# Timetable (CATW03)

## Computational complex analysis

Monday 9th December 2019 to Friday 13th December 2019

 09:30 to 09:50 Registration 09:50 to 10:00 Welcome from David Abrahams (Isaac Newton Institute) 10:00 to 11:00 Nick Trefethen (University of Oxford)Masterclass: contour integrals Part 1 of a three-hour series on Complex Computing in MATLAB and Chebfun INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Daan Huybrechs (KU Leuven)PathFinder: a toolbox for oscillatory integrals by deforming into the complex plane We present PathFinder, a Matlab toolbox developed by Andrew Gibbs for the computation of oscillatory integrals. The methods are based on the numerical evaluation of line integrals in the complex plane, which mostly follow paths of steepest descent. The toolbox aims for automation and robustness: the path deformation is found fully automatically, but the paths correspond to those of steepest descent only if doing so has guarantees of correctness. We face a trade-off between achieving asymptotic orders of accuracy, i.e. increasing accuracy with increasing frequency much like asymptotic expansions, and reliable accuracy. In the toolbox we aim for the former, but give precedence to the latter. We conclude with an example of scattering integrals with phase functions that may have clusters of coalescing saddle points. INI 1 12:30 to 13:30 Lunch at Murray Edwards College 13:30 to 14:00 Alex Townsend (Cornell University)Chebyshev to Zolotarev, Faber to Ganelius, and EIM to AAA In 1854, Chebyshev derived the Chebyshev polynomials via a minimax polynomial problem. About 20 years later, Zolotarev (a student in one of Chebyshev's courses) generalized the minimax problem to one involving rational functions. These minimax problems are now used to understand the convergence behavior of Krylov methods, the decay rate of singular values of structured matrices, and the development of fast PDE solvers. In this talk, we will survey the computational complex analysis techniques that can be used to solve Chebyshev's and Zolotarev's minimax problems and try to highlight the ongoing connections between polynomials and rationals. INI 1 14:00 to 14:30 Nick Hale (Stellenbosch University)Numerical Aspects of Quadratic Padé Approximation A classical (linear) Padé approximant is a rational approximation, F(x) = p(x)/q(x), of a given function, f(x), chosen so the Taylor series of F(x) matches that of f(x) to as many terms as possible. If f(x) is meromorphic, then F(x) often provides a good approximation of f(x) in the complex plane beyond the radius of convergence of the original Taylor series. A generalisation of this idea is quadratic Padé approximation, where now polynomials p(x), q(x), and r(x) are chosen so that p(x) + q(x)f(x) + r(x)f^2(x) = O(x^{max}). The approximant, F(x), can then be found by solving p(x) + q(x)F(x) + r(x)F^2(x) = 0 by, for example, the quadratic formula. Since F(x) now contains branch cuts, it typically provides better approximations than linear Padé approximants when f(x) is multi-sheeted, and may be used to estimate branch point locations as well as poles and roots of f(x). In this talk we focus not on approximation properties of Padé approximants, but rather on numerical aspects of their computation. In the linear case things are well-understood. For example, it is well-known that the ill conditioning in the linear system satisfied by p(x) and q(x) means that these are computed with poor relative error, but that in practice, F(x) itself still has good relative accuracy. Luke (1980) formalises this for linear Padé approximants, and we show how this analysis extends to the quadratic case. We discuss a few different algorithms for computing a quadratic Padé approximation, explore some of the problems which arise in the evaluation of the approximant, and demonstrate some example applications. INI 1 14:30 to 15:00 Andrew Gibbs (University College London)Numerical steepest descent for singular and oscillatory integrals Co-Authors: Daan Huybrechs, David Hewett When modelling high frequency scattering, a common approach is to enrich the approximation space with oscillatory basis functions. This can lead to a significant reduction in the DOFs required to accurately represent the solution, which is advantageous in terms of memory requirements and it makes the discrete system significantly easier to solve. A potential drawback is that the each element in the discrete system is a highly oscillatory, and sometimes singular, integral. Therefore an efficient quadrature rule for such integrals is essential for an efficient scattering model. In this talk I will present a new class of quadrature rule we have designed for this purpose, combining Numerical Steepest