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Seminars (GCS)

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Event When Speaker Title Presentation Material
GCSW01 8th July 2019
10:00 to 11:00
Hans Munthe-Kaas Why B-series, rooted trees, and free algebras? - 1
"We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures.   In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes.
GCSW01 8th July 2019
11:30 to 12:30
Charlie Elliott PDEs in Complex and Evolving Domains I
GCSW01 8th July 2019
14:00 to 15:00
Elizabeth Mansfield Introduction to Lie groups and algebras - 1
In this series of three lectures, I will give a gentle introduction to Lie groups and their associated Lie algebras, concentrating on the major examples of importance in applications. I will next discuss Lie group actions, their invariants, their associated infinitesimal vector fields, and a little on moving frames. The final topic will be Noether’s Theorem, which yields conservations laws for variational problems which are invariant under a Lie group action. Time permitting, I will show a little of the finite difference and finite element versions of Noether’s Theorem.

In this first talk, I will consider some of the simplest and most useful Lie groups. I will show how to derive their Lie algebras, and will further discuss the Adjoint action of the group on its Lie algebra, the Lie bracket on the algebra, and the exponential map.
GCSW01 8th July 2019
15:00 to 16:00
Christopher Budd Modified error estimates for discrete variational derivative methods

My three talks will be an exploration of geometric integration methods in the context of the numerical solution of PDEs. I will look both at discrete variational methods and their analysis using modified equations, and also of the role of adaptivity in studying and retaining qualitative features of PDEs.

GCSW01 8th July 2019
16:30 to 17:30
Paola Francesca Antonietti High-order Discontinuous Galerkin methods for the numerical modelling of earthquake ground motion
A number of challenging geophysical applications requires a flexible representation of the geometry and an accurate approximation of the solution field. Paradigmatic examples include seismic wave propagation and fractured reservoir simulations. The main challenges are i) the complexity of the physical domain, due to the presence of localized geological irregularities, alluvial basins, faults and fractures; ii) the heterogeneities in the medium, with significant and sharp contrasts; and iii) the coexistence of different physical models. The high-order discontinuous Galerkin FEM possesses the built-in flexibility to naturally accommodate both non-matching meshes, possibly made of polygonal and polyhedral elements, and high-order approximations in any space dimension. In this talk I will discuss recent advances in the development and analysis of high-order DG methods for the numerical approximation of seismic wave propagation phenomena. I will analyse the stability and the theoretical properties of the scheme and present some simulations of real large-scale seismic events in three-dimensional complex media.
GCSW01 9th July 2019
09:00 to 10:00
Douglas Arnold Finite Element Exterior Calculus - 1

These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.

GCSW01 9th July 2019
10:00 to 11:00
Elizabeth Mansfield Introduction to Lie groups and algebras - 2
I will discuss Lie group actions, their invariants, their associated infinitesimal vector fields, and a little on moving frames.
GCSW01 9th July 2019
11:30 to 12:30
Kurusch Ebrahimi-Fard Why B-series, rooted trees, and free algebras? - 2
"We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures.   In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes.
GCSW01 9th July 2019
14:00 to 15:00
Charlie Elliott PDEs in Complex and Evolving Domains II
GCSW01 9th July 2019
15:00 to 16:00
Christian Lubich Variational Gaussian wave packets revisited
The talk reviews Gaussian wave packets that evolve according to the Dirac-Frenkel time-dependent variational principle for the semi-classically scaled Schr\"odinger equation. Old and new results on the approximation to the wave function are given, in particular an $L^2$ error bound that goes back to Hagedorn (1980) in a non-variational setting, and a new error bound for averages of observables with a Weyl symbol, which shows the double approximation order in the semi-classical scaling parameter in comparison with the norm estimate.

The variational equations of motion in Hagedorn's parametrization of the Gaussian are presented. They show a perfect quantum-classical correspondence and allow us to read off directly that the Ehrenfest time is determined by the Lyapunov exponent of the classical equations of motion.

A variational splitting integrator is formulated and its remarkable conservation and approximation properties are discussed. A new result shows that the integrator approximates averages of observables with the full order in the time stepsize, with an error constant that is uniform in the semiclassical parameter.

The material presented here for variational Gaussians is part of an Acta Numerica review article on computational methods for quantum dynamics in the semi-classical regime, which is currently in preparation in joint work with Caroline Lasser.

GCSW01 10th July 2019
09:00 to 10:00
Christopher Budd Blow-up in PDES and how to compute it

My three talks will be an exploration of geometric integration methods in the context of the numerical solution of PDEs. I will look both at discrete variational methods and their analysis using modified equations, and also of the role of adaptivity in studying and retaining qualitative features of PDEs.

GCSW01 10th July 2019
10:00 to 11:00
Douglas Arnold Finite Element Exterior Calculus - 2

These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.

GCSW01 10th July 2019
11:30 to 12:30
Charlie Elliott PDEs in Complex and Evolving Domains III
GCSW01 11th July 2019
09:00 to 10:00
Charlie Elliott PDEs in Complex and Evolving Domains IV
GCSW01 11th July 2019
10:00 to 11:00
Beth Wingate An introduction to time-parallel methods
I will give an introduction to time-parallel methods and discuss how they apply to PDEs, in particular those with where resonance plays an important role. I’ll give some examples from ODEs before finally discussing PDEs.
GCSW01 11th July 2019
11:30 to 12:30
Douglas Arnold Finite Element Exterior Calculus - 3

These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.

GCSW01 11th July 2019
14:00 to 15:00
Kurusch Ebrahimi-Fard Why B-series, rooted trees, and free algebras? - 3
"We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures.   In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes.
GCSW01 11th July 2019
15:00 to 16:00
Elizabeth Mansfield Noether's conservation laws - smooth and discrete
The final topic will be Noether’s Theorem, which yields conservation laws for variational problems which are invariant under a Lie group action. . I will show a little of the finite difference and finite element versions of Noether’s Theorem.
GCSW01 11th July 2019
16:30 to 17:30
Panel
GCSW01 12th July 2019
09:00 to 10:00
Reinout Quispel Discrete Darboux polynomials and the preservation of measure and integrals of ordinary differential equations
Preservation of phase space volume (or more generally measure), first integrals (such as energy), and second integrals have been important topics in geometric numerical integration for more than a decade, and methods have been developed to preserve each of these properties separately. Preserving two or more geometric properties simultaneously, however, has often been difficult, if not impossible. Then it was discovered that Kahan’s ‘unconventional’ method seems to perform well in many cases [1]. Kahan himself, however, wrote: “I have used these unconventional methods for 24 years without quite understanding why they work so well as they do, when they work.” The first approximation to such an understanding in computational terms was: Kahan’s method works so well because
1.         It is very successful at preserving multiple quantities simultaneously, eg modified energy and modified measure.
2.         It is linearly implicit
3.         It is the restriction of a Runge-Kutta method
However, point 1 above raises a further obvious question: Why does Kahan’s method preserve both certain (modified) first integrals and certain (modified) measures? In this talk we invoke Darboux polynomials to try and answer this question. The method of Darboux polynomials (DPs) for ODEs was introduced by Darboux to detect rational integrals. Very recently we have advocated the use of DPs for discrete systems [2,3]. DPs provide a unified theory for the preservation of polynomial measures and second integrals, as well as rational first integrals. In this new perspective the answer we propose to the above question is: Kahan’s method works so well because it is good at preserving (modified) Darboux polynomials. If time permits we may discuss extensions to polarization methods.  

