University of Namur

Iterative algorithms for the solution of convex quadratic optimization problems are investigated, which exploit inaccurate matrix-vector products. Theoretical bounds on the performance of a Conjugate Gradients method are derived, the necessary quantities occurring in the theoretical bounds estimated and a new practical algorithm derived. Numerical experiments suggest that the new method has significant potential, including in the steadily more important context of multi-precision computations.

The structure of the state of art of scientific research is an important object of study motivated by the understanding of how research evolves and how new fields of study stem from existing research. In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks, in particular keyword co-occurrence networks proved useful to provide maps of knowledge inside a scientific domain. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose to adopt a simplicial complex approach to co-occurrence relations, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the shape of concepts in mathematical research, focusing on homological cycles, regions with low connectivity in the simplicial structure, and we discuss their role in the understanding of the evolution of scientific research. In addition, we map authors’ contribution to the conceptual space, and explore their role in the formation of homological cycles.

Authors: Daniele Cassese, Vsevolod Salnikov, Renaud Lambiotte
 

The talk will describe a new "Brute Force Optimizer" whose objective is to provide a very versatile derivative-free Matlab package for bound-constrained optimization, with the distinctive feature that it can be trained to improve its own performance on classes of problems specified by the user (rather than on a single-but-wide problem class chosen by the algorithm developer).  In addition, BFO can be used to optimize the performance of other algorithms and provides facilities for mixed-integer and multilevel problems, including constrained equilibrium calculations.

The talk will introduce the concepts of multilevel optimization and motivate them in the context of problems arising from the discretization of infinite dimensional applications. It will be shown how optimization methods can accomodate a number of useful (and classical) ideas from the multigrid
community, and thereby produce substantial efficiency improvements compared to existing large-scale minimization techniques.  Two different classes of multilevel methods will be discussed: trust-region and linesearch algorithms.
The first class will be described in the context of a multilevel generalization of the well-known trust-region-Newton method.  The second will focus on limited-memory quasi-Newton algorithms.  Preliminary numerical results will be presented which indicate that both types of multilevel algorithms may be practically very advantageous.

The context of data assimilation in oceanography will be described as well as the computational challenges associated with it. A class of numerical linear algebra methods is described whose purpose is to exploit the problem structure in order to reduce the computational burden and provide provable convergence results for what remains a (very large) nonlinear problem. This class belongs to the Krylov-space family of methods and the special structure used is the imbalance between the dimensions of the state space and the observation space. It is also shown how inexact matrix-vector products can be exploited. Finally, preconditioning issues and resulting adaptations of the trust-region methodology for nonlinear minimization will also be outlined.

By Serge Gratton, Selime Gurol, Philippe Toint, Jean Tshimanga and Anthony Weaver.

A new filter method will be presented that attempts to find a feasible

point for sets of nonlinear sets of equalities and inequalities. The

method is intended to work for problems where the number of variables

or the number of (in)equalities is large, or both. No assumption is

made about convexity. The technique used is that of maintaining a list

of multidimensional "filter entries", a recent development of ideas

introduced by Fletcher and Leyffer. The method will be described, as

well as large scale numerical experiments with the corresponding

Fortran 90 module, FILTRANE.

Many large-scale optimization problems arise in the context of the discretization of infinite dimensional applications. In such cases, the description of the finite-dimensional problem is not unique, but depends on the discretization used, resulting in a natural multi-level description. How can such a problem structure be exploited, in discretized problems or more generally? The talk will focus on discussing this issue in the context of unconstrained optimization and in relation with the classical multigrid approach to elliptic systems of partial differential equations. Both theoretical convergence properties of special purpose algorithms and their numerical performances will be discussed. Perspectives will also be given.

Collaboration with S. Gratton, A. Sartenaer and M. Weber.