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Permanent magnet optimization as sparse regression

Alan Kaptanoglu, Tony Qian,

Florian Wechsung, Matt Landreman

Permanent magnet optimization as sparse regression

(on arxiv next week!)

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Alan Kaptanoglu, Tony Qian, Florian Wechsung, Matt Landreman

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Why use permanent magnets for stage-2 optimization?

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  • Recent idea: reduce coil complexity by using permanent magnets instead of superconducting coils to do some of the magnetic field shaping.

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  • If permanent magnets are small compared to distance between the magnets and the plasma surface, can treat them all as independent dipoles.

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  • The plan now is to initialize big grid of possible locations for permanent magnets to reside (avoid shape optimization!).

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  • Still need optimization to determine dipole vector corresponding to each permanent magnet and ask for sparse solution.

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The idea: permanent magnet optimization as sparse regression

Constraints coming from maximum possible dipole strength of each of the individual permanent magnets.

Entirely encodes how well the permanent magnets reproduce the desired target plasma equilibrium fields.

The L0 “norm” (sparsity and grid-alignment!)

Optimization variables: each component of each dipole vector of D permanent magnets

Design new “relax-and-split” algorithm that satisfies:

  1. Solution is independent of the initial magnets.
  2. Explicitly enforces hard constraints on the dipole moment magnitudes.
  3. Sparse regression is prolific field of research – we can adapt this for PM optimization.
  4. Grid-aligned dipoles appear “for free”

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Algorithm progression

Algorithm iteration

Objective

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Results on MUSE are comparable to FAMUS + FICUS, and uses 7764 fewer magnets

FAMUS

SIMSOPT

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Future work

    • Permanent magnet configurations for some of the new high-quality stellarators being generated by the Simons collaboration

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    • Repeat with stochastic optimization

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    • Repeat with mixed-integer discrete optimization

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    • Repeat optimizations with correct near-fields for the permanent magnets

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Extra slides

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FAMUS

SIMSOPT

Grid-alignment in FAMUS and in SIMSOPT

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Solving nonconvex, constrained sparse regression with a relax-and-split algorithm

Can be computed analytically for certain nonconvex terms like L0!

Initialize initial guess w^(0) and provide a choice of hyperparameters

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Full relax-and-split algorithm