Optim: A mathematical optimization package for Julia

Mogensen, Patrick Kofod; Riseth, Asbjørn Nilsen

doi:10.21105/joss.00615

Cited by 382 publications

(248 citation statements)

References 6 publications

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“…where V k is defined in (6). The objective functions satisfy Assumption 1 by construction, and the constraint in (14) ensures that the spectral gap of the Laplacian matrix at each step is upperbounded by σ so that Assumption 2 holds. Therefore, the trajectory produced by (14) satisfies both our assumptions.…”

Section: Generating Worst-case Examplesmentioning

confidence: 99%

Analysis and Design of First-Order Distributed Optimization Algorithms Over Time-Varying Graphs

Sundararajan

Scoy

Lessard

2020

IEEE Trans. Control Netw. Syst.

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This work concerns the analysis and design of distributed first-order optimization algorithms over time-varying graphs. The goal of such algorithms is to optimize a global function that is the average of local functions using only local computations and communications. Several different algorithms have been proposed that achieve linear convergence to the global optimum when the local functions are strongly convex. We provide a unified analysis that yields a worstcase linear convergence rate as a function of the condition number of the local functions, the spectral gap of the graph, and the parameters of the algorithm. The framework requires solving a small semidefinite program whose size is fixed; it does not depend on the number of local functions or the dimension of the domain. The result is a computationally efficient method for distributed algorithm analysis that enables the rapid comparison, selection, and tuning of algorithms. Finally, we propose a new algorithm, which we call SVL, that is easily implementable and achieves the fastest possible worst-case convergence rate among all algorithms in the family we considered. We support our theoretical analysis with numerical experiments that generate worst-case examples demonstrating the effectiveness of SVL.

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Section: Generating Worst-case Examplesmentioning

confidence: 99%

Analysis and Design of First-Order Distributed Optimization Algorithms Over Time-Varying Graphs

Sundararajan

Scoy

Lessard

2020

IEEE Trans. Control Netw. Syst.

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“…To assess correctness of the T 1 and T 2 maps reconstructed with MR-STAT, data were also acquired using gold standard methods in the form of an inversion-recovery single spin-echo protocol with inversion times of [50,100,150,350,550,850, 1250] ms for T 1 mapping and a single-echo spin-echo protocol with echo times of [8,28,48,88,138, 188] ms for T 2 mapping.…”

Section: Gel Phantom Experimentsmentioning

confidence: 99%

High‐resolution in vivo MR‐STAT using a matrix‐free and parallelized reconstruction algorithm

et al. 2020

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MR‐STAT is a recently proposed framework that allows the reconstruction of multiple quantitative parameter maps from a single short scan by performing spatial localisation and parameter estimation on the time‐domain data simultaneously, without relying on the fast Fourier transform (FFT). To do this at high resolution, specialized algorithms are required to solve the underlying large‐scale nonlinear optimisation problem. We propose a matrix‐free and parallelized inexact Gauss–Newton based reconstruction algorithm for this purpose. The proposed algorithm is implemented on a high‐performance computing cluster and is demonstrated to be able to generate high‐resolution (1 mm × 1 mm in‐plane resolution) quantitative parameter maps in simulation, phantom, and in vivo brain experiments. Reconstructed T1 and T2 values for the gel phantoms are in agreement with results from gold standard measurements and, for the in vivo experiments, the quantitative values show good agreement with literature values. In all experiments, short pulse sequences with robust Cartesian sampling are used, for which MR fingerprinting reconstructions are shown to fail.

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“…As the results of many ParticleScattering computations can be recycled between optimization iterations, a large number of parameters can be optimized simultaneously in reasonable time. ParticleScattering performs gradient-based optimization of rotation angle for arbitrarily-shaped particles, and radius of circular particles, in conjunction with the Optim optimization package (Mogensen and Riseth 2018), where the objective is to minimize or maximize the electric field intensity at chosen points. Figure 1 shows an example of angle optimization of 20 particles, where the objective is the electric field intensity at the origin.…”

Section: Optimizationmentioning

confidence: 99%

ParticleScattering: Solving and optimizing multiple-scattering problems in Julia

Blankrot¹,

Heitzinger²

2018

JOSS

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SummaryParticleScattering is a Julia (Bezanson et al. 2017) package for computing the electromagnetic fields scattered by a large number of two-dimensional particles, as well as optimizing particle parameters for various applications. Such problems naturally arise in the design and analysis of metamaterials, including photonic crystals (Jahani and Jacob 2016). Unlike most solvers for these problems, ours does not require a periodic structure and is scalable to a large number of particles. In particular, this software is designed for scattering problems involving TM plane waves impinging on a collection of homogeneous dielectric particles with arbitrary smooth shapes. Our code performs especially well when the number of particles is substantially larger than the number of distinct shapes, where particles are considered indistinct if they are identical up to rotation. Solver overviewGiven a scattering problem consisting of a collection of penetrable particles in a homogeneous medium, the software performs the following steps to calculate the total electric field:• For each distinct non-circular shape, a single-and double-layer potential formulation is constructed.• The potential formulations are transformed to a multipole basis of Hankel functions, reducing the degrees of freedom by at least an order of magnitude.• Analytical multipole basis is computed for circular particles.• A multiple-scattering system of equations is constructed, and then solved with the Fast Multipole Method.• Electric field is computed at any point of interest.In addition, ParticleScattering can plot near-and far-field results using the popular framework PyPlot, create publication-level plots with PGFPlots integration, and compute minimum parameters for a desired error level. OptimizationParticleScattering is especially targeted at users who wish to design metamaterials belonging to the class described above. While the large number of variables such metamaterials contain allows for a variety of devices that meet different objectives, it also creates a large search space for choosing them. Therefore, a fast and automated approach can be beneficial for both inventing new designs and improving existing ones. As the results of many ParticleScattering computations can be recycled between optimization iterations, a large

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Optim: A mathematical optimization package for Julia

Cited by 382 publications

References 6 publications

Analysis and Design of First-Order Distributed Optimization Algorithms Over Time-Varying Graphs

Analysis and Design of First-Order Distributed Optimization Algorithms Over Time-Varying Graphs

High‐resolution in vivo MR‐STAT using a matrix‐free and parallelized reconstruction algorithm

ParticleScattering: Solving and optimizing multiple-scattering problems in Julia

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