Descent (which works well for oscillatory integrals) with Generalised Gaussian quadrature (which works well for singular integrals). INI 1 15:00 to 15:30 Matthew Colbrook (University of Cambridge)The Foundations of Infinite-Dimensional Spectral Computations Spectral computations in infinite dimensions are ubiquitous in the sciences and the problem of computing spectra is one of the most studied areas of computational mathematics over the last half-century. However, such computations are infamously difficult, since standard approaches do not, in general, produce correct solutions (the most famous problem in the self-adjoint case is spectral pollution in gaps of the essential spectrum). The goal of this talk is to introduce classes of resolvent based algorithms that compute spectral properties of operators on separable Hilbert spaces. As well as solving computational problems for the first time, these algorithms are proven to be optimal, and the computational problems themselves can be classed in a hierarchy (the SCI hierarchy) with ramifications beyond spectral theory. For concreteness, I shall focus on two problems for a very general class of operators on $l^2(\mathbb{N})$, where algorithms access the matrix elements of the operator: 1) Computing spectra of closed operators in the Attouch-Wets topology (local uniform convergence of closed sets). This algorithm uses estimates of the norm of the resolvent operator and a local minimisation scheme. As well as solving the long-standing computational spectral problem, this algorithm computes spectra with error control. It can also be extended to partial differential operators with coefficients of locally bounded total variation with algorithms point sampling the coefficients. 2) Computing (projection-valued) spectral measures of self-adjoint operators as given by the spectral theorem. This algorithm uses computation of the full resolvent operator (with asymptotic error control) to compute convolutions of rational kernels with the measure before taking a limit. I shall discuss local convergence properties and extensions to computing spectral decompositions (pure point, absolutely continuous and singular continuous parts). Finally, these algorithms are embarrassingly parallelisable. Numerical examples will be given, demonstrating efficiency, and tackling difficult problems taken from mathematics and other fields such as chemistry and physics. INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 16:30 Anastasia Kisil (University of Manchester)Effective ways of solving some PDEs with complicated Boundary Conditions on Unbounded domains This talk will be split into two parts. Firstly, I will talk about the situation when boundary conditions are given on many plates and the associated Wiener-Hopf equation. The Wiener-Hopf method can be considered as a Riemann-Hilbert method with added analytisity information allowing for full power of complex analysis methods to be exploited. I will explain the difficulties involved when matrix Wiener-Hopf equations are solved. I will describe how recent advances in computation of rational approximations and orthogonal polynomials can be used. The corresponding canonical acoustics scattering problems will be considered. This is joint work with Matthew J. Priddin and Lorna J. Ayton. Secondly, I will talk about how special functions called Mathieu function can be used to solve Helmholz equation with boundary conditions given on a plate. This allows to produce a fast and stable code for challenging conditions like varying elasticity and porosity on a plate. This is joint work with Matthew J. Colbrook. INI 1 16:30 to 17:00 Elena Luca (University of California, San Diego)Numerical methods for mixed boundary value problems in diffraction and homogenization theory In this talk, we present fast and accurate numerical methods for the solution of mixed boundary value problems and of the associated matrix Wiener–Hopf problems. The Wiener–Hopf problems are formulated as Riemann–Hilbert problems on the real line, and the numerical approach for such problems of Trogdon & Olver (2015) is employed. It is shown that the known far-field behaviour of the solutions can be exploited to construct tailor-made numerical schemes providing accurate results. A number of scalar and matrix Wiener–Hopf problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane, as well as problems arising in homogenization theory, are solved using the new approach. INI 1 17:00 to 18:00 Welcome Wine Reception at INI
