Page 443 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 443
MATRIX COMPUTATIONS AND NUMERICAL (MS-47)
A µ-mode-based integrator for solving evolution equations in Kronecker
form
Marco Caliari, marco.caliari@univr.it
University of Verona, Italy
In this talk, we propose a µ-mode integrator for computing the solution of stiff evolution equa-
tions. The integrator is based on a d-dimensional splitting approach and uses exact (usually
precomputed) one-dimensional matrix exponentials. We show that the action of the exponen-
tials, i.e. the corresponding matrix-vector products, can be implemented efficiently on modern
computer systems. We further explain how µ-mode products can be used to compute spectral
transformations efficiently even if no fast transform is available. We illustrate the performance
of the new integrator by solving three-dimensional Schrödinger equations, and we show that the
µ-mode integrator can significantly outperform numerical methods well established in the field.
This is a joint work with Fabio Cassini, Lukas Einkemmer, Alexander Ostermann, and Franco
Zivcovich.
Random multi-block ADMM: an ALM based view for the Quadratic
Programming case
Stefano Cipolla, stefano.cipolla87@gmail.com
The University of Edinburgh, United Kingdom
Coauthor: Jacek Gondzio
Embedding randomization procedures in the Alternating Direction Method of Multipliers
(ADMM) has recently attracted an increasing amount of interest as a remedy to the fact that
the direct multi-block generalization of ADMM is not necessarily convergent. Even if, in prac-
tice, the introduction of such techniques could mitigate the diverging behaviour of the multi-
block extension of ADMM, from the theoretical point of view, it can ensure just the conver-
gence in expectation, which may not be a good indicator of its robustness and efficiency. In
this work, analysing the strongly convex quadratic programming case, we interpret the block
Gauss-Seidel sweep performed by the multi-block ADMM in the context of the inexact Aug-
mented Lagrangian Method. Using the proposed analysis, we are able to outline an alternative
technique to those present in literature which, supported from stronger theoretical guarantees,
is able to ensure the convergence of the multi-block generalization of the ADMM method.
A Lanczos-like algorithm for time-ordered exponentials
Pierre Louis Giscard, giscard@univ-littoral.fr
Université du Littoral Côte d’Opale, France
Coauthor: Stefano Pozza
The time-ordered exponential is defined as the function that solves a system of coupled first-
order linear differential equations with generally non-constant coefficients. In spite of being at
the heart of much system dynamics, control theory, and model reduction problems, the time-
ordered exponential function remains elusively difficult to evaluate. In this presentation we
will present a Lanczos-like algorithm capable of evaluating it by producing a tridiagonalization
441
A µ-mode-based integrator for solving evolution equations in Kronecker
form
Marco Caliari, marco.caliari@univr.it
University of Verona, Italy
In this talk, we propose a µ-mode integrator for computing the solution of stiff evolution equa-
tions. The integrator is based on a d-dimensional splitting approach and uses exact (usually
precomputed) one-dimensional matrix exponentials. We show that the action of the exponen-
tials, i.e. the corresponding matrix-vector products, can be implemented efficiently on modern
computer systems. We further explain how µ-mode products can be used to compute spectral
transformations efficiently even if no fast transform is available. We illustrate the performance
of the new integrator by solving three-dimensional Schrödinger equations, and we show that the
µ-mode integrator can significantly outperform numerical methods well established in the field.
This is a joint work with Fabio Cassini, Lukas Einkemmer, Alexander Ostermann, and Franco
Zivcovich.
Random multi-block ADMM: an ALM based view for the Quadratic
Programming case
Stefano Cipolla, stefano.cipolla87@gmail.com
The University of Edinburgh, United Kingdom
Coauthor: Jacek Gondzio
Embedding randomization procedures in the Alternating Direction Method of Multipliers
(ADMM) has recently attracted an increasing amount of interest as a remedy to the fact that
the direct multi-block generalization of ADMM is not necessarily convergent. Even if, in prac-
tice, the introduction of such techniques could mitigate the diverging behaviour of the multi-
block extension of ADMM, from the theoretical point of view, it can ensure just the conver-
gence in expectation, which may not be a good indicator of its robustness and efficiency. In
this work, analysing the strongly convex quadratic programming case, we interpret the block
Gauss-Seidel sweep performed by the multi-block ADMM in the context of the inexact Aug-
mented Lagrangian Method. Using the proposed analysis, we are able to outline an alternative
technique to those present in literature which, supported from stronger theoretical guarantees,
is able to ensure the convergence of the multi-block generalization of the ADMM method.
A Lanczos-like algorithm for time-ordered exponentials
Pierre Louis Giscard, giscard@univ-littoral.fr
Université du Littoral Côte d’Opale, France
Coauthor: Stefano Pozza
The time-ordered exponential is defined as the function that solves a system of coupled first-
order linear differential equations with generally non-constant coefficients. In spite of being at
the heart of much system dynamics, control theory, and model reduction problems, the time-
ordered exponential function remains elusively difficult to evaluate. In this presentation we
will present a Lanczos-like algorithm capable of evaluating it by producing a tridiagonalization
441