[1] Petrera et al, Regular and Chaotic Dynamics 16 (2011), 245–289.
[2] Celledoni et al, arxiv:1902.04685.
[3] Celledoni et al, arxiv:1902.04715.
GCSW01 12th July 2019
10:00 to 10:30
James Jackaman Lie symmetry preserving finite element methods
Through the mathematical construction of finite element methods, the “standard” finite element method will often preserve the underlying symmetry of a given differential equation. However, this is not always the case, and while historically much attention has been paid to the preserving of conserved quantities the preservation of Lie symmetries is an open problem. We introduce a methodology for the design of arbitrary order finite element methods which preserve the underlying Lie symmetries through an equivariant moving frames based invariantization procedure.
GCSW01 12th July 2019
10:30 to 11:00
Candan Güdücü Port-Hamiltonian Systems
The framework of port-Hamiltonian systems (PH systems) combines both the Hamiltonian approach and the network approach, by associating with the interconnection structure of the network model a geometric structure given by a Dirac structure. In this talk, I introduce port-Hamiltonian (pH) systems and their underlying Dirac structures. Then, a coordinate-based representation of PH systems and some properties are shown. A Lanczos method for the solution of linear systems with nonsymmetric coefficient matrices and its application to pH systems are presenred.
GCSW01 12th July 2019
11:30 to 12:30
Christopher Budd Adaptivity and optimal transport
My three talks will be an exploration of geometric integration methods in the context of the numerical solution of PDEs. I will look both at discrete variational methods and their analysis using modified equations, and also of the role of adaptivity in studying and retaining qualitative features of PDEs.
GCSW01 12th July 2019
14:00 to 15:00
Douglas Arnold Finite element exterior calculus - 4
These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.
GCSW01 12th July 2019
15:00 to 16:00
Hans Munthe-Kaas Why B-series, rooted trees, and free algebras? - 2
"We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures.   In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes.
GCS 17th July 2019
11:00 to 12:00
Christian Offen Detection of high codimensional bifurcations in variational PDEs
We derive bifurcation test equations for A-series singularities of nonlinear functionals and, based on these equations, we propose a numerical method for detecting high codimensional bifurcations in parameter-dependent PDEs such as parameter-dependent semilinear Poisson equations. As an example, we consider a Bratu-type problem and show how high codimensional bifurcations such as the swallowtail bifurcation can be found numerically.

Lisa Maria Kreusser, Robert I McLachlan, Christian Offen





GCS 22nd July 2019
14:00 to 15:00
Ralf Hiptmair Exterior Shape Calculus
GCS 23rd July 2019
14:00 to 15:00
Annalisa Buffa Dual complexes and mortaring for regular approximations of electromagnetic fields
GCS 24th July 2019
15:00 to 16:00
Onno Bokhove A new wave-to-wire wave-energy model: from variational principle to compatible space-time discretisation
Amplification phenomena in a so-called bore-soliton-splash have led us to develop a novel wave-energy device with wave amplification in a contraction used to enhance wave-activated buoy motion and magnetically-induced energy generation. An experimental proof-of-principle shows that our wave-energy device works. Most importantly, we develop a novel wave-to-wire mathematical model of the combined wave hydrodynamics, wave-activated buoy motion and electric power generation by magnetic induction, from first principles, satisfying one grand variational principle in its conservative limit. Wave and buoy dynamics are coupled via a Lagrange multiplier, which boundary value at the waterline is subtly solved explicitly by imposing incompressibility in a weak sense. Dissipative features, such as electrical wire resistance and nonlinear LED-loads, are added a posteriori. New is also the intricate and compatible (finite-element) space-time discretisation of the linearised dynamics, guaranteeing numerical stability and the correct energy transfer between the three subsystems. Preliminary simulations of our simplified and linearised wave-energy model are encouraging, yet suboptimal, and involve a first study of the resonant behaviour and parameter dependence of the device.



GCS 30th July 2019
14:00 to 15:00
Melvin Leok The Connections Between Discrete Geometric Mechanics, Information Geometry and Machine Learning
GCS 30th July 2019
15:00 to 16:00
Benjamin Tapley Computational methods for simulating inertial particles in discrete incompressible flows.
GCS 31st July 2019
15:00 to 16:00
Anthony Bloch Optimal control and the geometry of integrable systems

In this talk we discuss a geometric approach to certain optimal control
problems and discuss the relationship of the solutions of these problem
to some classical integrable dynamical systems and their generalizations.
We consider the
so-called Clebsch optimal control problem and its relationship
to Lie group actions on manifolds. The integrable systems discussed include
the rigid body equations, geodesic flows on the ellipsoid, flows
on Stiefel manifolds, and the Toda lattice
flows. We discuss the Hamiltonian structure of these systems and relate
our work to some work of Moser. We also discuss the link to discrete dynamics
and symplectic integration.


GCS 7th August 2019
14:00 to 15:00
Arieh Iserles Fast approximation on the real line
Abstract: While approximation theory in an interval is thoroughly understood, the real line represents something of a mystery. In this talk we review the state of the art in this area, commencing from the familiar Hermite functions and moving to recent results  characterising all orthonormal sets on $L_2(-\infty,\infty)$ that have a skew-symmetric (or skew-Hermitian) tridiagonal differentiation matrix and such that their first $n$ expansion coefficients can be calculated in $O(n \log n)$ operations. In particular, we describe the generalised Malmquist–Takenaka system. The talk concludes with a (too!) long list of open problems and challenges.






GCS 7th August 2019
15:00 to 16:00
Robert McLachlan The Lie algebra of classical mechanics
Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket which is useful in geometric integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy.  Therefore, we study and give a complete description of the universal object in this setting, the ‘Lie algebra of classical mechanics’ modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket.
Joint work with Ander Murua.




GCS 14th August 2019
14:00 to 15:00
David Garfinkle Numerical General Relativity
This talk will cover the basic properties of the equations of General Relativity, and issues involved in performing numerical simulations of those equations.  Particular emphasis will be placed on three issues: (1) hyperbolicity of the equations.  (2) preserving constraints. (3) dealing with black holes and spacetime singularities.




GCS 14th August 2019
15:00 to 16:00
Ari Stern Finite element methods for Hamiltonian PDEs
Hamiltonian ODEs satisfy a symplectic conservation law, and there are many advantages to using numerical integrators that preserves this structure. This talk will discuss how the canonical Hamiltonian structure, and its preservation by a numerical method, can be generalized to PDEs. I will also provide a basic introduction to the finite element method and, time permitting, discuss how some classic symplectic integrators can be understood from this point of view.




GCS 21st August 2019
14:00 to 15:00
Fernando Casas More on composition methods: error estimation and pseudo-symmetry
In this talk I will review composition methods for the time integration of differential equations, 
paying special attention to two recent contributions in this area. The first one is the construction 
of a new local error estimator so that the additional computational effort required is almost insignificant. 
The second one is related to a new family of high-order methods obtained from a basic symmetric 
(symplectic) scheme in such a way that they are time-symmetric (symplectic) only up to a certain order.





GCS 21st August 2019
15:00 to 16:00
Yajuan Sun Contact Hamiltonian system and its application in solving Vlasov-Poisson Fokker-Planck system
The Vlasov-Poisson Fokker-Planck  system  is  a kinetic description of  Brownian motion for a large number of particles in a surrounding bath. In solving   this system,  the contact structure is investigated. This motivates the study for the  contact system and the construction of corresponding numerical methods. This talk will introduce the contact system and the construction of numerical methods based on generating function and variational principle.




GCS 28th August 2019
14:00 to 15:00
Gerard Awanou Computational geometric optics: Monge-Ampere
I will review recent developments in the numerical resolution of the second boundary value problem for Monge-Ampere type equations and their applications to the design of reflectors and refractors.




GCS 28th August 2019
15:00 to 16:00
Milo Viviani Lie-Poisson methods for isospectral flows and their application to long-time simulation of spherical ideal hydrodynamics
The theory of isospectral flows comprises a large class of continuous dynamical systems, particularly integrable systems and Lie–Poisson systems. Their discretization is a classical problem in numerical analysis. Preserving the spectra in the discrete flow requires the conservation of high order polynomials, which is hard to come by. Existing methods achieving this are complicated and usually fail to preserve the underlying Lie-Poisson structure. Here we present a class of numerical methods of arbitrary order for Hamiltonian and non-Hamiltonian isospectral flows, which preserve both the spectra and the Lie-Poisson structure. The methods are surprisingly simple, and avoid the use of constraints or exponential maps. Furthermore, due to preservation of the Lie–Poisson structure, they exhibit near conservation of the Hamiltonian function. As an illustration, we apply the methods to long-time simulation of the Euler equations on a sphere. Our findings suggest that our structure-preserving algorithms, on the one hand, perform at least as well as other popular methods (i.e. CLAM) without adding spurious hyperviscosity terms, on the other hand, show that the conservation of the Casimir functions can be actually used to predict the final state of the fluid




GCS 4th September 2019
14:00 to 15:00
Sina Ober-Blöbaum Variational formulations for dissipative systems
Variational principles are powerful tools for the modelling and simulation of conservative mechanical and electrical systems. As it is well-known, the fulfilment of a variational principle leads to the Euler-Lagrange equations of motion describing the dynamics of such systems. Furthermore, a variational discretisation directly yields unified numerical schemes with powerful structure-preserving properties. Since many years there have been several attempts to provide a variational description also for dissipative mechanical systems, a task that is addressed in the talk in order to construct both Lagrangian and Hamiltonian pictures of their dynamics. One way doing this is to use fractional terms in the Lagrangian or Hamiltonian function which allows for a purely variational derivation of dissipative systems. Another approach followed in this talk is to embed the non-conservative systems in larger conservative systems. These concepts are used to develop variational integrators for which superior qualitative numerical properties such as the correct energy dissipation rate are demonstrated.




