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Timetable (CATW03)

Computational complex analysis

Monday 9th December 2019 to Friday 13th December 2019

Monday 9th 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
Tuesday 10th December 2019
09:00 to 09:30 Maria Christina van der Weele (University of Cambridge)
Integrable Systems in Multidimensions
One of the main current topics in the field of integrable systems concerns the existence of nonlinear integrable evolution equations in more than two spatial dimensions. The fact that such equations exist has been proven by A.S. Fokas [1], who derived equations of this type in four spatial dimensions, which however had the disadvantage of containing two time dimensions. The associated initial value problem for such equations, where the dependent variables are specified for all space variables at t1 = t2 = 0, can be solved by means of a nonlocal d-bar problem. The next step in this program is to formulate and solve nonlinear integrable systems in 3+1 dimensions (i.e., with three space variables and a single time variable) in agreement with physical reality. The method we employ is to first construct a system in 4+2 dimensions, with the aim to reduce this then to 3+1 dimensions.

In this talk we focus on the Davey-Stewartson system [2] and the 3-wave interaction equations [3]. Both these integrable systems have their origins in fluid dynamics where they describe the evolution and interaction, respectively, of wave packets on e.g. a water surface. We start from these equations in their usual form in 2+1 dimensions (two space variables x, y and one time variable t) and we bring them to 4+2 dimensions by complexifying each of these variables. We solve the initial value problem of these equations in 4+2 dimensions. Subsequently, in the linear limit we reduce this analysis to 3+1 dimensions to comply with the natural world. Finally, we discuss the construction of the 3+1 reduction of the full nonlinear problem, which is currently under investigation.

This is joint work together with my PhD supervisor Prof. A.S. Fokas.

References
[1] A.S. Fokas, Integrable Nonlinear Evolution Partial Differential Equations in 4+2 and 3+1 Dimensions, Phys. Rev. Lett. 96 (2006), 190201.
[2] A.S. Fokas and M.C. van der Weele, Complexification and integrability in multidimensions, J. Math. Phys. 59 (2018), 091413.
[3] M.C. van der Weele and A.S. Fokas, Solving the Initial Value Problem for the 3-Wave Interaction Equations in Multidimensions (to be submitted, 2019).
INI 1
09:30 to 10:00 Percy Deift (Courant Institute of Mathematical Sciences); (New York University)
Universality in numerical analysis. Cyber algorithms
 It turns out that for a wide variety of numerical algorithms with random data, the stopping times for the algorithms to achieve a given accuracy, have universal fluctuations independent of the ensemble for the random data. The speaker will discuss various experimental and analytical results  illustrating universality, with particular emphasis on recent work on universality in cyber algorithms. 

This is  joint work at various stages with a number of authors, Christian Pfrang, Govind Menon, Sheehan Olver, Steven Miller, and particularly Tom Trogdon.
INI 1
10:00 to 11:00 Nick Trefethen (University of Oxford)
Masterclass: polynomial and rational approximation
Part 2 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 Pavel Lushnikov (University of New Mexico); (Landau Institute for Theoretical Physics)
Motion of complex singularities and Hamiltonian integrability of surface dynamics
A motion of fluid's free surface is considered in two dimensional
(2D) geometry. A time-dependent conformal transformation maps a
fluid domain into the lower complex half-plane of a new spatial
variable. The fluid dynamics is fully characterized by the complex
singularities in the upper complex half-plane of the conformal map
and the complex velocity. Both a single ideal fluid dynamics
(corresponds e.g. to oceanic waves dynamics) and a dynamics of
superfluid Helium 4 with two fluid components are considered. Both
systems share the same type of the non-canonical Hamiltonian
structure. A superfluid Helium case is shown to be completely
integrable for the zero gravity and surface tension limit with the
exact reduction to the Laplace growth equation which is completely
integrable through the connection to the dispersionless limit of the
integrable Toda hierarchy and existence of the infinite set of
complex pole solutions. A single fluid case with nonzero gravity and
surface tension turns more complicated with the infinite set of new
moving poles solutions found which are however unavoidably coupled
with the emerging moving branch points in the upper half-plane.
Residues of poles are the constants of motion. These constants
commute with each other in the sense of underlying non-canonical
Hamiltonian dynamics. It suggests that the existence of these extra
constants of motion provides an argument in support of the
conjecture of complete Hamiltonian integrability of 2D free surface
hydrodynamics.
INI 1
12:30 to 13:30 Lunch at Murray Edwards College
13:30 to 14:00 Yuji Nakatsukasa (University of Oxford); (National Institute of Informatics)
Vandermonde with Arnoldi
Vandermonde matrices are exponentially ill-conditioned, rendering the familiar “polyval(polyfit)” algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees. We show that Arnoldi orthogonalization fixes the problem.
INI 1
14:00 to 14:30 Andrew Horning (Cornell University)
Computing spectral measures of differential and integral operators
Unlike its matrix counterpart, the spectral measure of a self-adjoint operator may have an absolutely continuous component and an associated density function, e.g., in applications posed on unbounded domains. The state-of-the-art computational methods for these problems typically approximate the density function using a smoothed sum of Dirac measures, corresponding to the spectral measure of a matrix discretization of the operator. However, it is often difficult to determine the smoothing and discretization parameters that are necessary to accurately and efficiently resolve the density function. In this talk, we present an adaptive framework for computing the spectral measure of a self-adjoint operator that provides insight into the selection of smoothing and discretization parameters. We show how to construct local approximations to the density that converge rapidly when the density function is smooth and discuss possible connections with Pade approximation that could alleviate deteriorating convergence rates near non-smooth points.
INI 1
14:30 to 15:00 Abinand Gopal (University of Oxford)
Solving the Laplace and Helmholtz equation on domains with corners using rational functions and their analogs
The solutions to elliptic PDEs on domains with corners often exhibit singularities at the corners, which pose difficulties for numerical methods. Recently, it was found that the approximation power of rational functions can be used in this context to obtain higher-order convergence, in a manner related to the method of fundamental solutions. In this talk, we focus on the extension of these ideas to the case of Helmholtz equation.
INI 1
15:00 to 15:30 Marcus Webb (University of Manchester)
Approximation Theory for a Rational Orthogonal Basis on the Real Line

