An Extrinsic Look at the Riemannian Hessian

Absil, Pierre-Antoine; Mahony, Robert; Trumpf, Jochen

doi:10.1007/978-3-642-40020-9_39

Cited by 75 publications

(98 citation statements)

References 16 publications

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“…We will show that G(·) and H(·) are Riemannian gradient and Riemannian Hessian of f over Ω in the following lemma. A similar argument can be found in [17] and [18] Lemma 5. G(·) and H(·) defined in 3.2 and 3.3 are the Riemannian gradient and Riemannian Hessian of f (·) over Ω.…”

Section: Riemannian Gradient and Riemannian Hessiansupporting

confidence: 77%

Using negative curvature in solving nonlinear programs

Goldfarb

Wright

et al. 2017

Comput Optim Appl

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Minimization methods that search along a curvilinear path composed of a non-ascent negative curvature direction in addition to the direction of steepest descent, dating back to the late 1970s, have been an effective approach to finding a stationary point of a function at which its Hessian is positive semidefinite. For constrained nonlinear programs arising from recent applications, the primary goal is to find a stationary point that satisfies the second-order necessary optimality conditions. Motivated by this, we generalize the approach of using negative curvature directions from unconstrained optimization to nonlinear ones. We focus on equality constrained problems and prove that our proposed negative curvature method is guaranteed to converge to a stationary point satisfying second-order necessary conditions. A possible way to extend our proposed negative curvature method to general nonlinear programs is also briefly discussed.

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Section: Riemannian Gradient and Riemannian Hessiansupporting

confidence: 77%

Using negative curvature in solving nonlinear programs

Goldfarb

Wright

et al. 2017

Comput Optim Appl

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“…13, the covariant second derivatives, valid at all points of the phase space, can be calculated using a projector-based approach. 64 The corresponding 2N × 2N Hessian matrix can be represented as…”

Section: F Minimum Mode Following Methodsmentioning

confidence: 99%

Spirit : Multifunctional framework for atomistic spin simulations

et al. 2019

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The Spirit framework is designed for atomic scale spin simulations of magnetic systems of arbitrary geometry and magnetic structure, providing a graphical user interface with powerful visualizations and an easy to use scripting interface. An extended Heisenberg type spin-lattice Hamiltonian including competing exchange interactions between neighbors at arbitrary distance, higher-order exchange, Dzyaloshinskii-Moriya and dipole-dipole interactions is used to describe the energetics of a system of classical spins localised at atom positions. A variety of common simulations methods are implemented including Monte Carlo and various time evolution algorithms based on the Landau-Lifshitz-Gilbert equation of motion, which can be used to determine static ground state and metastable spin configurations, sample equilibrium and finite temperature thermodynamical properties of magnetic materials and nanostructures or calculate dynamical trajectories including spin torques induced by stochastic temperature or electric current. Methods for finding the mechanism and rate of thermally assisted transitions include the geodesic nudged elastic band method, which can be applied when both initial and final states are specified, and the minimum mode following method when only the initial state is given. The lifetime of magnetic states and rate of transitions can be evaluated within the harmonic approximation of transition-state theory. The framework offers performant CPU and GPU parallelizations. All methods are verified and applications to several systems, such as vortices, domain walls, skyrmions and bobbers are described.

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“…In this larger space, second order derivatives are easily performed. The Hessian in the physical subspace can then be reconstructed by a projection operator approach [28]. For any scalar function f on the manifold M phys , this true Hessian is defined as…”

mentioning

confidence: 99%

Duplication, Collapse, and Escape of Magnetic Skyrmions Revealed Using a Systematic Saddle Point Search Method

et al. 2018

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Various transitions that a magnetic skyrmion can undergo are found in calculations using a method for climbing up the energy surface and converging onto first order saddle points. In addition to collapse and escape through a boundary, the method identifies a transition where the skyrmion divides and forms two skyrmions. The activation energy for this duplication process can be similar to that of collapse and escape. A tilting of the external magnetic field for a certain time interval is found to induce the duplication process in a dynamical simulation. Such a process could turn out to be an important avenue for the creation of skyrmions in future magnetic devices.Localized, non-collinear magnetic states are receiving a great deal of attention, where skyrmions have come under special focus. Along with interesting transport properties, skyrmions exhibit particle-like behaviour and carry a topological charge enhancing their stability with respect to uniform ferromagnetic background. In addition to the interest in their intriguing properties, they have been suggested as a basis for technological applications e.g. data storage or even data processing devices [1, 2]. A racetrack design of a memory device has been outlined where a spin polarized current drives a chain of skyrmions past a reading device [3, 4]. The effect of temperature and external magnetic field on the stability of the skyrmions need to be studied, as well as ways to generate and manipulate them. The effect of defects is also an important consideration [5, 6]. Two mechanisms for the annihilation of skyrmions have been characterized by theoretical calculations of atomic scale systems: Collapse of a skyrmion to form ferromagnetic state [7][8][9][10][11] and escape of a skyrmion through the boundary of the magnetic domain [9][10][11][12]. The effect of a non-magnetic impurity has also been calculated [11]. By using harmonic transition state theory for magnetic systems [14,15], the lifetime of skyrmions has been estimated [11,12]. Parameter values obtained from density functional theory [13] are found to give results that are consistent with experimental observations [16,17]. The challenge is to design materials where magnetic skyrmions are small enough while being sufficiently stable at ambient temperature, and to develop methods for manipulating them.Theoretical calculations can help accelerate this development by identifying the various possible transformations that a skyrmion can undergo at a finite temperature on a laboratory time scale. This can be achieved by the use of rate theory where the major challenge is to find the relevant transition mechanisms. If the final state of a transition is specified, in addition to the initial state, the geodesic nudged elastic band (GNEB) method [7,18] can be used to find the minimum energy path of the transition and, thereby, the activation energy which is the highest rise in energy along the path. However, the final states of possible transitions are not always known. Another category of methods for identifying ...

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An Extrinsic Look at the Riemannian Hessian

Cited by 75 publications

References 16 publications

Using negative curvature in solving nonlinear programs

Using negative curvature in solving nonlinear programs

Spirit : Multifunctional framework for atomistic spin simulations

Duplication, Collapse, and Escape of Magnetic Skyrmions Revealed Using a Systematic Saddle Point Search Method

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