GCS 4th September 2019
15:00 to 16:00
Sigrid Leyendecker Mixed order and multirate variational integrators for the simulation of dynamics on different time scales
Mechanical systems with dynamics on varying time scales, e.g. including highly oscillatory motion, impose challenging questions for numerical integration schemes. High resolution is required to guarantee a stable integration of the fast frequencies. However, for the simulation of the slow dynamics, integration with a lower resolution is accurate enough - and computationally cheaper, especially for costly function evaluations. Two approaches are presented, a mixed order Galerkin variational integrator and a multirate variational integrator, and analysed with respect to the preservation of invariants, computational costs, accuracy and linear stability.




GCS 5th September 2019
14:15 to 15:15
Chus Sanz-Serna Numerical Integrators for the Hamiltonian Monte Carlo Method
GCS 11th September 2019
14:00 to 15:00
Peter Hydon Conservation laws and Euler operators
A (local) conservation law of a given system of differential or difference equations is a divergence expression that is zero on all solutions. The Euler operator is a powerful tool in the formal theory of conservation laws that enables key results to be proved simply, including several generalizations of Noether's theorems.  This talk begins with a short survey of the main ideas and results.   The current method for inverting the divergence operator generates many unnecessary terms by integrating in all directions simultaneously. As a result, symbolic algebra packages create over-complicated representations of conservation laws, making it difficult to obtain efficient conservative finite difference approximations symbolically. A new approach resolves this problem by using partial Euler operators to construct near-optimal representations. The talk explains this approach, which was developed during the GCS programme.




GCS 11th September 2019
15:00 to 16:00
Gianluca Frasca-Caccia Numerical preservation of local conservation laws
In the numerical treatment of partial differential equations (PDEs), the benefits of preserving global integral invariants are well-known. Preserving the underlying local conservation law gives, in general, a stricter constraint than conserving the global invariant obtained by integrating it in space. Conservation laws, in fact, hold throughout the domain and are satisfied by all solutions, independently of initial and boundary conditions. A new approach that uses symbolic algebra to develop bespoke finite difference schemes that preserve multiple local conservation laws has been recently applied to PDEs with polynomial nonlinearity. The talk illustrates this new strategy using some well-known equations as benchmark examples and shows comparisons between the obtained schemes and other integrators known in literature.




GCS 18th September 2019
14:05 to 14:50
Blanca Ayuso De Dios Constructing Discontinuous Galerkin methods for Vlasov-type systems
The Vlasov-Poisson and the Vlasov-Maxwell systems are two classical models in collisionless kinetic theory. They are both derived as mean-field limit description of a large ensemble of interacting particles by  electrostatic and electro-magnetic forces, respectively.
In this talk we describe how to design (semi-discrete!) discontinuous Galerkin finite element methods for approximating such Vlasov-type systems. We outline the error analysis of the schemes and discuss further properties of the proposed schemes, as well as their shortcomings.
If time allows, we discuss further endeavours in alleviating the drawbacks of the schemes. 




GCS 18th September 2019
15:05 to 15:50
Linyu Peng Variational systems on the variational bicomplex

 It is well know that symplecticity plays a fundamentally important role in Lagrangian and Hamiltonian systems. Numerical methods preserving symplecticity (or multisymplecticity for PDEs) have been greatly developed and applied during last decades.  In this talk, we will show how the variational bicomplex, a double cochain complex on jet manifolds, provides a natural framework for understanding multisymplectic systems. The discrete counterpart, discrete multisymplectic systems on the difference variational bicomplex will briefly be introduced if time permits.



 

GCS 19th September 2019
16:00 to 17:00
Marcus Webb Energy preserving spectral methods on the real line whose analysis strays into the complex plane (copy)
GCS 25th September 2019
14:05 to 14:50
Martin Licht Newest Results in Newest Vertex Bisection
The algorithmic refinement of triangular meshes is an important component in numerical simulation codes. Newest vertex bisection is one of the most popular methods for geometrically stable local refinement. Its complexity analysis, however, is a fairly intricate recent result and many combinatorial aspects of this method are not yet fully understood. In this talk, we access newest vertex bisection from the perspective of theoretical computer science. We outline the amortized complexity analysis over generalized triangulations. An immediate application is the convergence and complexity analysis of adaptive finite element methods over embedded surfaces and singular surfaces. Moreover, we "combinatorialize" the complexity estimate and remove any geometry-dependent constants, which is only natural for this purely combinatorial algorithm and improves upon prior results. This is joint work with Michael Holst and Zhao Lyu.




GCSW02 30th September 2019
09:30 to 10:30
Deirdre Shoemaker Numerical Relativity in the Era of Gravitational Wave Observations
The birth and future of gravitational wave astronomy offers new opportunities and challenges for numerical methods in general relativity.
Numerical relativity in particular provides critical support to detect and interpret gravitational wave measurements. In this talk, I’ll discuss the role numerical relativity is playing in the observed black hole binaries by LIGO and Virgo and its future potential for unveiling strong-field gravity in future detections with an emphasis on the computational challenges. I'll frame a discussion about what demands will be placed on this field to maximize the science output of the new era.
GCSW02 30th September 2019
11:00 to 12:00
Michael Holst Some Research Problems in Mathematical and Numerical General Relativity
The 2017 Nobel Prize in Physics was awarded to three of the key scientists involved in the development of LIGO and its eventual successful first detections of gravitational waves. How do LIGO (and other gravitational wave detector) scientists know what they are detecting? The answer is that the signals detected by the devices are shown, after extensive data analysis and numerical simulations of the Einstein equations, to be a very close match to computer simulations of wave emission from very particular types of binary collisions.

In this lecture, we begin with a brief overview of the mathematical formulation of Einstein (evolution and constraint) equations, and then focus on some fundamental mathematics research questions involving the Einstein constraint equations. We begin with a look at the most useful mathematical formulation of the constraint equations, and then summarize the known existence, uniqueness, and multiplicity results through 2009. We then present a number of new existence and multiplicity results developed since 2009 that substantially change the solution theory for the constraint equations. In the second part of the talk, we consider approaches for developing "provably good" numerical methods for solving these types of geometric PDE systems on 2- and 3-manifolds. We examine how one proves rigorous error estimates for particular classes of numerical methods, including both classical finite element methods and newer methods from the finite element exterior calculus.

This lecture will touch on several joint projects that span more than a decade, involving a number of collaborators. The lecture is intended both for mathematicians interested in potential research problems in mathematical and numerical general relativity, as well as physicists interested in relevant new developments in mathematical and numerical methods for nonlinear geometric PDE.
GCSW02 30th September 2019
13:30 to 14:30
Pau Figueras numerical relativity beyond astrophysics: new challenges and new dynamics
Motivated by more fundamental theories of gravity such as string theory, in recent years there has been a growing interesting in solving the Einstein equations numerically beyond the traditional astrophysical set up. For instance in spacetime dimensions higher than the four that we have observed, or in exotic spaces such as anti-de Sitter spaces. In this talk I will give an overview of the challenges that are often encountered when solving the Einstein equations in these new settings. In the second part of the talk I will provide some examples, such as the dynamics of unstable black holes in higher dimensions and gravitational collapse in anti-de Sitter spaces.
GCSW02 30th September 2019
14:30 to 15:30
Ari Stern Structure-preserving time discretization: lessons for numerical relativity?
In numerical ODEs, there is a rich literature on methods that preserve certain geometric structures arising in physical systems, such as Hamiltonian/symplectic structure, symmetries, and conservation laws. I will give an introduction to these methods and discuss recent work extending some of these ideas to numerical PDEs in classical field theory.
GCSW02 30th September 2019
16:00 to 17:00
Frans Pretorius Computational Challenges in Numerical Relativity
I will give a brief overview of the some of the challenges in computational solution of the Einstein field equations.I will then describe the background error subtraction technique, designed to allow for more computationally efficient solution of scenarios where a significant portion of the domain is close to a know, exact solution. To demonstrate, I will discuss application to tidal disruption of a star by a supermassive black hole, and studies of black hole superradiance.
GCSW02 1st October 2019
09:30 to 10:30
Lee Lindblom Solving PDEs Numerically on Manifolds with Arbitrary Spatial Topologies
GCSW02 1st October 2019
11:00 to 12:00
Melvin Leok Variational discretizations of gauge field theories using group-equivariant interpolation spaces
Variational integrators are geometric structure-preserving numerical methods that preserve the symplectic structure, satisfy a discrete Noether's theorem, and exhibit exhibit excellent long-time energy stability properties. An exact discrete Lagrangian arises from Jacobi's solution of the Hamilton-Jacobi equation, and it generates the exact flow of a Lagrangian system. By approximating the exact discrete Lagrangian using an appropriate choice of interpolation space and quadrature rule, we obtain a systematic approach for constructing variational integrators. The convergence rates of such variational integrators are related to the best approximation properties of the interpolation space.

