commutative algebra and a division ring of numbers introduced by Hamilton in 1843. The main obstacle in deriving eigenvalue algorithms for matrices of quaternions, due to non-commutativity, is the efficient implementation of shifts. Other linear algebra concepts naturally carry over from real or complex numbers. Reduced biquaternions are a...

609. Pollution free eigenvalue bounds for the Gramian operator

Luka Grubišić

09/04/2025, 18:10

In this talk we will present eigenvalue estimates for the solutions of operator Lyapunov equations with a non-compact (but relatively Hilbert Schmidt) control operator. We compute eigenvalue estimates from Galerkin discretizations of Lyapunov equations and discuss the appearance of spurious (nonconvergent) discrete eigenvalues. This phenomenon is called the spectral pollution. Our main tools,...