Past Computational Mathematics and Applications Seminar

6 February 2020
14:00
Raphael Hauser
Abstract

(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.

  • Computational Mathematics and Applications Seminar
30 January 2020
14:00
Abstract

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 ´ ∗iain.duff@stfc.ac.uk 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.

  • Computational Mathematics and Applications Seminar
23 January 2020
14:00
Timo Betcke
Abstract

Boundary integral equations are an elegant tool to model and simulate a range of physical phenomena in bounded and unbounded domains.

While mathematically well understood, the numerical implementation (e.g. via boundary element methods) still poses a number of computational challenges, from the efficient assembly of the underlying linear systems up to the fast preconditioned solution in complex applications. In this talk we provide an overview of some of these challenges and demonstrate the efficient implementation of boundary element methods on modern CPU and GPU architectures. As part of the talk we will present a number of practical examples using the Bempp-cl boundary element software, our next generation boundary element package, that has been developed in Python and supports modern vectorized CPU instruction sets and a number of GPU types.

  • Computational Mathematics and Applications Seminar
28 November 2019
14:00
Philippe Toint
Abstract

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.

  • Computational Mathematics and Applications Seminar
21 November 2019
14:00
Abstract

Everybody is familiar with the concept of eigenvalues of a matrix. In this talk, we consider the nonlinear eigenvalue problem. These are problems for which the eigenvalue parameter appears in a nonlinear way in the equation. In physics, the Schroedinger equation for determining the bound states in a semiconductor device, introduces terms with square roots of different shifts of the eigenvalue. In mechanical and civil engineering, new materials often have nonlinear damping properties. For the vibration analysis of such materials, this leads to nonlinear functions of the eigenvalue in the system matrix.

One particular example is the sandwhich beam problem, where a layer of damping material is sandwhiched between two layers of steel. Another example is the stability analysis of the Helmholtz equation with a noise excitation produced by burners in a combustion chamber. The burners lead to a boundary condition with delay terms (exponentials of the eigenvalue).


We often receive the question: “How can we solve a nonlinear eigenvalue problem?” This talk explains the different steps to be taken for using Krylov methods. The general approach works as follows: 1) approximate the nonlinearity by a rational function; 2) rewrite this rational eigenvalue problem as a linear eigenvalue problem and then 3) solve this by a Krylov method. We explain each of the three steps.

  • Computational Mathematics and Applications Seminar
14 November 2019
14:00
Massimiliano Ferronato
Abstract

The fully coupled numerical simulation of different physical processes, which can typically occur
at variable time and space scales, is often a very challenging task. A common feature of such models is that
their discretization gives rise to systems of linearized equations with an inherent block structure, which
reflects the properties of the set of governing PDEs. The efficient solution of a sequence of systems with
matrices in a block form is usually one of the most time- and memory-demanding issue in a coupled simulation.
This effort can be carried out by using either iteratively coupled schemes or monolithic approaches, which
tackle the problem of the system solution as a whole.

This talk aims to discuss recent advances in the monolithic solution of coupled multi-physics problems, with
application to poromechanical simulations in fractured porous media. The problem is addressed either by proper
sparse approximations of the Schur complements or special splittings that can partially uncouple the variables
related to different physical processes. The selected approaches can be included in a more general preconditioning
framework that can help accelerate the convergence of Krylov subspace solvers. The generalized preconditioner
relies on approximately decoupling the different processes, so as to address each single-physics problem
independently of the others. The objective is to provide an algebraic framework that can be employed as a
general ``black-box'' tool and can be regarded as a common starting point to be later specialized for the
particular multi-physics problem at hand.

Numerical experiments, taken from real-world examples of poromechanical problems and fractured media, are used to
investigate the behaviour and the performance of the proposed strategies.

  • Computational Mathematics and Applications Seminar
7 November 2019
14:00
Abstract

Domain decomposition methods are widely employed for the numerical solution of partial differential equations on parallel computers. We develop an adjoint-based a posteriori error analysis for overlapping multiplicative Schwarz domain decomposition and for overlapping additive Schwarz. In both cases the numerical error in a user-specified functional of the solution (quantity of interest), is decomposed into a component that arises due to the spatial discretization and a component that results from of the finite iteration between the subdomains. The spatial discretization error can be further decomposed in to the errors arising on each subdomain. This decomposition of the total error can then be used as part of a two-stage approach to construct a solution strategy that efficiently reduces the error in the quantity of interest.

  • Computational Mathematics and Applications Seminar
Niall Bootland
Abstract

The development of effective solvers for high frequency wave propagation problems, such as those described by the Helmholtz equation, presents significant challenges. One promising class of solvers for such problems are parallel domain decomposition methods, however, an appropriate coarse space is typically required in order to obtain robust behaviour (scalable with respect to the number of domains, weakly dependant on the wave number but also on the heterogeneity of the physical parameters). In this talk we introduce a coarse space based on generalised eigenproblems in the overlap (GenEO) for the Helmholtz equation. Numerical results within FreeFEM demonstrate convergence that is effectively independent of the wave number and contrast in the heterogeneous coefficient as well as good performance for minimal overlap.

  • Computational Mathematics and Applications Seminar
24 October 2019
14:00
Fredrik Johansson
Abstract

Can we get rigorous answers when computing with real and complex numbers? There are now many applications where this is possible thanks to a combination of tools from computer algebra and traditional numerical computing. I will give an overview of such methods in the context of two projects I'm developing. The first project, Arb, is a library for arbitrary-precision ball arithmetic, a form of interval arithmetic enabling numerical computations with rigorous error bounds. The second project, Fungrim, is a database of knowledge about mathematical functions represented in symbolic form. It is intended to function both as a traditional reference work and as a software library to support symbolic-numeric methods for problems involving transcendental functions. I will explain a few central algorithmic ideas and explain the research goals of these projects.

  • Computational Mathematics and Applications Seminar

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