Many gauge field theories can be formulated variationally using a multisymplectic Lagrangian formulation, and we will present a characterization of the exact generating functionals that generate the multisymplectic relation. By discretizing these using group-equivariant spacetime finite element spaces, we obtain methods that exhibit a discrete multimomentum conservation law. We will then briefly describe an approach for constructing group-equivariant interpolation spaces that take values in the space of Lorentzian metrics that can be efficiently computed using a generalized polar decomposition. The goal is to eventually apply this to the construction of variational discretizations of general relativity, which is a second-order gauge field theory whose configuration manifold is the space of Lorentzian metrics.
GCSW02 1st October 2019
13:30 to 14:30
Oscar Reula Hyperbolicity and boundary conditions.
Abstract: (In collaboration with Fernando Abalos.) Very often in physics, the evolution systems we have to deal with are not purely hyperbolic, but contain also constraints and gauge freedoms. After fixing these gauge freedoms we obtain a new system with constraints which we want to solve subject to initial and boundary values. In particular, these values have to imply the correct propagation of constraints. In general, after fixing some reduction to a purely evolutionary system, this is asserting by computing by hand what is called the constraint subsidiary system, namely a system which is satisfied by the constraints quantities when the fields satisfy the reduced evolution system.
If the subsidiary system is also hyperbolic then for the initial data case the situation is clear: we need to impose the constraints on the initial data and then they will correctly propagate along evolution. For the boundary data, we need to impose the constraint for all incoming constraint modes. These must be done by fixing some of the otherwise free boundary data, that is the incoming modes. Thus, there must be a relation between some of the incoming modes of the evolution system and all the incoming modes of the constraint subsidiary system. Under certain conditions on the constraints, this relation is known and understood, but those conditions are very restrictive. In this talk, we shall review the known results and discuss what is known so far for the general case and what are the open questions that still remain.
GCSW02 1st October 2019
14:30 to 15:30
Snorre Christiansen Compatible finite element spaces for metrics with curvature
I will present some new finite element spaces for metrics with integrable curvature. These were obtained in the framework of finite element systems, developed for constructing differential complexes with adequate gluing conditions between the cells of a mesh. The new spaces have a higher regularity than those of Regge calculus, for which the scalar curvature contains measures supported on lower dimensional simplices (Dirac deltas). This is joint work with Kaibo Hu.
GCSW02 1st October 2019
16:00 to 17:00
Second Chances
The formalism of discrete differential forms has been used very successfully in computational electrodynamics. It is based on the idea that only the observables (i.e., the electromagnetic field) should be discretised and that coordinates should not possess any relevance in the numerical method. This is reflected in the fact that Maxwell's theory can be written entirely in geometric terms using differential forms. Einstein's theory is entirely geometric as well and can also be written in terms of differential forms. In this talk I will describe an attempt to discretise Einstein's theory in a way similar to Maxwell's theory. I will describe the advantages and point out disadvantages. I will conclude with some remarks about more general discrete structures on manifolds.
GCSW02 2nd October 2019
09:30 to 10:30
Anil Hirani Discrete Vector Bundles with Connection and the First Chern Class
The use of differential forms in general relativity requires ingredients like the covariant exterior derivative and curvature. One potential approach to numerical relativity would require discretizations of these ingredients. I will describe a discrete combinatorial theory of vector bundles with connections. The main operator we develop is a discrete covariant exterior derivative that generalizes the coboundary operator and yields a discrete curvature and a discrete Bianchi identity. We test this theory by defining a discrete first Chern class, a topological invariant of vector bundles. This discrete theory is built by generalizing discrete exterior calculus (DEC) which is a discretization of exterior calculus on manifolds for real-valued differential forms. In the first part of the talk I will describe DEC and its applications to the Hodge-Laplace problem and Navier-Stokes equations on surfaces, and then I will develop the discrete covariant exterior derivative and its implications. This is joint work with Daniel Berwick-Evans and Mark Schubel.
GCSW02 2nd October 2019
11:00 to 12:00
Soeren Bartels Approximation of Harmonic Maps and Wave Maps
Partial differential equations with a nonlinear pointwise constraint defined by a manifold occur in a variety of applications: the magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rod-like molecules that constitute a liquid crystal is often modeled by a restricted vector field. Other applications arise in geometric modeling, nonlinear bending of solids, and quantum mechanics. Nodal finite element methods have to appropriately relax the pointwise constraint leading to a variational crime. Since exact solutions are typically nonunique and do not admit higher regularity properties, the correctness of discretizations has to be established by weaker means avoiding unrealistic conditions. The iterative solution of the nonlinear systems of equations can be based on linearizations of the constraint or by using appropriate constraint-preserving reformulations. The talk focusses on the approximation of harmonic maps and wave maps. The latter arise as a model problem in general relativity.
GCSW02 2nd October 2019
13:30 to 14:30
Helvi Witek New prospects in numerical relativity
Both observations and deeply theoretical considerations indicate that general relativity, our elegant standard model of gravity, requires modifications at high curvatures scales. Candidate theories of quantum gravity, in their low-energy limit, typically predict couplings to additional fields or extensions that involve higher curvature terms.

At the same time, the breakthrough discovery of gravitational waves has provided a new channel to probe gravity in its most extreme, strong-field regime. Modelling the expected gravitational radiation in these extensions of general relativity enables us to search for - or place novel observational bounds on - deviations from our standard model. In this talk I will give an overview of the recent progress on simulating binary collisions in these situations and address renewed mathematical challenges such as well-posedness of the underlying initial value formulation.
GCSW02 3rd October 2019
09:30 to 10:30
Fernando Abalos On necessary and sufficient conditions for strong hyperbolicity in systems with differential constraints
In many physical applications, due to the presence of constraints, the number of equations in the partial differential equation systems is larger than the number of unknowns, thus the standard Kreiss conditions can not be directly applied to check whether the system admits a well posed initial value formulation. In this work we show necessary and sufficient conditions such that there exists a reduced set of equations, of the same dimensionality as the set of unknowns, which satisfy Kreiss conditions and so are well defined and properly behaved evolution equations. We do that by decomposing the systems using the Kronecker decomposition of matrix pencils and, once the conditions are met, we look for specific families of reductions. We show the power of the theory in the densitized, pseudo-differential ADM equations.
GCSW02 3rd October 2019
11:00 to 12:00
David Hilditch Putting Infinity on the Grid
I will talk about an ongoing research program relying on a dual frame approach to treat numerically the field equations of GR (in generalized harmonic gauge) on compactified hyperboloidal slices. These slices terminate at future-null infinity, and the hope is to eventually extract gravitational waves from simulations there. The main obstacle to their use is the presence of 'infinities' coming from the compactified coordinates, which have to somehow interact well with the assumption of asymptotic flatness so that we may arrive at regular equations for regular unknowns. I will present a new 'subtract the logs' regularization strategy for a toy nonlinear wave equation that achieves this goal.
GCSW02 3rd October 2019
13:30 to 14:30
Warner Miller General Relativity: One Block at a Time
This talk will provide an overview and motivation for Regge calculus (RC). We will highlight our insights into unique features of building GR on a discrete geometry in regards to structure preservation, and highlight some relative strengths and weaknesses of RC. We will review some numerical applications of RC, including our more recent work on discrete Ricci flow.
GCSW02 3rd October 2019
14:30 to 15:30
Ragnar Winther Finite element exterior calculus as a tool for compatible discretizations

The purpose of this talk is to review the basic results of finite element exterior calculus (FEEC) and to illustrate how the set up gives rise to
to compatible discretizations of various problems. In particular, we will recall how FEEC, combined with the Bernstein-Gelfand-Gelfand framework,
gave new insight into the construction of stable schemes for elasticity methods based on the Hellinger-Reissner variational principle.

