L4

Whereas the spectral properties of random matrices has been the subject of numerous studies and is well understood, the statistical properties of the corresponding eigenvectors has only been investigated in the last few years. We will review several recent results and emphasize their importance for cleaning empirical covariance matrices, a subject of great importance for financial applications.

The finite element method (FEM) is one of the great triumphs of applied mathematics, numerical analysis and software development. Recent developments in sensor and signalling technologies enable the phenomenological study of systems. The connection between sensor data and FEM is restricted to solving inverse problems placing unwarranted faith in the fidelity of the mathematical description of the system. If one concedes mis-specification between generative reality and the FEM then a framework to systematically characterise this uncertainty is required. This talk will present a statistical construction of the FEM which systematically blends mathematical description with observations.

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The task of solving large scale linear algebraic problems such as factorising matrices or solving linear systems is of central importance in many areas of scientific computing, as well as in data analysis and computational statistics. The talk will describe how randomisation can be used to design algorithms that in many environments have both better asymptotic complexities and better practical speed than standard deterministic methods.

The talk will in particular focus on randomised algorithms for solving large systems of linear equations. Both direct solution techniques based on fast factorisations of the coefficient matrix, and techniques based on randomised preconditioners, will be covered.

Note: There is a related talk in the Random Matrix Seminar on Tuesday Feb 25, at 15:30 in L4. That talk describes randomised methods for computing low rank approximations to matrices. The two talks are independent, but the Tuesday one introduces some of the analytical framework that supports the methods described here.

Systems of nonlinear polynomial equations arise in a variety of fields in mathematics, science, and engineering.  Many numerical techniques for solving and analyzing solution sets of polynomial equations over the complex numbers, collectively called numerical algebraic geometry, have been developed over the past several decades.  However, since real solutions are the only solutions of interest in many applications, there is a current emphasis on developing new methods for computing and analyzing real solution sets.  This talk will summarize some numerical real algebraic geometric approaches as well as recent successes of these methods for solving a variety of problems in science and engineering.

(Joint work with: Jüri Lember, Heinrich Matzinger, Raul Kangro)

Principal component analysis is an important pattern recognition and dimensionality reduction tool in many applications and are computed as eigenvectors

of a maximum likelihood covariance that approximates a population covariance. The eigenvectors are often used to extract structural information about the variables (or attributes) of the studied population. Since PCA is based on the eigen-decomposition of the proxy covariance rather than the ground-truth, it is important to understand the approximation error in each individual eigenvector as a function of the number of available samples. The combination of recent results of Koltchinskii & Lounici [8] and Yu, Wang & Samworth [11] yields such bounds. In the presented work we sharpen these bounds and show that eigenvectors can often be reconstructed to a required accuracy from a sample of strictly smaller size order.

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.

A well-known theorem of Choquet-Bruhat and Geroch states that for given smooth initial data for the Einstein equations there exists a unique maximal globally hyperbolic development. In particular, time evolution of globally hyperbolic solutions is unique. This talk investigates whether the same result holds for quasilinear wave equations defined on a fixed background. After recalling the notion of global hyperbolicity, we first present an example of a quasilinear wave equation for which unique time evolution in fact fails and contrast this with the Einstein equations. We then proceed by presenting conditions on quasilinear wave equations which ensure uniqueness. This talk is based on joint work with Harvey Reall and Felicity Eperon.
 

A polynomial form is established for the off-shell CHY scattering equations proposed by Lam and Yao. Re-expressing this in terms of independent Mandelstam invariants provides a new expression for the polynomial scattering equations, immediately valid off shell, which makes it evident that they yield the off-shell amplitudes given by massless ϕ3 Feynman graphs. A CHY expression for individual Feynman graphs, valid even off shell, is established through a recurrence relation.

The Camassa-Holm (CH) equation is a nonlinear nonlocal dispersive equation which arises as a model for the propagation of unidirectional shallow water waves over a flat bottom. One of the most important features of the CH equation is the existence of peaked travelling waves, also called peakons. The aim of this talk is to review some asymptotic stability result for peakon solutions for CH-type equations as well as to present some new result for higher-order generalization of the CH equation.