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Magnetic confinement devices for nuclear fusion can be large and expensive. Compact stellarators are promising candidates for costreduction, but introduce new difficulties: confinement in smaller volumes requires higher magnetic field, which calls for higher coil-currents and ultimately causes higher Laplace forces on the coils-if everything else remains the same. This motivates the inclusion of force reduction in stellarator coil optimization. In the present paper we consider a coil winding surface, we prove that there is a natural and rigorous way to define the Laplace force (despite the magnetic field discontinuity across the current-sheet), we provide examples of cost associated (peak force, surface-integral of the force squared) and discuss easy generalizations to parallel and normal force-components, as these will be subject to different engineering constraints. Such costs can then be easily added to the figure of merit in any multi-objective stellarator coil optimization code. We demonstrate this for a generalization of the REGCOIL code [1], which we rewrote in python, and provide numerical examples for the NCSX (now QUASAR) design. We present results for various definitions of the cost function, including peak force reductions by up to 40 %, and outline future work for further reduction.
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We consider a problem relating to magnetic confinement devices known as stellarators. Plasma is confined by magnetic fields generated by current-carrying coils, and here we investigate how closely to the plasma they need to be positioned. Current-carrying coils are represented as singularities within the magnetic field and therefore this problem can be modelled mathematically as finding how far we can harmonically extend a vector field from the boundary of a domain. For this paper we consider two-dimensional domains with real analytic boundary, and prove that a harmonic extension exists if and only if the boundary data satisfies a combined compatibility and regularity condition. Our method of proof uses a generalisation of a result of Hadamard on the Cauchy problem for the Laplacian. We then provide a lower bound on how far we can harmonically extend the vector field from the boundary via the Cauchy--Kovalevskaya Theorem.
A neoclassically optimized compact stellarator with simple coils has been designed. The magnetic field of the new stellarator is generated by only four planar coils including two interlocking coils of elliptical shape and two circular poloidal field coils. The interlocking coil topology is the same as that of the Columbia Non-neutral Torus (CNT). The new configuration was obtained by minimizing the effective helical ripple directly via the shape of the two interlocking coils. The optimized compact stellarator has very low effective ripple in the plasma core implying excellent neoclassical confinement. This is confirmed by the results of the drift-kinetic code SFINCS showing that the particle diffusion coefficient of the new configuration is one order of magnitude lower than CNTs.
Entropy functionals (i.e. convex integral functionals) and extensions of these functionals are minimized on convex sets. This paper is aimed at reducing as much as possible the assumptions on the constraint set. Dual equalities and characterizations of the minimizers are obtained with weak constraint qualifications.
We propose faster methods for unconstrained optimization of emph{structured convex quartics}, which are convex functions of the form begin{equation*} f(x) = c^top x + x^top mathbf{G} x + mathbf{T}[x,x,x] + frac{1}{24} mathopen| mathbf{A} x mathclose|_4^4 end{equation*} for $c in mathbb{R}^d$, $mathbf{G} in mathbb{R}^{d times d}$, $mathbf{T} in mathbb{R}^{d times d times d}$, and $mathbf{A} in mathbb{R}^{n times d}$ such that $mathbf{A}^top mathbf{A} succ 0$. In particular, we show how to achieve an $epsilon$-optimal minimizer for such functions with only $O(n^{1/5}log^{O(1)}(mathcal{Z}/epsilon))$ calls to a gradient oracle and linear system solver, where $mathcal{Z}$ is a problem-dependent parameter. Our work extends recent ideas on efficient tensor methods and higher-order acceleration techniques to develop a descent method for optimizing the relevant quartic functions. As a natural consequence of our method, we achieve an overall cost of $O(n^{1/5}log^{O(1)}(mathcal{Z} / epsilon))$ calls to a gradient oracle and (sparse) linear system solver for the problem of $ell_4$-regression when $mathbf{A}^top mathbf{A} succ 0$, providing additional insight into what may be achieved for general $ell_p$-regression. Our results show the benefit of combining efficient higher-order methods with recent acceleration techniques for improving convergence rates in fundamental convex optimization problems.
We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term. Properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem. Preliminary computational results on both convex and nonconvex examples are promising.
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