GCSW02 3rd October 2019
16:00 to 17:00
Douglas Arnold FEEC 4 GR?
The finite element exterior calculus (FEEC) has proven to be a powerful tool for the design and understanding of numerical methods for solving PDEs from many branches of physics: solid mechanics, fluid flow, electromagnetics, etc. Based on preserving crucial geometric and topological structures underlying the equations, it is a prime example of a structure-preserving numerical method. It has organized many known finite element methods resulting in the periodic table of finite elements. For elasticity, which is not covered by the table, it led to new methods with long sought-after properties. Might the FEEC approach lead to better numerical solutions of the Einstein equations as well? This talk will explore this question through two examples: the Einstein--Bianchi formulation of the Einstein equations based on Bel decomposition of the Weyl tensor, and the Regge elements, a family of finite elements inspired by Regge calculus. Our goal in the talk is to raise questions and inspire future work; we do not purport to provide anything near definitive answers.
GCSW02 4th October 2019
11:00 to 12:00
Pablo Laguna Inside the Final Black Hole from Black Hole Collisions
Modeling black hole singularities as punctures in space-time is common in binary black hole simulations. As the punctures approach each other, a common apparent horizon forms, signaling the coalescence of the black holes and the formation of the final black hole. I will present results from a study that investigates the fate of the punctures and in particular the dynamics of the trapped surfaces on each puncture.

Co-authors: Christopher Evans, Deborah Ferguson, Bhavesh Khamesra and Deirdre Shoemaker
GCSW02 4th October 2019
13:30 to 14:30
Charalampos Markakis On numerical conservation of the Poincaré-Cartan integral invariant in relativistic fluid dynamics
The motion of strongly gravitating fluid bodies is described by the Euler-Einstein system of partial differential equations. We report progress on formulating well-posed, acoustical and canonical hydrodynamic schemes, suitable for binary inspiral simulations and gravitational-wave source modelling. The schemes use a variational principle by Carter-Lichnerowicz stating that barotropic fluid motions are conformally geodesic, a corollary to Kelvin's theorem stating that initially irrotational flows remain irrotational, and Christodoulou's acoustic metric approach adopted to numerical relativity, in order to evolve the canonical momentum of a fluid element via Hamilton or Hamilton-Jacobi equations. These mathematical theorems leave their imprints on inspiral waveforms from binary neutron stars observed by the LIGO-Virgo detectors. We describe a constraint damping scheme for preserving circulation in numerical general relativity, in accordance with Helmholtz's third theorem.
GCSW02 4th October 2019
14:30 to 15:30
Luis Lehner tba
GCSW02 4th October 2019
16:00 to 17:00
David Garfinkle Tetrad methods in numerical relativity
Most numerical relativity simulations use the usual coordinate methods to put the Einstein field equations in the form of partial differential equations (PDE), which are then handled using more or less standard numerical PDE methods, such as finite differences. However, there are some advantages to instead using a tetrad (orthonormal) basis rather than the usual coordinate basis. I will present the tetrad method and its numerical uses, particularly for simulating the approach to a spacetime singularity. I will end with open questions about which tetrad systems are suitable for numerical simulations.
GCS 9th October 2019
14:05 to 14:50
Richard Falk Numerical Computation of Hausdorff Dimension
We show how finite element approximation theory can be combined with theoretical results about the properties of the eigenvectors of a class of linear Perron-Frobenius operators to obtain accurate approximations of the Hausdorff dimension of some invariant sets arising from iterated function systems.

The theory produces rigorous upper and lower bounds on the Hausdorff dimension. Applications to the computation of the Hausdorff dimension of some Cantor sets arising from real and complex continued fraction expansions are described.





GCS 9th October 2019
15:05 to 15:50
Daniele Boffi Approximation of eigenvalue problems arising from partial differential equations: examples and counterexamples
We discuss the finite element approximation of eigenvalue problems arising from elliptic partial differential equations. We present various examples of non-standard schemes, including mixed finite elements, approximation of operators related to the least-squares finite element method, parameter dependent formulations such as those produced by the virtual element method. Each example is studied theoretically; advantages and disadvantages of
each approach are pointed out.






GCS 21st October 2019
16:00 to 17:00
Donatella Marini Kirk Lecture: A recent technology for Scientific Computing: the Virtual Element Method

The Virtual Element Method (VEM) is a recent technology for the numerical solution of boundary value problems for Partial Differential Equations. It could be seen as a generalization of the Finite Element Method (FEM). With FEM the computational domain is typically split in triangles/quads (tetrahedra/hexahedra). VEM responds to the recent interest in using decompositions into polygons/polyhedra of very general shape, whenever more convenient for the approximation of problems of practical interest. Indeed,the possibility of using general polytopal meshes opens up a new range of opportunities in terms of accuracy, efficiency and flexibility. This is for instance reflected by the fact that various (commercial and free) codes recently included and keep developing polytopal meshes, showing in selected applications an improved computational efficiency with respect to tetrahedral or hexahedral grids. In this talk, after a general description of the use and potential of Scientific Computing, basic ideas of conforming VEM will be described on a simple model problem. Numerical results on more general problems in two and three dimension will be shown. Hints on Serendipity versions will be given at the end. These procedures allow to decrease significantly the number of degrees of freedom, that is, to reduce the dimension of the final linear system.

GCS 22nd October 2019
09:05 to 09:50
Franco Brezzi Serendipity Virtual Elements

After a brief reminder of classical ("plain vanilla") Virtual Elements  we will see the general philosophy of "enhanced Virtual Elements" and the various types of Serendipity spaces as particular cases. The construction will always be conceptually simple (and extremely powerful, in particular for polygons with many edges), but a code exploiting the full advantage of  having many edges might become difficult in the presence of non convex polygons, and in particular for complicated shapes. We shall also discuss different choices ensuring various advantages for different amounts of work. 



 

GCS 22nd October 2019
09:55 to 10:40
Andrea Cangiani A posteriori error estimation for discontinuous Galerkin methods on general meshes and adaptivity
The application and a priori error analysis of discontinuous Galerkin (dG) methods for general classes of PDEs under general mesh assumptions is by now well developed. dG methods naturally permits local mesh and order adaptivity. However, deriving robust error estimators allowing for curved/degenerating mesh interfaces as well as developing adaptive algorithms able to exploit such flexibility is non-trivial. In this talk we present recent work on a posteriori error estimates for dG methods of interior penalty type which hold on general mesh settings, including elements with degenerating and curved boundaries. The exploitation of general meshes within mesh adaptation algorithms applied to a few challenging problems will also be discussed.
GCS 22nd October 2019
11:10 to 11:55
Emmanuil Georgoulis Discontinuous Galerkin methods on arbitrarily shaped elements.
We extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for curved element shapes, without the use of (iso-)parametric elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity-penal-ization parameter, which turns out to be essentially independent on the particular element shape. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach. Moreover, we shall discuss a number of perspectives on the possible applications of the proposed framework in parabolic problems on moving domains as well as on multiscale problems. The above is an overview of results from joint works with A. Cangiani (Nottingham, UK), Z. Dong (FORTH, Greece / Cardiff UK) and T. Kappas (Leicester, UK).




GCS 22nd October 2019
13:05 to 13:50
Paul Houston An Agglomeration-Based, Massively Parallel Non-Overlapping Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Methods on Polytopic Grids
In this talk we design and analyze a class of two-level non-overlapping additive Schwarz preconditioners for the solution of the linear system of equations stemming from high-order/hp version discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polytopic meshes. The preconditioner is based on a coarse space and a non-overlapping partition of the computational domain where local solvers are applied in parallel. In particular, the coarse space can potentially be chosen to be non-embedded with respect to the finer space; indeed it can be obtained from the fine grid by employing agglomeration and edge coarsening techniques. We investigate the dependence of the condition number of the preconditioned system with respect to the diffusion coefficient and the discretization
parameters, i.e., the mesh size and the polynomial degree of the fine and coarse spaces. Numerical examples are presented which confirm the theoretical bounds.





