TBA

## Past Seminars 2019-2020

 5 June, 2020 Cancelled Title : TBA Dr Sheehan Olver Reader in department of Mathematics, Imperial College. Abstract : TBA Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building. 22 May, 2020 Cancelled Title : TBA Dr Oliver Rhodes Lecturer in department of Computer Science, The University of Manchester. Abstract : TBA Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building. 20 March, 2020 Cancelled Title : Quasi-Monte Carlo methods for the uncertainty quantification of eigenvalue problems Dr. Alexander Gilbert Institute for Applied Mathematics, Heidelberg University. Abstract : Eigenvalue problems are useful for modelling many important physical phenomena, ranging from photonic crystal structures to quantum mechanics to the neutron diffusion criticality problem. In many of these applications the model parameters are unknown, in which case one aims to quantify the uncertainty in the eigenproblem model. With this as motivation, we will study an elliptic eigenvalue problem with coefficients that depend on infinitely many stochastic parameters. The stochasticity in the coefficients causes the eigenvalues and eigenfunctions to also be stochastic, and so our goal will be to compute the expectation of the minimal eigenvalue. In practice, to approximate this expectation one must: 1) truncate the stochastic dimension; 2) discretise the eigenvalue problem in space (e.g., by finite elements); and 3) apply a quadrature rule to estimate the expected value.In this talk, we will present a multilevel quasi-Monte Carlo method for approximating the expectation of the minimal eigenvalue, which is based on a hierarchy of finite element meshes and truncation dimensions. To improve the sampling efficiency over Monte Carlo we will use a quasi-Monte Carlo rule to generate the sampling points. Quasi-Monte Carlo rules are deterministic (or quasi-random) quadrature rules that are well-suited to high-dimensional integration and can converge at a rate of 1/N, which is faster than the rate for Monte Carlo. Also, to make each eigenproblem solve on a given level more efficient, we utilise the two-grid scheme from [Xu & Zhou 1999] to obtain the eigenvalue on the fine mesh from the coarse eigenvalue (and eigenfunction) with a single linear solve. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building. 27 March, 2020 Cancelled Title : A new framework for solving Lyapunov (and other matrix) equations  Prof. Heike Fassbender Professor of Mathematics, Institut Computational Mathematics, AG Numerik Technische Universität Braunschweig. Abstract : We will consider model order reduction for stable linear time-invariant (LTI) systems $\dot{x}=Ax+Bu,\quad y=Cx$ with real, large and sparse system matrices. In particular, $A$ is a square $n \times n$ matrix, $B$ is rectangular $n \times m,$ and $C$ is $p \times n.$ Among the many existing model order reduction methods our focus will be on (approximate) balanced truncation. The method makes use of the two Lyapunov equations $A\mathfrak{P}+\mathfrak{P}A^T=-BB^T,$ and $A^T \mathfrak{Q}+\mathfrak{Q}A=-C^TC.$ The solutions $\mathfrak{P}$ and $\mathfrak{Q}$ of these equations are called the controllability and observability Gramians, respectively. The balanced truncation method transforms the LTI system into a balanced form whose controllability and observability Gramians become diagonal and equal, together with a truncation of those states that are both difficult to reach and to observe. One way to solve these large-scale Lyapunov equations is via the Cholesky factor–alternating direction implicit (CF–ADI) method which provides a low rank approximation to the exact solution matrix $\mathfrak{P}$, $\mathfrak{Q}$ resp.. After reviewing existing solution techniques, in particular the CF-ADI method, we will present and analyze a system of ODEs, whose solution for $t \rightarrow \infty$ is the Gramian $\mathfrak{P}.$ We will observe that the solution evolves on a manifold and will characterize numerical methods whose approximate low-rank solution evolves on this manifold as well. This will allow us to give a new interpretation of the ADI method. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building. 29 Nov, 2019 Title : Reduced Precision Computing for Weather and Climate Models Dr. Jan Ackmann Postdoctoral Research Associate, Department of Physics, University of Oxford. Abstract : Weather and Climate (W&C) prediction models are required to satisfy strict time-to-solution and energy-to-solution constraints and thus need to be as computationally efficient as possible on modern supercomputers. To explore possible gains in computational efficiency, we follow various approaches: Precision reduction for floating point operations, exploring alternative number formats, and replacing model components with machine-learned surrogates. Resulting computational savings could then be reinvested where they are needed more urgently.The justification for these approaches – that inevitably introduce additional model errors – is motivated by the presence of irreducible uncertainties in W&C model predictions. These uncertainties are due to the interplay of the W&C models’ two main components, the dynamical core, a discretization of the Navier-Stokes equations, and the so-called model physics – a collection of stochastic parametrizations for the unresolved subgrid-scale processes (turbulence, cloud physics, convection,…). In the presence of the resulting uncertainties, high precision is deemed unnecessary for many computational operations and model components.The first part of the talk will be about the group’s work on the use of low precision arithmetic for various model components (Spectral dynamical cores, Adjoint calculation, Data Assimilation, Legendre Transforms, and model physics), where often a level of half precision is found feasible. Also, alternative number formats such as Posits (emulated in software) and neural network approaches that replace parts of the model physics are discussed.The second part of the talk will be a more detailed account on reduced-precision preconditioned elliptic solvers in dynamical cores. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building. 22 Nov, 2019 Title : Compact Finite Differences and Cubic Splines Prof. Robert M. Corless Professor in School of Mathematics, and StatisticalComputational SciencesMathematics, University of Western Ontario. Abstract : In this talk I uncover and explain—using contour integrals and residues—a connection between cubic splines and a popular compact finite difference formula. The connection is that on a uniform mesh the simplest Pad\’e scheme for generating fourth-order accurate compact finite differences gives exactly the derivatives at the interior nodes needed to guarantee twice-continuous differentiability for cubic splines. I also introduce an apparently new spline-like interpolant that I call a compact cubic interpolant; this is similar to one introduced in 1972 by Swartz and Varga, but has higher order accuracy at the edges. I argue that for mildly nonuniform meshes the compact cubic approach offers some potential advantages, and even for uniform meshes offers a simple way to treat the edge conditions, relieving the user of the burden of deciding to use one of the three standard options: free (natural), complete (clamped), or “not-a-knot” conditions. Finally, I establish that the matrices defining the compact cubic splines (equivalently, the fourth-order compact finite difference formulas) are positive definite, and in fact totally nonnegative, if all mesh widths are the same sign. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building. 8 Nov, 2019 Title : Rayleigh quotient optimizations and eigenvalue problems Prof. Zhaojun Bai Professor of Computer Science and Mathematics, University of California, Davis. Abstract : Many computational science and data analysis techniques lead to optimizing Rayleigh quotient (RQ) and RQ type objective functions, such as computing excitation states (energies) of electronic structures, robust classification to handle uncertainty and constrained data clustering to incorporate a prior information. We will discuss origins of recently emerging RQ optimization problem, variational principles, and reformulations to algebraic linear and nonlinear eigenvalue problems. We will show how to exploit underlying properties of eigenvalue problems for designing eigensolvers, and illustrate the efficacy of these solvers in applications. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building. 25 Oct, 2019 Title : Beyond Chebyshev Technology Dr Marcus Webb Lecturer in Department of Mathematics, The University of Manchester. Abstract : Chebfun is a MATLAB software package for computing numerically with functions, whose inner workings boil down essentially to approximating functions by Chebyshev polynomial expansions. In this talk we’ll discuss problems in which Chebyshev polynomials are not the best basis to use, necessitating the transformation to other bases such as Legendre and other Jacobi polynomials (and back again to Chebyshev). The main part of the talk will be on the state-of-the-art algorithm for transforming between different families of Jacobi polynomials, due to Townsend, myself, and Olver (https://doi.org/10.1090/mcom/3277), which involves Toeplitz matrices, Hankel matrices, low-rank matrix approximation, and the FFT. The analysis involves some rational approximation problems of Zolotarev (https://doi.org/10.1137/19M1244433). We conclude with interesting related miscellanea. Time : 2:00 PM to 3:00 PM. Venue : Frank Adams 1, Alan Turing Building.