We discuss the design of algorithms and codes for the solution of large sparse systems of linear equations on extreme scale computers that are characterized by having many nodes with multi-core CPUs or GPUs. We first use two approaches to get good single node performance. For symmetric systems we use task-based algorithms based on an assembly tree representation of the factorization. We then use runtime systems for scheduling the computation on both multicore CPU nodes and GPU nodes [6]. In this work, we are also concerned with the efficient parallel implementation of the solve phase using the computed sparse factors, and we show impressive results relative to other state-of-the-art codes [3]. Our second approach was to design a new parallel threshold Markowitz algorithm [4] based on Luby’s method [7] for obtaining a maximal independent set in an undirected graph. This is a significant extension since our graph model is a directed graph. We then extend the scope of both these approaches to exploit distributed memory parallelism. In the first case, we base our work on the block Cimmino algorithm [1] using the ABCD software package coded by Zenadi in Toulouse [5, 8]. The kernel for this algorithm is the direct factorization of a symmetric indefinite submatrix for which we use the above symmetric code. To extend the unsymmetric code to distributed memory, we use the Zoltan code from Sandia [2] to partition the matrix to singly bordered block diagonal form and then use the above unsymmetric code on the blocks on the diagonal. In both cases, we illustrate the added parallelism obtained from combining the distributed memory parallelism with the high single-node performance and show that our codes out-perform other state-of-the-art codes. This work is joint with a number of people. We developed the algorithms and codes in an EU Horizon 2020 Project, called NLAFET, that finished on 30 April 2019. Coworkers in this were: Sebastien Cayrols, Jonathan Hogg, Florent Lopez, and Stojce ´ ∗@email 1 Nakov. Collaborators in the block Cimmino part of the project were: Philippe Leleux, Daniel Ruiz, and Sukru Torun. Our codes available on the github repository https://github.com/NLAFET.

References [1] M. ARIOLI, I. S. DUFF, J. NOAILLES, AND D. RUIZ, A block projection method for sparse matrices, SIAM J. Scientific and Statistical Computing, 13 (1992), pp. 47–70. [2] E. BOMAN, K. DEVINE, L. A. FISK, R. HEAPHY, B. HENDRICKSON, C. VAUGHAN, U. CATALYUREK, D. BOZDAG, W. MITCHELL, AND J. TERESCO, Zoltan 3.0: Parallel Partitioning, Load-balancing, and Data Management Services; User’s Guide, Sandia National Laboratories, Albuquerque, NM, 2007. Tech. Report SAND2007-4748W http://www.cs.sandia. gov/Zoltan/ug_html/ug.html. [3] S. CAYROLS, I. S. DUFF, AND F. LOPEZ, Parallelization of the solve phase in a task-based Cholesky solver using a sequential task flow model, Int. J. of High Performance Computing Applications, To appear (2019). NLAFET Working Note 20. RAL-TR-2018-008. [4] T. A. DAVIS, I. S. DUFF, AND S. NAKOV, Design and implementation of a parallel Markowitz threshold algorithm, Technical Report RAL-TR-2019-003, Rutherford Appleton Laboratory, Oxfordshire, England, 2019. NLAFET Working Note 22. Submitted to SIMAX. [5] I. S. DUFF, R. GUIVARCH, D. RUIZ, AND M. ZENADI, The augmented block Cimmino distributed method, SIAM J. Scientific Computing, 37 (2015), pp. A1248–A1269. [6] I. S. DUFF, J. HOGG, AND F. LOPEZ, A new sparse symmetric indefinite solver using a posteriori threshold pivoting, SIAM J. Scientific Computing, To appear (2019). NLAFET Working Note 21. RAL-TR-2018-012. [7] M. LUBY, A simple parallel algorithm for the maximal independent set problem, SIAM J. Computing, 15 (1986), pp. 1036–1053. [8] M. ZENADI, The solution of large sparse linear systems on parallel computers using a hybrid implementation of the block Cimmino method., These de Doctorat, ´ Institut National Polytechnique de Toulouse, Toulouse, France, decembre 2013.

Molecules are dynamical systems that can adopt a variety of three dimensional conformations which, in general, differ in energy and physical properties. The identification of energetically favourable conformations is fundamental in molecular physics and computational chemistry, since it is closely related to important open problems such as the prediction of the folding of proteins and virtual screening for drug design.
In this talk I will present theoretical and data-driven approaches to the study of molecular conformational spaces and their associated energy landscapes. I will show that the topology of the internal molecular conformational space might change after taking its quotient by the group action of a discrete group of symmetries. I will also show that geometric and topological tools for data analysis such as procrustes analysis, local dimensionality reduction, persistent homology and discrete Morse theory provide with efficient methods to study the mathematical structures underlying the molecular conformational spaces and their energy landscapes.
 

The reach is an important geometric invariant of submanifolds of Euclidean space. It is a real-valued global invariant incorporating information about the second fundamental form of the embedding and the location of the first critical point of the distance from the submanifold. In the subject of geometric inference, the reach plays a crucial role. I will give a new method of estimating the reach of a submanifold, developed jointly with Clément Berenfeld, Marc Hoffmann and Krishnan Shankar.

The talk will introduce two general models of random simplicial complexes which extend the highly studied Erdös-Rényi model for random graphs. These models include the well known probabilistic models of random simplicial complexes from Costa-Farber, Kahle, and Linial-Meshulam as special cases. These models turn out to have a satisfying Alexander duality relation between them prompting the hope that information can be transferred for free between them. This turns out to not quite be the case with vanishing probability parameters, but when all parameters are uniformly bounded the duality relation works a treat. Time permitting I may talk about the Rado simplicial complex, the unique (with probability one) infinite random simplicial complex.
This talk is based on various bits of joint work with Michael Farber, Tahl Nowik, and Lewin Strauss.

In this talk, we will provide an overview of results in the setting of granular crystals, consisting of spherical beads interacting through nonlinear elastic spring-like forces. These crystals are used in numerous engineering applications including, e.g., for the production of "sound bullets'' or the examination of bone quality. In one dimension we show that there exist three prototypical types of coherent nonlinear waveforms: shock waves, traveling solitary waves and discrete breathers. The latter are time-periodic, spatially localized structures. For each one, we will analyze the existence theory, presenting connections to prototypical models of nonlinear wave theory, such as the Burgers equation, the Korteweg-de Vries equation and the nonlinear Schrodinger (NLS) equation, respectively. We will also explore the stability of such structures, presenting some explicit stability criteria for traveling waves in lattices. Finally, for each one of these structures, we will complement the mathematical theory and numerical computations with state-of-the-art experiments, allowing their quantitative identification and visualization. Finally, time permitting, ongoing extensions of these themes will be briefly touched upon, most notably in higher dimensions, in heterogeneous or disordered chains and in the presence of damping and driving; associated open questions will also be outlined.

The optimal matching problem is about the rate of convergence
in Wasserstein distance of the empirical measure of iid uniform points
to the Lebesgue measure. We will start by reviewing the macroscopic
behaviour of the matching problem and will then report on recent results
on the mesoscopic behaviour in the thermodynamic regime. These results
rely on a quantitative large-scale linearization of the Monge-Ampere
equation through the Poisson equation. This is based on joint work with
Michael Goldman and Felix Otto.