 09:00 to 10:00 Sheehan Olver (Imperial College London); Thomas Trogdon (University of Washington)Masterclass: Riemann–Hilbert problems Riemann–Hilbert problems are complex analytical problems where a jump is specified on a complicated contour, often with multiple self-intersections and open endpoints. The numerical methods for singular integrals in Part I lead naturally to an effective collocation scheme provided extra care is taken at self-intersections to ensure the solution is sectionally analytic and smoothness is preserved. Applications discussed include special functions, integrable PDEs, computing orthogonal polynomials, and computing random matrix statistics. We will demonstrate these results in Julia using RiemannHilbert.jl. INI 1 10:00 to 11:00 Robert Corless (University of Western Ontario)Masterclass: The Computer Algebra System Maple, in 2019 I'm tempted to advertise this as "Not Your Parent's Maple", but on the off-chance that Pedro Gonnet drops in (or watches the recording), I won't. Maple is a computer algebra system and a PSE (Problem Solving Environment) and has been in wide use since its invention in the early 1980's. A lot has changed since then. In this first hour I will give an overview: Getting Started, What you can and can't do in Computer Algebra; Linear Algebra (actually I am going to "punt" that and let you read our chapter in the CRC Handbook of Linear Algebra, Leslie Hogben ed, 2nd ed; there is an online copy you can read in the Betty and Gordon Moore Library); and choose from about a dozen other topics including integration, differential equations, solving multivariate polynomial systems, and a listing of the current packages in Maple (more than a hundred and forty of them). INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Maria Lopez-Fernandez (Università degli Studi di Roma La Sapienza); (Universidad de Málaga)Efficient algorithms for convolutions based on contour integral methods We propose an efficient family of algorithms for the approximation of Volterra integral equations of convolution type arising in two different applications. The first application we consider is the approximation of the fractional integral and associated fractional differential equations. The second application is the resolution of Schrödinger equations with concentrated potentials, which admit a formulation as systems of integral equations. In both cases, we are able to derive fast implementations of Lubich's Convolution Quadrature with very much reduced memory requirements and very easy to implement. Our algorithms are based on special contour integral representations of the Convolution Quadrature weights, according to the application, and special quadratures to compute them. Numerical experiments showing the performance of our methods will be shown. This is joint work with Lehel Banjai (Heriot-Watt University) INI 1 12:30 to 13:30 Lunch at Murray Edwards College 13:30 to 14:00 Ted Johnson (University College London)Vortical sources on the walls of rotating containers: a model for oceanic outflows Fluid of uniform vorticity is expelled from a line source against a wall. An exact analytical solution is obtained for the nonlinear problem determining the final steady state. Sufficiently close to the source, the flow is irrotational and isotropic, turning on the vortical scale $\sqrt{Q/\omega}$ (for area flux $Q$ and vorticity $\omega$) to travel along the wall to the right (for $\omega>0$). The flow is linearly stable with perturbations propagating unattenuated along the interface between vortical and irrotational fluid. Fully nonlinear numerical integrations of the time-dependent equations of motion show that flow started from rest does indeed closely approach the steady state. Superposing an opposing dipole at the origin changes the momentum flux of the flow and leads to the growth of a bulge in the flow near the origin, a phenomenon seen in many rotating outflow experiments. When density variations are allowed the governing equation is no longer Laplace's equation but solutions can be obtained through a long-wave theory.Co-authors: Sean Jamshidi (UCL), Robb McDonald (UCL) INI 1 14:00 to 14:30 Sonia Mogilevskaya (University of Minnesota)Asymptotic modeling of composite materials with thin coatings by using complex variables Co-Authors: Svetlana Baranova (University of Minnesota) Dominik Schillinger and Hoa Nguyen (Leibniz Universität Hannover�) Recent advances in surface chemistry made it possible to create materials with ultrathin high-performance coating layers. Numerical modeling of such structures is a challenging task, as accurate resolution of thin layers with standard continuum-based numerical methods, e.g. FEM or BEM, would require prohibitively fine mesh sizes. To avoid this, it has been proposed in the literature to replace a finite-thickness coating layer by an interface of zero thickness and model the associated jump conditions in the relevant fields. The