The Malmquist-Takenaka basis is a rational orthogonal basis constructed by mapping the Laurent basis from the unit circle to the real line by a Möbius transformation and multiplying by a weight to ensure orthogonality. Over the last century its properties have piqued the interest of various researchers including Boyd, Weideman, Christov, and Wiener. Despite this history, the approximation theory of this basis still defies straightforward description. For example, it was shown by Boyd and Weideman that for entire functions the convergence of approximation is superalgebraic, but that exponential convergence is only possible if the function is analytic at infinity (i.e. at the top of the Riemann sphere --- quite a strong condition). Nonetheless, convergence can be surprisingly quick, and the main body of this talk will be the result that wave packets clearly cannot have exponentially convergent approximations, but they /initially/ exhibit exponential convergence for large wave packet frequencies with exponential convergence rate proportional to said frequency. Hence, O(log(|eps|) omega) coefficients are required to resolve a wave packet to an error of O(eps). The proof is by the method of steepest descent in the complex plane. This is joint work with Arieh Iserles and Karen Luong (Cambridge).

INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 16:30 Mikael Slevinsky (University of Manitoba)
Fast and stable rational approximation of generalized hypergeometric functions

Generalized hypergeometric functions are a central tool in the theory of special functions and complex analysis. Rational approximation permits analytic continuation of formal power series well beyond their radii of convergence. In this talk, we will use sequence transformations to convert successive partial sums of generalized hypergeometric series into rational approximants, which are in certain cases Padé approximants. We will describe algorithms for their computation in linear time. This improvement, over traditional algorithms with quadratic complexity, also increases their numerical stability and offers insight into the localization of their poles.

INI 1
16:30 to 17:00 Birgit Schörkhubre (Karlsruhe Institute of Technology (KIT))
Blowup for the supercritical cubic wave equation
This talk is concerned with recent results on singularity formation for the focussing cubic wave equation. We found that in the energy supercritical regime this equation admits an explicit self-similar blowup solution, which is stable along a codimension one manifold. Furthermore, based on numerical experiments, we conjecture that this manifold is a threshold between finite-time blowup and dispersion. This is joint work with Irfan Glogić and Maciej Maliborski (University of Vienna).
INI 1
Wednesday 11th December 2019
09:00 to 10:00 Sheehan Olver (Imperial College London); Thomas Trogdon (University of Washington)
Masterclass: singular integrals and orthogonal polynomials
Orthogonal polynomials are fundamental tools in numerical methods, including for numerical methods for singular integral equations. A known result is that Cauchy transforms of weighted orthogonal polynomials satisfy the same three-term recurrences as the orthogonal polynomials themselves for n > 0. This basic fact leads to extremely effective schemes of calculating singular integrals that converge spectrally fast (faster than any algebraic power), uniformly in the complex plane. Closed formulae for Cauchy transforms on more complicated geometries are derivable using the Plemelj lemma. These techniques extend to other singular integrals such as those with logarithmic kernels.