GCS 22nd October 2019
13:55 to 14:40
Jinchao Xu UPWIND FINITE ELEMENT METHODS FOR H(grad), H(curl) AND H(div) CONVECTION-DIFFUSION PROBLEMS
This talk is devoted to the construction and analysis of the finite element approximations for the H(grad), H(curl) and H(div) convection-diffusion problems. An essential feature of these constructions is to properly average the PDE coefficients on sub-simplexes from the underlying simplicial finite element meshes. The schemes are of the class of exponential fitting methods that result in special upwind schemes when the diffusion coefficient approaches to zero. Their well-posedness are established for sufficiently small mesh size assuming that the convection-diffusion problems are uniquely solvable. Convergence of first order is derived under minimal smoothness of the solution. Some numerical examples are given to demonstrate the robustness and effectiveness for general convection-diffusion problems. This is a joint work with Shounan Wu.
GCS 28th October 2019
10:00 to 10:45
Erwan Faou High-order splitting for the Vlasov-Poisson equation
We consider the Vlasov{Poisson equation in a Hamiltonian framework and
derive time splitting methods based on the decomposition of the Hamiltonian functional
between the kinetic and electric energy. We also apply a similar strategy to the Vlasov{
Maxwell system. These are joint works with N. Crouseilles, F. Casas, M. Mehrenberger
and L. Einkemmer.
GCS 28th October 2019
10:45 to 11:30
Katharina Kormann On structure-preserving particle-in-cell methods for the Vlasov-Maxwell equations
Numerical schemes that preserve the structure of the kinetic equations can
provide new insight into the long time behavior of fusion plasmas. An electromagnetic
particle-in-cell solver for the Vlasov{Maxwell equations that preserves at the discrete
level the non-canonical Hamiltonian structure of the Vlasov{Maxwell equations has
been presented in [1]. In this talk, the framework of this geometric particle-in-cell
method will be presented and extension to curvilinear coordinates will be discussed.
Moreover, various options for the temporal discretizations will be proposed and compared.
[1] M. Kraus, K. Kormann, P. J. Morrison, and E. Sonnendrucker. GEMPIC: geometric electromag-
netic particle-in-cell methods. Journal of Plasma Physics, 83(4), 2017.
GCS 28th October 2019
12:00 to 12:45
Ernst Hairer Numerical treatment of charged particle dynamics in a magnetic field

Combining the Lorentz force equations with Newton's law gives a second
order dierential equation in space for the motion of a charged particle in a magnetic
eld. The most natural and widely used numerical discretization is the Boris algorithm,
which is explicit, symmetric, volume-preserving, and of order 2.
In a rst part we discuss geometric properties (long-time behaviour, and in particular
near energy conservation) of the Boris algorithm. This is achieved by applying standard
backward error analysis. Near energy conservation can be obtained also in situations,
where the method is not symplectic.
In a second part we consider the motion of a charged particle in a strong magnetic eld.
Backward error analysis can no longer be applied, and the accuracy (order 2) breaks
down. To improve accuracy we modify the Boris algorithm in the spirit of exponential
integrators. Theoretical estimates are obtained with the help of modulated Fourier
expansions of the exact and numerical solutions.
This talk is based on joint work with Christian Lubich, and Bin Wang.
Related publications (2017{2019) can be downloaded from

GCS 28th October 2019
14:00 to 14:45
Wayne Arter Challenges for modelling fusion plasmas

Modelling fusion plasmas presents many challenges, so that it is reasonable
that many modelling codes still use simple nite dierence representations that make
it relatively easy to explore new physical processes and preserve numerical stability [1].
However, October 2019 announcements by UK government have given UKAEA the
challenge of designing a nuclear fusion reactor in the next 5 years, plus a funding element
for upgrading existing software both for Exascale and to meet the design challenge. One
option under examination is the use of high order 'spectrally accurate' elements.
The biggest modelling problem is still that of turbulence mostly at relatively low plasma
collisionality. Specically non-dissipative issues are the tracking of plasma particle orbits
between collisions, sometimes reducing to tracing lines of divergence-free magnetic
eld. These particles then build into a Maxwell{Vlasov solver, for which many dierent
numerical representations, exploiting low collisonality and the presence of a strong,
directed magnetic eld have been explored [2]. One such is ideal MHD, where I have
explored the use of the Lie derivative [3, 4]. Some further speculations as to the likely
role of Lie (and spectral accuracy) in solving Vlasov{Maxwell, its approximations and
their ensembles, and interactions between the dierent approximations, in the Exascale
era will be presented.
This work was funded by the RCUK Energy Programme and the European Communities under the
contract of Association between EURATOM and CCFE.
[1] B.D. Dudson, A. Allen, G. Breyiannis, E. Brugger, J. Buchanan, L. Easy, S. Farley, I. Joseph,
M. Kim, A.D. McGann, et al. BOUT++: Recent and current developments. Journal of Plasma
Physics, 81(01):365810104, 2015.
[2] W. Arter. Numerical simulation of magnetic fusion plasmas. Reports on Progress in Physics,
58:1{59, 1995.
[3] W. Arter. Potential Vorticity Formulation of Compressible Magwnetohydrodynamics. Physical
Review Letters, 110(1):015004, 2013.
[4] W. Arter. Beyond Linear Fields: the Lie{Taylor Expansion. Proc Roy Soc A, 473:20160525, 2017.

GCS 28th October 2019
15:15 to 16:00
Jitse Niesen Spectral deferred correction in particle-in-cell methods
Particle-in-cell methods solve the Maxwell equations for the electromagnetic
eld in combination with the equation of motion for the charged particles in a
plasma. The motion of charegd particles is usually computed using the Boris algorithm,
a variant of Stormer{Verlet for Lorentz force omputations, which has impressive
performance and order two (like Stormer{Verlet). Spectral deferred correction is an
iterative time stepping method based on collocation, which in each time step performs
multiple sweeps of a low-order method (here, the Boris method) in order to obtain a
high-order approximation. This talk describes the ongoing eorts of Kristoer Smedt,
Daniel Ruprecht, Steve Tobias and the speaker to embed a spectral deferred correction
time stepper based on the Boris method in a particle-in-cell method.
GCS 30th October 2019
14:05 to 15:05
Jinchao Xu Deep Neural Networks and Multigrid Methods
In this talk, I will first give an introduction to several models and algorithms from two different fields: (1) machine learning, including logistic regression, support vector machine and deep neural networks, and (2) numerical PDEs, including finite element and multigrid methods.  I will then explore mathematical relationships between these models and algorithms and demonstrate how such relationships can be used to understand, study and improve the model structures, mathematical properties and relevant training algorithms for deep neural networks. In particular, I will demonstrate how a new convolutional neural network known as MgNet, can be derived by making very minor modifications of a classic geometric multigrid method for the Poisson equation and then explore the theoretical and practical potentials of MgNet.




GCS 31st October 2019
16:00 to 17:00
Patrick Farrell A Reynolds-robust preconditioner for the 3D stationary Navier-Stokes equations
When approximating PDEs with the finite element method, large sparse linear systems must be solved. The ideal preconditioner yields convergence that is algorithmically optimal and parameter robust, i.e.  the number of Krylov iterations required to solve the linear system to a given accuracy does not grow substantially as the mesh or problem parameters are changed.   Achieving this for the stationary Navier-Stokes has proven challenging: LU factorisation is Reynolds-robust but scales poorly with degree of freedom count, while Schur complement approximations such as PCD and LSC degrade as the Reynolds number is increased.   Building on ideas of Schöberl, Xu, Zikatanov, Benzi & Olshanskii, in this talk we present the first preconditioner for the Newton linearisation of the stationary Navier–Stokes equations in three dimensions that achieves both optimal complexity and Reynolds-robustness. The scheme combines augmented Lagrangian stabilisation to control the Schur complement, the convection stabilisation proposed by Douglas & Dupont, a divergence-capturing additive Schwarz relaxation method on each level, and a specialised prolongation operator involving non-overlapping local Stokes solves. The properties of the preconditioner are tailored to the divergence-free CG(k)-DG(k-1) discretisation and the appropriate relaxation is derived from considerations of finite element exterior calculus.   We present 3D simulations with over one billion degrees of freedom with robust performance from Reynolds numbers 10 to 5000.




GCS 11th November 2019
14:00 to 15:00
Shi Jin Random Batch Methods for Interacting Particle Systems and Consensus-based Global Non-convex Optimization in High-dimensional Machine Learning (copy)
We develop random batch methods for interacting particle systems with large number of particles. These methods
use small but random batches for particle interactions,
thus the computational cost is reduced from O(N^2) per time step to O(N), for a
system with N particles with binary interactions.
For one of the methods, we give a particle number independent error estimate under some special interactions.
Then, we apply these methods
to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian
motion from random matrix theory, Thomson's problem,
distribution of wealth, opinion dynamics and clustering. Numerical results show that
the methods can capture both the transient solutions and the global equilibrium in
these problems.

We also apply this method and improve the consensus-based global optimization algorithm for high
dimensional machine learning problems. This method does not require taking gradient in finding global
minima for non-convex functions in high dimensions.
GCS 11th November 2019
16:00 to 17:00
Sergio Blanes Magnus, splitting and composition techniques for solving non-linear Schrödinger equations
In this talk I will consider several non-autonomous non-linear Schrödinger equations
(the Gross-Pitaevskii equation, the Kohn-Sham equation and an Quantum Optimal Control equation)
and some of the numerical methods that have been used to solve them.

With a proper linearization of these equations we end up with non-autonomous linear systems
where many of the algebraic techniques from Magnus, splitting and composition algorithms can be used.
This will be an introductory talk to stimulate some collaboration between participants of the program
at the INI.


GCS 12th November 2019
16:00 to 17:00
Erwan Faou Some results in the long time analysis of Hamiltonian PDEs and their numerical approximations
I will review some results concerning the long time behavior of Hamiltonian PDEs, and address
similar questions for their numerical approximation. I will show numerical resonances can appear
both in space and time. I will also discuss the long time stability of solitary waves evolving
on a discret set of lattice points. 

GCS 13th November 2019
14:00 to 15:00
Caroline Lasser What it takes to catch a wave packet
Wave packets describe the quantum vibrations of a molecule. They are highly oscillatory,
highly localized and move in high dimensional configuration spaces. The talk addresses
three meshless numerical methods for catching them: single Gaussian beams,
superpositions of them, and the so-called linearized initial value representation.
GCS 13th November 2019
16:00 to 17:00
Alexander Ostermann Low-regularity time integrators
Nonlinear Schrödinger equations are usually solved by pseudo-spectral methods, where the time integration is performed by splitting schemes or exponential integrators. Notwithstanding the benefits of this approach, its successful application requires additional regularity of the solution. For instance, second-order Strang splitting requires four additional derivatives for the solution of the cubic nonlinear Schrödinger equation. Similar statements can be made about other dispersive equations like the Korteweg-de Vries or the Boussinesq equation.

In this talk, we introduce low-regularity Fourier integrators as an alternative. They are obtained from Duhamel's formula in the following way: first, a Lawson-type transformation eliminates the leading linear term and second, the dominant nonlinear terms are integrated exactly in Fourier space. For cubic nonlinear Schrödinger equations, first-order convergence of such methods only requires the boundedness of one additional derivative of the solution, and second-order convergence the boundedness of two derivatives. Similar improvements can also be obtained for other dispersive problems.

This is joint work with Frédéric Rousset (Université Paris-Sud), Katharina Schratz (Hariot-Watt, UK), and Chunmei Su (Technical University of Munich).
GCS 20th November 2019
13:05 to 13:45
CANCELLED
GCS 20th November 2019
13:50 to 14:30
Tony Lelievre title tba
Various applications require the sampling of probability measures restricted to submanifolds defined as the level set of some functions, in particular in computational statistical physics. We will present recent results on so-called Hybrid Monte Carlo methods, which consists in adding an extra momentum variable to the state of the system, and discretizing the associated Hamiltonian dynamics with some stochastic perturbation in the extra variable. In order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed Hamiltonian dynamics, a Metropolis rejection procedure can be considered. The so-obtained scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms, and we will discuss how to ensure that the sampling method is unbiased in practice.

References:
- T. Lelièvre, M. Rousset and G. Stoltz, Langevin dynamics with constraints and computation of free energy differences, Mathematics of Computation, 81(280), 2012.
- T. Lelièvre, M. Rousset and G. Stoltz, Hybrid Monte Carlo methods for sampling probability measures on submanifolds, to appear in Numerische Mathematik, 2019.
- E. Zappa, M. Holmes-Cerfon, and J. Goodman. Monte Carlo on manifolds: sampling densities and integrating functions. Communications in Pure and Applied Mathematics, 71(12), 2018.





GCS 20th November 2019
15:05 to 15:35
Jonathan Goodman Step size control for Newton type MCMC samplers Jonathan Goodman
ABSTRACT: MCMC sampling can use ideas from the optimization community.  Optimization via Newton’s method can fail without line search, even for smooth strictly convex problems.  Affine invariant Newton based MCMC sampling uses a Gaussian proposal based on a quadratic model of the potential using the local gradient and Hessian.  This can fail (conjecture: give a transient Markov chain) even for smooth strictly convex potentials.  We describe a criterion that allows a sequence of proposal distributions from X_n with decreasing “step sizes” until (with probability 1) a proposal is accepted.  “Very detailed balance” allows the whole process to preserve the target distribution.  The method works in experiments but the theory is missing.




GCS 20th November 2019
15:40 to 16:10
Miranda Holmes-Cefron A Monte Carlo method to sample a Stratification
Many problems in materials science and biology involve particles interacting with strong, short-ranged bonds, that can break and form on experimental timescales. Treating such bonds as constraints can significantly speed up sampling their equilibrium distribution, and there are several methods to sample subject to fixed constraints. We introduce a Monte Carlo method to handle the case when constraints can break and form. Abstractly, the method samples a probability distribution on a stratification: a collection of manifolds of different dimensions, where the lower-dimensional manifolds lie on the boundaries of the higher-dimensional manifolds. We show several applications in polymer physics, self-assembly of colloids, and volume calculation.




GCS 21st November 2019
13:05 to 13:45
Alessandro Barp Hamiltonian Monte Carlo on Homogeneous Manifolds for QCD and Statistics.
GCS 21st November 2019
13:50 to 14:30
Benedict Leimkuhler Some thoughts about constrained sampling algorithms
I will survey our work on algorithms for sampling diffusions on manifolds, including isokinetic methods and constrained Langevin dynamics methods.   These have mostly been introduced and tested in the setting of molecular dynamics.  It is interesting to consider possible uses of these ideas in other types of sampling computations, like neural network parameterization and training of generative models.




GCS 21st November 2019
15:05 to 15:45
Elena Celledoni Deep learning as optimal control problems and Riemannian discrete gradient descent.
We consider recent work where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We review the first order conditions for optimality, and the conditions ensuring optimality after discretisation. This leads to a class of algorithms for solving the discrete optimal control problem which guarantee that the corresponding discrete necessary conditions for optimality are fulfilled. The differential equation setting lends itself to learning additional parameters such as the time discretisation. We explore this extension alongside natural constraints (e.g. time steps lie in a simplex). We compare these deep learning algorithms numerically in terms of induced flow and generalisation ability.   References   - M Benning, E Celledoni, MJ Ehrhardt, B Owren, CB Schönlieb, Deep learning as optimal control problems: models and numerical methods, JCD.




GCS 21st November 2019
16:00 to 17:00
Peter Clarkson Symmetric Orthogonal Polynomials
In this talk I will discuss symmetric orthogonal polynomials on the real line. Such polynomials give rise to orthogonal systems which have important applications in spectral methods, with several important advantages if their differentiation matrix is skew-symmetric and highly structured. Such orthogonal systems, where the differentiation matrix is skew-symmetric, tridiagonal and irreducible, have recently been studied by Iserles and Webb. The symmetric orthogonal polynomials studied will include generalisations of the classical Hermite weight and generalisations of the Freud weight.
GCS 26th November 2019
14:05 to 14:50
Evelyne Hubert Symmetry Preserving Interpolation
In this talk I choose to present the PhD work of Erick Rodriguez Bazan. We address multivariate interpolation in the presence of symmetry as given by a finite group. Interpolation is a prime tool in algebraic computation while symmetry is a qualitative feature
that can  be more relevant to a mathematical model than the numerical accuracy of the parameters. Beside its preservation, symmetry shall also be exploited to alleviate the computational cost.

We revisit minimal degree and least interpolation spaces [de Boor & Ron 1990] with symmetry adapted bases (rather than the usual monomial bases). In these bases, the multivariate Vandermonde matrix  (a.k.a colocation matrix) is block diagonal as soon as the set of nodes is invariant. These blocks capture the inherent redundancy in the computations. Furthermore any equivariance an interpolation problem might have will be automatically preserved : the output interpolant will have the same equivariance property.

The special case of multivariate Hermite interpolation leads us to question the representation of polynomial ideals. Gröbner bases, the preferred tool for algebraic computations, breaks any kind of symmetry. The prior notion of H-Bases, introduced by Macaulay, appears as more suitable.

Reference:
  https://dl.acm.org/citation.cfm?doid=3326229.3326247
  https://hal.inria.fr/hal-01994016
  Joint work with  Erick Rodriguez Bazan




GCS 26th November 2019
15:05 to 16:05
Anders Hansen On the Solvability Complexity Index (SCI) hierarchy - Establishing the foundations of computational mathematics
There are four areas in computational mathematics that have been intensely investigated over more than half a century: Spectral problems, PDEs, optimisation and inverse problems. However, despite the matureness of these fields, the foundations are far from known. Indeed, despite almost 90 years of quantum mechanics, it is still unknown whether it is possible to compute the spectrum of a self-adjoint Schrodinger operator with a bounded smooth potential. Similarly, it is not known which time dependent Schrodinger equations can be computed (despite well posedness of the equation). Linear programs (LP) can be solved with rational inputs in polynomial time, but can LPs be solved with irrational inputs? Problems in signal and image processing tend to use irrational numbers, so what happens if one plugs in the discrete cosine transform in one's favourite LP solver? Moreover, can one always compute the solutions to well-conditioned infinite-dimensional inverse problems, and if not, which inverse problems can then be solved? In this talk we will discuss solutions to many of the questions above, and some of the results may seem paradoxical. Indeed, despite being an open problem for more than half a century, computing spectra of Schrodinger operators with a bounded potential is not harder than computing spectra of infinite diagonal matrices, the simplest of all infinite-dimensional spectral problems. Moreover, computing spectra of compact operators, for which the method has been known for decades, is strictly harder than computing spectra of such Schrodinger operators. Regarding linear programs (and basis pursuit, semidefinite programs and LASSO) we have the following. For any integer K > 2 and any norm, there exists a family of well conditioned inputs containing irrational numbers so that no algorithm can compute K correct digits of a minimiser, however, there exists an algorithm that can compute K-1 correct digits. But any algorithm producing K-1 correct digits will need arbitrarily long time. Finally, computing K-2 correct digits can be done in polynomial time in the number of variables. As we will see, all of these problems can be solved via the the Solvability Complexity Index (SCI) hierarchy, which is a theoretical program for establishing the boundaries of what computers can achieve in the sciences. 




GCS 28th November 2019
16:00 to 17:00
Elizabeth Mansfield On the nature of mathematical joy
Elizabeth Mansfield will discuss seven levels of mathematical joy based on her mathematical travels. This is a talk for a general audience.
GCS 4th December 2019
14:05 to 14:50
Balázs Kovács Energy estimates: proving stability for evolving surface PDEs and geometric flows
In this talk we will give some details on the main steps and ideas behind energy estimates used to prove stability of backward difference semi- and full discretisations of parabolic evolving surface problems, or geometric flows (e.g. mean curvature flow). We will give details on how the G-stability result of Dahlquist and the multiplier techniques of Nevanlinna and Odeh will be used.




GCS 5th December 2019
13:30 to 14:15
Vanessa Styles Numerical approximations of a tractable mathematical model for tissue growth
We consider a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We derive sharp interface and diffuse interface finite element approximations of this model and present some numerical results




GCS 5th December 2019
14:15 to 15:00
Bjorn Stinner Phase field modelling of free boundary problems
Diffuse interface models based on the phase field methodology have been developed and investigated in various applications such as solidification processes, tumour growth, or multi-phase flow. The interfaces are represented by thin layers, across which quantities rapidly but smoothly change their values. These interfacial layers are described in terms of order parameters, the equations for which can be solved using relatively straightforward methods, such as finite elements with adaptive mesh refinement, as no tracking of any interface is required. The interface motion is usually coupled to other fields and equations adjacent or on the interface, for instance, diffusion equations in alloys or the momentum equation in fluid flow. We discuss how such systems can be incorporated into phase field models in a generic way. Furthermore, we present a computational framework where specific models can be implemented and later on conveniently amended, if desired, in a high-level language, and which then bind to efficient software backends. A couple of code listings and numerical simulations serve to illustrate the approach




GCS 5th December 2019
15:15 to 16:00
Bertram Düring Structure-preserving variational schemes for nonlinear partial differential equations with a Wasserstein gradient flow structure
A wide range of diffusion equations can be interpreted as gradient flow with respect to Wasserstein distance of an energy functional. Examples include the heat equation, the porous medium equation, and the fourth-order Derrida-Lebowitz-Speer-Spohn equation. When it comes to solving equations of gradient flow type numerically, schemes that respect the equation's special structure are of particular interest. The gradient flow structure gives rise to a variational scheme by means of the minimising movement scheme (also called JKO scheme, after the seminal work of Jordan, Kinderlehrer and Otto) which constitutes a time-discrete minimization problem for the energy.   While the scheme has been used originally for analytical aspects, more recently a number of authors have explored the numerical potential of this scheme. In this talk we present some results on Lagrangian schemes for Wasserstein gradient flows in one spatial dimension and then discuss extensions to higher approximation order and to higher spatial dimensions




GCS 6th December 2019
16:00 to 17:00
Chus Sanz-Serna Rothschild Lecture: Hamiltonian Monte Carlo and geometric integration

Many application fields require samples from an arbitrary probability distribution. Hamiltonian Monte Carlo is a sampling algorithm that originated in the physics literature and has later gained much popularity among statisticians. This is a talk addressed to a general audience, where I will describe the algorithm and some of its applications. The exposition requires basic ideas from different fields, from statistical physics to geometric integration of differential equations and from Bayesian statistics to Hamiltonian dynamics and I will provide the necessary background, albeit superficially.



 

GCS 11th December 2019
14:05 to 14:50
Antonella Zanna On the construction of some symplectic P-stable additive Runge—Kutta methods
Symplectic partitioned Runge–Kutta methods can be obtained from a variational formulation treating all the terms in the Lagrangian with the same quadrature formula. We construct a family of symplectic methods allowing the use of different quadrature formula for different parts of the Lagrangian. In particular, we study a family of methods using Lobatto quadrature (with corresponding Lobatto IIIA-IIIB symplectic method) and Gauss–Legendre quadrature combined in an appropriate way. The resulting methods are similar to additive Runge-Kutta methods. The IMEX method, using the Verlet and IMR combination is a particular case of this family. The methods have the same favourable implicitness as the underlying Lobatto IIIA-IIIB pair. Differently from the Lobatto IIIA-IIIB, which are known not to be P-stable, we show that the new methods satisfy the requirements for P-stability.




GCS 11th December 2019
15:05 to 15:50
Brynjulf Owren Equivariance and structure preservation in numerical methods; some cases and viewpoints
Our point of departure is the situation when there is a group of transformations acting both on our problem space and on the space in which our computations are produced. Equivariance happens when the map from the problem space to the computation space, i.e. our numerical method, commutes with the group action. This is a rather general and vague definition, but we shall make it precise and consider a few concrete examples in the talk. In some cases, the equivariance property is natural, in other cases it is something that we want to impose in the numerical method in order to obtain computational schemes with certain desired structure preserving qualities. Many of the examples we present will be related to the numerical solution of differential equations and we may also present some recent examples from artificial neural networks and discrete integrable systems. This is work in progress and it summarises some of the ideas the speaker has been discussing with other participants this autumn.




University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons