Permanent magnet optimization for stellarators as sparse regression

A common scientific inverse problem is the placement of magnets that produce a desired magnetic field inside a prescribed volume. This is a key component of stellarator design, and recently permanent magnets have been proposed as a potentially useful tool for magnetic field shaping. Here, we take a closer look at possible objective functions for permanent magnet optimization, reformulate the problem as sparse regression, and propose an algorithm that can efficiently solve many convex and nonconvex variants. The algorithm generates sparse solutions that are independent of the initial guess, explicitly enforces maximum strengths for the permanent magnets, and accurately produces the desired magnetic field. The algorithm is flexible, and our implementation is open-source and computationally fast. We conclude with two new permanent magnet configurations for the NCSX and MUSE stellarators. Our methodology can be additionally applied for effectively solving permanent magnet optimizations in other scientific fields, as well as for solving quite general high-dimensional, constrained, sparse regression problems, even if a binary solution is required

Minimax Current Density Coil Design

'Coil design' is an inverse problem in which arrangements of wire are designed to generate a prescribed magnetic field when energized with electric current. The design of gradient and shim coils for magnetic resonance imaging (MRI) are important examples of coil design. The magnetic fields that these coils generate are usually required to be both strong and accurate. Other electromagnetic properties of the coils, such as inductance, may be considered in the design process, which becomes an optimization problem. The maximum current density is additionally optimized in this work and the resultant coils are investigated for performance and practicality. Coils with minimax current density were found to exhibit maximally spread wires and may help disperse localized regions of Joule heating. They also produce the highest possible magnetic field strength per unit current for any given surface and wire size. Three different flavours of boundary element method that employ different basis functions (triangular elements with uniform current, cylindrical elements with sinusoidal current and conic section elements with sinusoidal-uniform current) were used with this approach to illustrate its generality.Comment: 24 pages, 6 figures, 2 tables. To appear in Journal of Physics D: Applied Physic

Greedy permanent magnet optimization

A number of scientific fields rely on placing permanent magnets in order to produce a desired magnetic field. We have shown in recent work that the placement process can be formulated as sparse regression. However, binary, grid-aligned solutions are desired for realistic engineering designs. We now show that the binary permanent magnet problem can be formulated as a quadratic program with quadratic equality constraints (QPQC), the binary, grid-aligned problem is equivalent to the quadratic knapsack problem with multiple knapsack constraints (MdQKP), and the single-orientation-only problem is equivalent to the unconstrained quadratic binary problem (BQP). We then provide a set of simple greedy algorithms for solving variants of permanent magnet optimization, and demonstrate their capabilities by designing magnets for stellarator plasmas. The algorithms can a-priori produce sparse, grid-aligned, binary solutions. Despite its simple design and greedy nature, we provide an algorithm that outperforms the state-of-the-art algorithms while being substantially faster, more flexible, and easier-to-use

Topology optimization for inverse magnetostatics as sparse regression: application to electromagnetic coils for stellarators

Topology optimization, a technique to determine where material should be placed within a predefined volume in order to minimize a physical objective, is used across a wide range of scientific fields and applications. A general application for topology optimization is inverse magnetostatics; a desired magnetic field is prescribed, and a distribution of steady currents is computed to produce that target field. In the present work, electromagnetic coils are designed by magnetostatic topology optimization, using volume elements (voxels) of electric current, constrained so the current is divergence-free. Compared to standard electromagnet shape optimization, our method has the advantage that the nonlinearity in the Biot-Savart law with respect to position is avoided, enabling convex cost functions and a useful reformulation of topology optimization as sparse regression. To demonstrate, we consider the application of designing electromagnetic coils for a class of plasma experiments known as stellarators. We produce topologically-exotic coils for several new stellarator designs and show that these solutions can be interpolated into a filamentary representation and then further optimized

Accelerated partial separable model using dimension-reduced optimization technique for ultra-fast cardiac MRI

Objective. Imaging dynamic object with high temporal resolution is challenging in magnetic resonance imaging (MRI). Partial separable (PS) model was proposed to improve the imaging quality by reducing the degrees of freedom of the inverse problem. However, PS model still suffers from long acquisition time and even longer reconstruction time. The main objective of this study is to accelerate the PS model, shorten the time required for acquisition and reconstruction, and maintain good image quality simultaneously. Approach. We proposed to fully exploit the dimension reduction property of the PS model, which means implementing the optimization algorithm in subspace. We optimized the data consistency term, and used a Tikhonov regularization term based on the Frobenius norm of temporal difference. The proposed dimension-reduced optimization technique was validated in free-running cardiac MRI. We have performed both retrospective experiments on public dataset and prospective experiments on in-vivo data. The proposed method was compared with four competing algorithms based on PS model, and two non-PS model methods. Main results. The proposed method has robust performance against shortened acquisition time or suboptimal hyper-parameter settings, and achieves superior image quality over all other competing algorithms. The proposed method is 20-fold faster than the widely accepted PS+Sparse method, enabling image reconstruction to be finished in just a few seconds. Significance. Accelerated PS model has the potential to save much time for clinical dynamic MRI examination, and is promising for real-time MRI applications.Comment: 23 pages, 11 figures. Accepted as manuscript on Physics in Medicine & Biolog

Real-time Feedback of B0 Shimming at Ultra High Field MRI

Magnetic resonance imaging(MRI) is moving towards higher and higher field strengths. After 1.5T MRI scanners became commonplace, 3T scanners were introduced and once 3T scanners became commonplace, ultra high field (UHF) scanners were introduced. UHF scanners typically refer to scanners with a field strength of 7T or higher. The number of sites that utilise UHF scanners is slowly growing and the first 7T MRI scanners were recently CE certified for clinical use. Although UHF scanners have the benefit of higher signal-to-noise ratio (SNR), they come with their own challenges. One of the many challenges is the problem of inhomogeneity of the main static magnetic field(B0 field). This thesis addresses multiple aspects associated with the problem of B0 inhomogeneity. The process of homogenising the field is called "shimming". The focus of this thesis is on active shimming where extra shim coils drive DC currents to generate extra magnetic fields superimposed on the main magnetic field to correct for inhomogeneities. In particular, we looked at the following issues: algorithms for calculating optimal shim currents; global static shimming using very high order/degree spherical harmonic-based (VHOS) coils; dynamic slice-wise shimming using VHOS coils compared to a localised multi-coil array shim system; B0 field monitoring using an NMR field camera; characterisation of the shim system using a field camera; and designing a controller based on the shim system model for real-time feedback

Spatio-temporal wavelet regularization for parallel MRI reconstruction: application to functional MRI

Parallel MRI is a fast imaging technique that enables the acquisition of highly resolved images in space or/and in time. The performance of parallel imaging strongly depends on the reconstruction algorithm, which can proceed either in the original k-space (GRAPPA, SMASH) or in the image domain (SENSE-like methods). To improve the performance of the widely used SENSE algorithm, 2D- or slice-specific regularization in the wavelet domain has been deeply investigated. In this paper, we extend this approach using 3D-wavelet representations in order to handle all slices together and address reconstruction artifacts which propagate across adjacent slices. The gain induced by such extension (3D-Unconstrained Wavelet Regularized -SENSE: 3D-UWR-SENSE) is validated on anatomical image reconstruction where no temporal acquisition is considered. Another important extension accounts for temporal correlations that exist between successive scans in functional MRI (fMRI). In addition to the case of 2D+t acquisition schemes addressed by some other methods like kt-FOCUSS, our approach allows us to deal with 3D+t acquisition schemes which are widely used in neuroimaging. The resulting 3D-UWR-SENSE and 4D-UWR-SENSE reconstruction schemes are fully unsupervised in the sense that all regularization parameters are estimated in the maximum likelihood sense on a reference scan. The gain induced by such extensions is illustrated on both anatomical and functional image reconstruction, and also measured in terms of statistical sensitivity for the 4D-UWR-SENSE approach during a fast event-related fMRI protocol. Our 4D-UWR-SENSE algorithm outperforms the SENSE reconstruction at the subject and group levels (15 subjects) for different contrasts of interest (eg, motor or computation tasks) and using different parallel acceleration factors (R=2 and R=4) on 2x2x3mm3 EPI images.Comment: arXiv admin note: substantial text overlap with arXiv:1103.353