publications

2023

1. 

   An adjoint-based method for optimizing MHD equilibria against the infinite-n, ideal ballooning mode

   Rahul Gaur, Stefan Buller, Maximilian E. Ruth, and 3 more authors

   Feb 2023

   arXiv:2302.07673 [physics]

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   We demonstrate a fast adjoint-based method to optimize tokamak and stellarator equilibria against a pressure-driven instability known as the infinite-\n ideal ballooning mode. We present three finite-{}beta (the ratio of thermal to magnetic pressure) equilibria: one tokamak equilibrium and two stellarator equilibria that are unstable against the ballooning mode. Using the self-adjoint property of ideal MHD, we construct a technique to rapidly calculate the change in the growth rate, a measure of ideal ballooning instability. Using the~}texttt{SIMSOPT} framework, we then implement our fast adjoint gradient-based optimizer to minimize the growth rate and find stable equilibria for each of the three initially unstable equilibria.

2. 

   Parallel Algorithms for Computing the Tensor-Train Decomposition

   Tianyi Shi, Maximilian Ruth, and Alex Townsend

   SIAM Journal on Scientific Computing, Jun 2023

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   A simple nonrecursive form of the tensor decomposition in d dimensions is presented. It does not inherently suffer from the curse of dimensionality, it has asymptotically the same number of parameters as the canonical decomposition, but it is stable and its computation is based on low-rank approximation of auxiliary unfolding matrices. The new form gives a clear and convenient way to implement all basic operations efficiently. A fast rounding procedure is presented, as well as basic linear algebra operations. Examples showing the benefits of the decomposition are given, and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19-dimensional operator.

3. 

   Optimal Control of Nonholonomic Systems via Magnetic Fields

   Maria Oprea, Max Ruth, Dora Kassabova, and 1 more author

   IEEE Control Systems Letters, Jun 2023

   Conference Name: IEEE Control Systems Letters

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   Geometric optimal control utilizes tools from differential geometry to analyze the structure of a problem to determine the control and state trajectories to reach a desired outcome while minimizing some cost function. For a controlled mechanical system, the control usually manifests as an external force which, if conservative, can be added to the Hamiltonian. In this letter, we focus on mechanical systems with controls added to the symplectic form rather than the Hamiltonian. In practice, this translates to controlling the magnetic field for an electrically charged system. We develop a basic theory deriving necessary conditions for optimality of such a system subjected to nonholonomic constraints. We consider the representative example of a magnetically charged Chaplygin Sleigh, whose resulting optimal control problem is completely integrable.

2018

1. The Role of Topology in Magnetic Solitary Wave Dynamics

   Maximilian E. Ruth

   University of Colorado Boulder, Department of Applied Mathematics, Jun 2018

2. 

   Transverse instabilities of stripe domains in magnetic thin films with perpendicular magnetic anisotropy

   Max E. Ruth, Ezio Iacocca, Panayotis G. Kevrekidis, and 1 more author

   Physical Review B, Mar 2018

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   Stripe domains are narrow, elongated, reversed regions that exist in magnetic materials with perpendicular magnetic anisotropy. They appear as a pair of domain walls that can exhibit topology with a nonzero chirality. Recent experimental and numerical investigations identify an instability of stripe domains along the long direction as a means of nucleating isolated magnetic skyrmions. Here, the onset and nonlinear evolution of transverse instabilities for a dynamic stripe domain known as the bion stripe are investigated. Both nontopological and topological variants of the bion stripe are shown to exhibit a long-wavelength transverse instability with different characteristic features. In the former, small transverse variations in the stripe’s width lead to a neck instability that eventually pinches the nontopological stripe into a chain of two-dimensional breathers composed of droplet soliton pairs. In the latter case, small variations in the stripe’s center result in a snake instability whose topological structure leads to the nucleation of dynamic magnetic skyrmions and antiskyrmions as well as perimeter-modulated droplets. Quantitative, analytical predictions for both the early, linear evolution and the long-time, nonlinear evolution are achieved using an averaged Lagrangian approach that incorporates both exchange (dispersion) and anisotropy (nonlinearity). The method of analysis is general and can be applied to other filamentary structures.