Past Computational Mathematics and Applications Seminar

28 May 2020
14:00
Patrick Farrell
Abstract

We discuss two recent extensions of work on Reynolds-robust preconditioners for the Navier-Stokes equations, to non-Newtonian fluids and to the equations of magnetohydrodynamics.  We model non-Newtonian fluids by means of an implicit constitutive relation between stress and strain. This framework is broadly applicable and allows for proofs of convergence under quite general assumptions. Since the stress cannot in general be solved for in terms of the strain, a three-field stress-velocity-pressure formulation is adopted. By combining the augmented Lagrangian approach with a kernel-capturing space decomposition, we derive a preconditioner that is observed to be robust to variations in rheological parameters in both two and three dimensions.  In the case of magnetohydrodynamics, we consider the stationary incompressible resistive Newtonian equations, and solve a four-field formulation for the velocity, pressure, magnetic field and electric field. A structure-preserving discretisation is employed that enforces both div(u) = 0 and div(B) = 0 pointwise. The basic idea of the solver is to split the fluid and electromagnetic parts and to employ our existing Navier-Stokes solver in the Schur complement. We present results in two dimensions that exhibit robustness with respect to both the fluids and magnetic Reynolds numbers, and describe ongoing work to extend the solver to three dimensions.

[To be added to our seminars mailing list, or to receive a Zoom invitation for a particular seminar, please contact trefethen@maths.ox.ac.uk.]

  • Computational Mathematics and Applications Seminar
21 May 2020
14:00
Mark Embree
Abstract

In 2007, Andrew Mayo and Thanos Antoulas proposed a rational interpolation algorithm to solve a basic problem in control theory: given samples of the transfer function of a dynamical system, construct a linear time-invariant system that realizes these samples.  The resulting theory enables a wide range of data-driven modeling, and has seen diverse applications and extensions.  We will introduce these ideas from a numerical analyst's perspective, show how the selection of interpolation points can be guided by a Sylvester equation and pseudospectra of matrix pencils, and mention an application of these ideas to a contour algorithm for the nonlinear eigenvalue problem. (This talk involves collaborations with Michael Brennan (MIT), Serkan Gugercin (Virginia Tech), and Cosmin Ionita (MathWorks).)

[To be added to our seminars mailing list, or to receive a Zoom invitation for a particular seminar, please contact helen.mcgregor@maths.ox.ac.uk.]

  • Computational Mathematics and Applications Seminar
12 March 2020
14:00
Mark Girolami
Abstract

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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  • Computational Mathematics and Applications Seminar
27 February 2020
14:00
Gunnar Martinsson
Abstract

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.

  • Computational Mathematics and Applications Seminar
Abstract

In this talk I will present a Perron-Frobenius type result for nonlinear eigenvector problems which allows us to compute the global maximum of a class of constrained nonconvex optimization problems involving multihomogeneous functions.

I will structure the talk into three main parts:

First, I will motivate the optimization of homogeneous functions from a graph partitioning point of view, showing an intriguing generalization of the famous Cheeger inequality.

Second, I will define the concept of multihomogeneous function and I will state our main Perron-Frobenious theorem. This theorem exploits the connection between optimization of multihomogeneous functions and nonlinear eigenvectors to provide an optimization scheme that has global convergence guarantees.

Third, I will discuss a few example applications in network science and machine learning that require the optimization of multihomogeneous functions and that can be solved using nonlinear Perron eigenvectors.

 

 

  • Computational Mathematics and Applications Seminar
13 February 2020
14:00
Jonathan Hauenstein
Abstract

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.

  • Computational Mathematics and Applications Seminar

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