existing models, however, are low order accurate with respect to the thickness of the layer. The reasons for this considerable limitation are related to theoretical difficulties in constructing accurate higher-order interface models and to computational difficulties in integrating these models into standard FEM formulations characterized by low regularity conditions for the involved fields and geometry. This talk presents a) a new complex variables based approach in developing arbitrary orders interface models for two-dimensional potential problems involving thin isotropic interphase layers and b) a new variationally consistent FEM discretization framework to naturally deal with higher-order derivatives on complex surfaces. Theoretical and computational benefits of the proposed approach will be discussed. INI 1 14:30 to 15:00 Anna Zemlyanova (Kansas State University)Fracture mechanics with Steigmann-Ogden surface energy It is known that surface energy plays an increasingly important role in modeling of objects at nano-scales. In this talk, surface energy model proposed by Steigmann and Ogden will be applied to fracture problems in two-dimensional setting. Introduction of surface energy leads to non-classical boundary conditions incorporating both stresses and derivatives of displacements. The mechanical problem will be reduced to systems of singular integro-differential equations which are further reduced to systems of weakly-singular equations. The existence and uniqueness of the solution will be discussed. It is shown that taking into account surface mechanics leads to the size-dependent solutions. A numerical scheme of the solution of the systems of singular integro-differential equations based on B-splines is suggested, and the numerical results are presented for different values of the mechanical and the geometric parameters. INI 1 15:00 to 15:30 Irina Mitrea (Temple University)Validated Numerics Techniques for Singular Integrals In this talk I will discuss spectral properties of singular integral operators of layer potential type in the class of curvilinear polygons in two dimensions and illustrate how a combination of Harmonic Analysis techniques and Validated Numerics methods can be successfully implemented for establishing well-posedness results for boundary value problems for second order elliptic partial differential operators with constant coefficients in this geometric setting. This is based on joint work with H. Awala, T. Johnson, K. Ott and W. Tucker. INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 16:30 Shari Moskow (Drexel University)Reduced order models for spectral domain inversion: embedding into the continuous problem and generation of internal data.* We generate data-driven reduced order models (ROMs) for inversion of the one and two dimensional Schr\"odinger equation in the spectral domain given boundary data at a few frequencies. The ROM is the Galerkin projection of the Schr\"odinger operator onto the space spanned by solutions at these sample frequencies, and corresponds to a rational interpolant of the Neumann to Dirichlet map. The ROM matrix is in general full, and not good for extracting the potential. However, using an orthogonal change of basis via Lanczos iteration, we can transform the ROM to a block triadiagonal form from which it is easier to extract the unknown coefficient. In one dimension, the tridiagonal matrix corresponds to a three-point staggered finite-difference system for the Schr\"odinger operator discretized on a so-called spectrally matched grid which is almost independent of the medium. In higher dimensions, the orthogonalized basis functions play the role of the grid steps. The orthogonalized basis functions are localized and also depend only very weakly on the medium, and thus by embedding into the continuous problem, the reduced order model yields highly accurate internal solutions. That is to say, we can obtain, just from boundary data, very good approximations of the solution of the Schr\"odinger equation in the whole domain for a spectral interval that includes the sample frequencies. We present inversion experiments based on the internal solutions in one and two dimensions. * joint with L. Borcea, V. Druskin, A. Mamonov, M. Zaslavsky INI 1 16:30 to 17:00 Manfred Trummer (Simon Fraser University)The Szegö Kernel and Oblique Projections: Conformal Mapping of Non-smooth Regions A method for computing the Riemann mapping function of a smooth domain is extended to include the case of simply connected convex regions with corners, in particular convex polygons. The method expresses the Szegö kernel as the solution of an integral equation; the equation is modified to allow for corners in the region. INI 1