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Timetable (GCSW02)

Structure preservation and general relativity

Monday 30th September 2019 to Friday 4th October 2019

Monday 30th September 2019
09:00 to 09:20 Registration
09:20 to 09:30 Welcome from Christie Marr (INI Deputy Director)
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.
10:30 to 11:00 Morning Coffee
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.
12:00 to 13:30 Lunch at Murray Edwards College
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.
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.
15:30 to 16:00 Afternoon Tea
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.
17:00 to 18:00 Welcome drinks reception & poster session
Tuesday 1st October 2019
09:30 to 10:30 Lee Lindblom
Solving PDEs Numerically on Manifolds with Arbitrary Spatial Topologies
10:30 to 11:00 Morning Coffee
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.
12:00 to 13:30 Lunch at Murray Edwards College
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.
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.
15:30 to 16:00 Afternoon Tea
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.
Wednesday 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.
10:30 to 11:00 Morning Coffee
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.
12:00 to 13:30 Lunch at Murray Edwards College
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.
19:30 to 20:30 Formal Dinner at Emmanuel College
Thursday 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.
10:30 to 11:00 Morning Coffee
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.
12:00 to 13:30 Lunch at Murray Edwards College
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.
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.

15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Douglas Arnold
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.
Friday 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
12:00 to 13:30 Lunch at Murray Edwards College
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.
14:30 to 15:30 Luis Lehner
15:30 to 16:00 Afternoon Tea
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.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons