Abstract
This paper studies the J-method of [E. Jarlebring, S. Kvaal, W. Michiels. SIAM J. Sci. Comput. 36-4:A1978-A2001, 2014] for nonlinear eigenvector problems in a general Hilbert space framework. This is the basis for variational discretization techniques and a mesh-independent numerical analysis. A simple modification of the method mimics an energy-decreasing discrete gradient flow. In the case of the Gross–Pitaevskii eigenvalue problem, we prove global convergence towards an eigenfunction for a damped version of the J-method. More importantly, when the iterations are sufficiently close to an eigenfunction, the damping can be switched off and we recover a local linear convergence rate previously known from the discrete setting. This quantitative convergence analysis is closely connected to the J-method’s unique feature of sensitivity with respect to spectral shifts. Contrary to classical gradient flows, this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems. These advantageous convergence properties are demonstrated in a series of numerical experiments involving exponentially localized states under disorder potentials and vortex lattices in rotating traps.
Highlights
This paper studies eigenvalues of nonlinear differential operators A on a real Hilbert space V of the formA(v) = A(v, v), where A : V × V → V ∗ is a continuous, bounded mapping that is invariant under scaling of the first argument and real-linear in the second argument
Another important class of iterative methods is based on gradient flows for the energy functional associated with (1), where we mention the Discrete Normalized Gradient Flow (DNGF), cf. [11,12,13,15], which is based on an implicit Euler discretization of the L2-gradient flow
We mention the Projected Sobolev Gradient Flows (PSGFs), cf. [27,33,34,36,39,46,47,57], which form a subclass of the gradient flow methods
Summary
An improvement of PSGF by using Riemannian conjugate gradients was suggested in [28] Another strategy to solve the GPEVP involves a direct minimization of the energy functional, cf [14,18], which means that (1) is written as a nonlinear saddle point problem which is solved by a Newtontype method. The resulting iteration scheme is based on a variational formulation which is mesh-independent (cf [2] for a similar approach in electromagnetism) This allows the application of any spatial discretization, including finite elements and spectral methods. As usual for nonlinear problems, the quantitative convergence results are of local nature in the sense that they require a sufficiently accurate initial approximation In practical computations this may be a rather challenging task.
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