Simons Collaboration on Hidden Symmetries and Fusion Energy Annual Meeting 2020
Deadline to Register
Friday February 28, 2020
Invitation Only
The most compelling transformational use of magnetically confined, hightemperature plasma is to realize sustained fusion energy. Despite impressive progress, net energy production has not yet been achieved. The tokamak, which is the leading magnetic confinement concept in the world today, has the topology of a torus and continuous symmetry with respect to the toroidal angle, giving it good confinement properties. In the stellarator, which is the leading alternative to the tokamak, the confining magnetic field is mostly produced by external currentcarrying coils. In contrast to the tokamak, stellarators rely on symmetry breaking to realize the magnetic field needed to confine particles.
Over the last few decades, a new concept has emerged in the design of stellarators, giving rise to a renaissance—the remarkable discovery that it is possible to design 3D magnetic confinement devices with hidden symmetries that can have the same virtues as tokamaks while overcoming some of the inherent drawbacks of the latter. An example of a hidden symmetry, known as “quasisymmetry,” is that the magnitude B of the magnetic vector field B has a negligible coordinate dependence (in a special curvilinear coordinate system) even though B does not. The primary purpose of this Simons Collaboration in Mathematical and Physical Sciences is to create and exploit an effective mathematical and computational framework for the design of stellarators with hidden symmetries.
The challenge of finding 3D optimum magnetic fields with hidden symmetries encompasses mathematical and computational problems of great subtlety, straddling optimization theory, plasma physics, dynamical systems, and the analysis of partial differential equations. In this Workshop, we are bringing together a diverse interdisciplinary group of applied mathematicians, computer scientists, and plasma physicists to discuss the fundamental challenges and transformational potential of the stellarator approach to magnetic confinement. Presentations at the Annual Meeting, which is by invitation only, include eight plenary lectures, a poster session, and ample time for discussion.

Agenda
Thursday, March 26
8:30 AM CHECKIN & BREAKFAST 9:30 AM Amitava Bhattacharjee  Hidden Symmetries and Fusion Energy — A Year Later 10:30 AM BREAK 11:00 AM Aaron Bader  Stellarator Optimization in Practice — And Where We Can Improve 12:00 PM LUNCH 1:00 PM Elizabeth Paul  Efficient Stellarator Shape Optimization and Sensitivity Analysis 2:00 PM BREAK 2:30 PM Andrew Giuliani  AdjointBased VacuumField Stellarator Optimization 3:30 PM BREAK 4:00 PM Dhairya Malhotra  Integral Equation Methods for Computing SteppedPressure Equilibrium in Stellarators 5:00 PM DAY ONE CONCLUDES Friday, March 27
8:30 AM CHECKIN & BREAKFAST 9:30 AM Peter Constantin  Deformation and Rigidity of Equilibria 10:30 AM BREAK 11:00 AM Per Helander  Stellarators with Permanent Magnets 12:00 PM LUNCH 1:00 PM David Keyes  Hierarchical Algorithms on Hierarchical Architectures” 2:00 PM MEETING CONCLUDES 
Abstracts
Amitava Bhattacharjee
Hidden Symmetries and Fusion Energy — A Year LaterThe inaugural Annual Meeting of the Collaboration on Hidden Symmetries and Fusion Energy, supported by the Simons Foundation, occurred a year ago, inspired by the beautiful idea that magnetic fields with hidden symmetries provide a unique pathway to realizing thermonuclear fusion under controlled laboratory conditions. An example of a hidden symmetry, known as ‘quasisymmetry,’ is that the magnitude B of the vector magnetic field B has an ignorable coordinate in a special curvilinear coordinate system, even though B may not. (The ‘tokamak,’ which is the world’s most advanced fusion concept, is axisymmetric and is a special case in that both B and B exhibit the same symmetry.) The quest for designing quasisymmetric ‘stellarators’ is a computational grand challenge that is confronted by this collaboration, which brings together a team of applied mathematicians, computer scientists and plasma physicists drawn from multiple institutions. During the last year, this team has broken new ground on elucidating fundamental mathematical and physical implications of quasisymmetry and producing new methodologies for various components of the Simons Optimization (SIMSOPT) code, which we envision as a stateoftheart computational optimization tool that can potentially exploit the power of supercomputers in the exascale era. In this talk, we will provide an overview of the new developments during the first year and a half of the collaboration. We will also describe work carried out by the Simons team in collaboration with scientists in the Princeton Plasma Physics Laboratory on a possible new experiment on stellarators with permanent magnets, which, if successful, could lead to dramatic reductions in the cost for fusion power.
Aaron Bader
Stellarator Optimization in Practice — And Where We Can ImproveOptimization of stellarator configurations has met with practical success through the experiments of HSX and W7X. The basic principle of optimization consists of three parts: 1) the solution of a magnetic equilibrium given a boundary or set of coils, 2) the metrics with which to evaluate that equilibrium and 3) an optimization scheme to vary the boundary or coils to find a minimum in the metrical evaluation. In this talk, Bader will look at each of those three parts in turn: equilibrium solution, metrics and optimization algorithms. We will describe the current state of the art, discuss some of the shortfalls and look toward future possible improvements.
Elizabeth Paul
Efficient Stellarator Shape Optimization and Sensitivity AnalysisStellarators are a class of magnetic confinement devices without continuous toroidal symmetry. The design of modern stellarators often employs gradientbased optimization to navigate the nonconvex, highdimensional spaces used to describe their geometry. Computing the numerical gradient of a target function with respect to many parameters can be expensive. The adjoint method allows these gradients to be calculated at a much lower cost and without the noise associated with finite differences. In this talk, Paul will provide an introduction to the adjoint method and its application to several problems in stellarator optimization. Examples will be given for the design of the magnetohydrodynamic equilibria for good confinement and stability properties, as well as the design of electromagnetic coils that are consistent with the desired plasma equilibria. As the relevant optimization spaces are often shapes, such as the outer boundary of the plasma domain or the filamentary lines of electromagnetic coils, Paul will describe how tools from the field of shape optimization can be applied for such problems. A top priority for the stellarator design community is the identification and reduction of engineering tolerances, as tight electromagnetic coil tolerances have historically been a larger driver of cost for the stellarator program. This sensitivity information can also be evaluated from adjointbased derivatives using the shape gradient.
Andrew Giuliani
AdjointBased VacuumField Stellarator OptimizationA standard approach to stellarator optimization separates design into two stages. The first stage determines a target magnetic field with desired physical quantities of interest. The second stage optimizes a set of coils such that they reproduce the target magnetic field as accurately as possible. Small errors in the coil optimization stage can lead to errors in the physical properties of the field generated by the coils. In this talk, Giuliani presents a combined approach to stellarator optimization whereby he optimizes directly the coil geometry to generate a magnetic field optimized for quasisymmetry and other physical properties. He defines an objective function for which exact gradients are obtained using adjoint methods. Finally, he presents a stellarator design that is obtained with his approach.
Dhairya Malhotra
Integral Equation Methods for Computing SteppedPressure Equilibrium in StellaratorsThis talk will describe the numerical construction of ideal magnetohydrodynamic (MHD) steppedpressure equilibrium in stellarators. In the steppedpressure equilibrium problem, the entire plasma volume is partitioned into regions separated by ideal MHD interfaces. Each region has constant pressure and is assumed to have undergone Taylor relaxation to a minimum energy state. In the equilibrium configuration, the forcebalance condition must be satisfied at each point on the interface.
Malhotra will present a fast, highorder accurate boundary integral equation (BIE) solver for computing Taylor relaxed states in each constant pressure region. Since BIE formulations only discretize the boundary of the domain, his method requires significantly fewer unknowns compared to volume discretizing schemes. In addition, the representation leads to a wellconditioned (away from physical interior resonances) secondkind integral equation that can be numerically inverted to highprecision. Malhotra used a gradient descent algorithm to determine the position of the ideal MHD interfaces in the equilibrium configuration. Each iteration of this scheme requires computing shape derivatives of an objective function, and he gives an efficient algorithm for computing such shape derivatives for BIE formulations.
Peter Constantin
Deformation and Rigidity of EquilibriaConstantin discusses possibility and the impossibility of smooth deformations of solutions of equilibrium equations in the presence of symmetry.
Per Helander
Stellarators with Permanent MagnetsStellarators, tokamaks and other devices for fusion plasma confinement use electromagnets to create the magnetic field. In the case of stellarators, the required magneticfield coils can be very complicated and contribute significantly to the overall cost of the device. It has recently been suggested that permanent magnets could be used to shape the plasma and drastically simplify the coils. This talk will discuss mathematical aspects of this idea, in particular the problem of how to arrange the magnets.
Permanent magnets cannot create toroidal magnetic flux, but they can create poloidal flux and thus produce rotational transform. Toroidalfield coils are, therefore, unavoidable, but much (or all) of the plasma shaping could be accomplished by permanent magnets. In contrast to coils, they can easily be arranged in complicated patterns and do not require power supplies or cooling, but suffer from other disadvantages, such as limitations in field strength, nontunability and the possibility of demagnetization.
The central mathematical question is how to arrange permanent magnets to produce a desired magnetic field inside the plasma. It will be shown that this problem is no more difficult — in fact, it is in a certain sense easier — than that of finding suitable coils in ordinary stellarator design. Moreover, there is great freedom in choosing the magnetization M(r) in the region occupied by the magnets since the magnetic field they produce is invariant under the gauge transformation M(r) → M(r) + grad χ (r) where χ is any function that vanishes on the boundary of the region. This freedom can be used to minimize the maximum value of M(r), which is limited to about 1.4 T /µ_{0} for commercially available magnets.
Several concrete examples of stellarator fields with permanent magnets will be shown and discussed in the talk, including a quasiaxisymmetric stellarator using only eight identical circular coils.
David Keyes
Hierarchical Algorithms on Hierarchical ArchitecturesA traditional goal of algorithmic optimality, squeezing out flops, has been superseded by evolution in architecture. Flops no longer serve as a reasonable proxy for all aspects of complexity. Instead, algorithms must now squeeze memory, data transfers and synchronizations, while extra flops on locally cached data represent only small costs in time and energy. Hierarchically lowrank matrices realize a rarely achieved combination of optimal storage complexity and highcomputational intensity for a wide class of formally dense linear operators that arise in applications for which exascale computers are being constructed. They may be regarded as algebraic generalizations of the fast multipole method. Methods based on these hierarchical data structures and their simpler cousins, tile lowrank matrices, are well proportioned for early exascale computer architectures, which are provisioned for high processing power relative to memory capacity and memory bandwidth. Hierarchically lowrank matrices are ushering in a renaissance of computational linear algebra. A challenge is that emerging hardware architecture possesses hierarchies of its own that do not generally align with those of a given algorithmapplication pair. Keyes describes modules of a software tool kit, hierarchical computations on manycore architectures (HiCMA), that illustrate these features and are intended as building blocks of applications, such as matrixfree higherorder methods in optimization and largescale spatial statistics. Some modules of this opensource project have been adopted in the software libraries of major vendors.

Public Lecture
Simons Foundation Lecture
Wednesday, March 25, 2020Amitava Bhattacharjee, Princeton University
Tea 4:155:00 PM
Lecture 5:006:15 PMParticipation is optional; separate registration is required. Registration is expected to open on Wednesday, February 12, 2020.
https://www.simonsfoundation.org/simonsfoundationlectures/

Travel
Air and Train
Groups A & B
The foundation will arrange and pay for all air and train travel to the conference for those in Groups A and B. Please provide your travel specifications by clicking the registration link above. If you are unsure of your group, please refer to your invitation sent via email.Group C
Individuals in Group C will not receive financial support. Please register at the link above so we can capture your dietary requirements. If you are unsure of your group, please refer to your invitation sent via email.Personal Car
For participants driving to Manhattan, The James Hotel New York, NoMad offers valet parking. Please note there are no inandout privileges when using the hotel’s garage, therefore it is encouraged that participants walk or take public transportation to the Simons Foundation. 
Hotel
Participants in Groups A & B who require accommodations are hosted by the foundation for a maximum of three nights at The James Hotel New York, NoMad. Hotel nights must directly correspond to the dates of the meeting. Any additional nights are at the attendee’s own expense.
The James Hotel New York, NoMad
22 East 29th Street
New York, NY 10016
(between 28^{th} and 29^{th} Streets)To arrange accommodations, please register here.
For driving directions to The James, please click here.

Contacts
Travel and Hotel Assistance
Elise Volpe, CCE Global Meetings & Incentives
simons@cceglobal.com
6319236757Registration and General Meeting Assistance
Emily Klein
MPS Event Coordinator, Simons Foundation
eklein@simonsfoundation.org
(646) 7511262 
Travel & Expense Policy
Expense reimbursement will be handled online via our reimbursement platform hosted by Concur. Additional information in this regard will be sent at the conclusion of the annual meeting.