We will demonstrate these results in Julia using ApproxFun.jl and SingularIntegralEquations.jl.
INI 1
10:00 to 11:00 Nick Trefethen (University of Oxford)
Masterclass: conformal mapping
Part 3 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 Folkmar Bornemann (Technische Universität München)
Numerical Generatingfunctionology: Counting with Toeplitz Determinants, Hayman-Admissibility, and the Wiener-Hopf-Factorization
Counting related to representation theory and symmetric functions can be framed as generating functions given by Toeplitz determinants. Prime examples are counting all permutations with no long increasing subsequence or lattice paths in last passage percolation. Intricate scaling limits of those generating functions have been used, e.g., in the seminal work by Baik/Deift/Johansson, to obtain asymptotic formulae in terms of random matrix theory. In this talk, we address the question whether generating functions can be used to numerically extract the counts in a mesoscopic regime where combinatorial methods are already infeasible and the random matrix asymptotics is still too inaccurate. The stable computation of the counts by means of complex analysis is possible, indeed, and can be explained by the theory of Hayman admissibility. As a bonus track from complex analysis, the numerical evaluation of the Toeplitz determinant itself has to be stabilized by a variant of the Borodin-Okounkov formula based on the Wiener-Hopf factorization. This way, we obtain, e.g., exact 1135-digit counts in permutations of order 500 or, by taking Hayman’s famous generalization of Stirling’s formula at face value, a blazingly fast, surprisingly robust and accurate numerical asymptotics.
INI 1
12:30 to 13:30 Lunch at Murray Edwards College
13:30 to 18:00 Free afternoon
19:30 to 22:00 Formal Dinner at Christ's College

 

LOCATION
Christ's College (Hall)
St Andrew's Street, CB2 3BU - map

DRESS CODE
Smart casual

Thursday 12th December 2019
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
Friday 13th December 2019
09:00 to 10:00 Sheehan Olver (Imperial College London); Thomas Trogdon (University of Washington)
Masterclass: oscillatory Riemann-Hilbert problems
Riemann-Hilbert problems arising in applications are often oscillatory presenting challenges to their numerical solution. An effective scheme for determining their asymptotic behaviour is Deift-Zhou steepest descent, which mirrors steepest descent for oscillatory integrals by deforming to paths that turn oscillations to exponential decay. This is a fundamental result that lead to numerous important rigorous asymptotic results over the last 35+ years. This technique proves useful for numerics as well providing a convergent approach that is accurate both in the asymptotic and non-asymptotic regime. Recent progress on going beyond steepest descent and solving oscillatory problems without deformation using GMRES is also discussed.
INI 1
10:00 to 11:00 Robert Corless (University of Western Ontario)
Masterclass: Programming in Maple: an extended example using Bohemians

A Bohemian is a BOunded HEight Matrix of Integers (BOHEMI, close enough). Recently I have become very interested in such things; see bohemianmatrices.com for some reasons why. In this hour we will look at a collection of Maple procedures designed (this month!) to answer questions about complex symmetric, tridiagonal, irreducible, zero-diagonal Bohemians (a new class, chosen just for this workshop). This means that there will be (#P)^(m-1) m-dimensional such matrices for a given population of elements P, which for this class cannot contain zero. We will look at fast ways to generate these matrices, how to generate fast(ish) code to compute the characteristic polynomials (and why), and generally use this topic as an excuse to learn some Maple programming.

The hour will assume some familiarity with programming; for instance, if you know Matlab, then you very nearly know Maple already (in some ways they are similar enough that it causes confusion, unfortunately). But it will not be necessary; I hope to encourage a friendly atmosphere and we'll generate some interesting (I hope) images, and perhaps some interesting mathematical conjectures. But even if you know Maple well, you might learn something interesting. All the scripts/worksheets/workbooks have been made available at http://publish.uwo.ca/~rcorless/Maple2019/ so you may download them and run and modify the examples yourself, and generate your own Bohemian images.

Indeed I believe that it is entirely likely that you will be able to formulate your own Bohemian conjectures during this activity; and it has been known for participants to prove theorems about them, during the lecture. Who knows, perhaps your next paper will get its main result during this activity.

Licences for Maple valid for one month have been generously provided for participants by Maplesoft. There will be a representative from Maplesoft here to answer any questions.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Christian Klein (Université de Bourgogne)
Computational approach to compact Riemann surfaces
A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the de ning equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw-Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw-Curtis algorithm and contour integrals. A special approach is presented for hyperelliptic curves in Weierstrass normal form.
INI 1
12:30 to 13:30 Lunch at Murray Edwards College
13:30 to 14:00 Bernard Deconinck (University of Washington)
The numerical solution of BVPs using the Unified Transform Method
Using a semi-discrete analog of the Unified Transform Method of Fokas, we explore the possibility of devising finite-difference schemes that take boundary conditions into account properly. This is done by combining the exact solution for the linear problem with the solution of the nonlinear problem in a split-step approach. In this talk, I will discuss the first step of this project, where the solution of the linear problem is explored.

Co-author: 
Jorge Cisneros (University of Washington)
INI 1
14:00 to 14:30 Tom DeLillo (Wichita State University)
Remarks on numerical methods for conformal mapping of multiply connected domains and applications

Conformal maps from multiply connected domains with circular boundaries to physical domains and complex velocity potentials for potential flow problems can be represented by Laurent series in the circle domains. The linear systems arising in the computation of the truncated series have a block structure in the form of the discrete Fourier transform plus low rank matrices representing the interaction of the circles plus certain auxiliary
parameters such as the conformal moduli or the circulations of the flow. The conjugate gradient method can be used to solve these systems efficiently. We will give an outline of this approach. Time permitting, we will also give several examples of conformal map calculations.

INI 1
14:30 to 15:00 Olivier Sète (Technische Universität Berlin)
Computing Walsh's conformal map onto lemniscatic domains
The conformal mapping of multiply connected domains onto lemniscatic domains was introduced by Walsh in 1956. Walsh's map is a direct generalization of the Riemann mapping. Several important quantities, such as the Green's function with pole at infinity of the multiply connected domain, or the logarithmic capacity of its complemet, can be read of the lemniscatic domain and Walsh's map. In this talk, we discuss the computation of this conformal mapping and lemniscatic domain. This is based on joint work with Jörg Liesen and Mohamed Nasser.
INI 1
15:00 to 15:30 Peter Baddoo (Imperial College London)
Computing periodic conformal mappings
Conformal mappings are used to model a range of phenomena in the physical sciences. Although the Riemann mapping theorem guarantees the existence of a mapping between two conformally equivalent domains, actually constructing these mappings is extremely challenging. Moreover, even when the mapping is known in principle, an efficient representation is not always available. Accordingly, we present techniques for rapidly computing the conformal mapping from a multiply connected canonical circular domain to a periodic array of polygons. The boundary correspondence function is found by solving the parameter problem for a new periodic Schwarz--Christoffel formula. We then represent the mapping using rational function approximation. To this end, we present a periodic analogue of the adaptive Antoulas--Anderson (AAA) algorithm to obtain the relevant support points and weights. The procedure is extremely fast; evaluating the mappings typically takes around 10 microseconds. Finally, we leverage the new algorithms to solve problems in fluid mechanics in periodic domains.
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 16:30 Alfredo Deaño (University of Kent)
Computational aspects of complex orthogonal polynomials

Complex orthogonal polynomials, and associated quantities such as recurrence coefficients, have been recently used in problems related to highly oscillatory quadrature (construction of complex Gaussian quadrature rules), in the analysis of non-Hermitian random matrix ensembles and also in the study of solutions of Painlevé differential equations. In this talk we will present examples and propose several options for their numerical evaluation.

INI 1
16:30 to 17:00 Philippe Trinh (University of Bath)
The search for complex singularities in exponential asymptotics
In the last two decades, the development of specialized techniques in mathematics known as exponential asymptotics has led to the successful resolution of long-standing problems in topics as varied as quantum mechanics, crystal growth, dislocations, pattern formation, turbulence, thin film flow, and hydrodynamics. These developments have emerged from the realization that in many such problems, exponentially small effects, linked to the presence of singularities in the analytic continuation of a perturbative solution, can significantly change the solutions of the underlying mathematical models.

However, the majority of problems studied have been one-dimensional in nature (essentially ordinary differential or difference equations). The extension of exponential asymptotics to multi-dimensional problems requires some fundamental advances in our ability to compute numerical analytic continuations of single- and multi-variate complex functions. In this talk, I shall present the basic theory of exponential asymptotics and discuss these open challenges in the context of problems in fluid mechanics.
INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons