6,042 research outputs found
A conjugate gradient like method for p-norm minimization in functional spaces.
We develop an iterative algorithm to recover the minimum p-norm solution of the functional linear equation Ax=b, where A:Xâ¶Y is a continuous linear operator between the two Banach spaces X=Lp, 11, with xâX and bâY. The algorithm is conceived within the same framework of the Landweber method for functional linear equations in Banach spaces proposed by Schöpfer et al. (Inverse Probl 22:311â329, 2006). Indeed, the algorithm is based on using, at the n-th iteration, a linear combination of the steepest current âdescent functionalâ AâJ(bâAxn) and the previous descent functional, where J denotes a duality map of the Banach space Y. In this regard, the algorithm can be viewed as a generalization of the classical conjugate gradient method on the normal equations in Hilbert spaces. We demonstrate that the proposed iterative algorithm converges strongly to the minimum p-norm solution of the functional linear equation Ax=b and that it is also a regularization method, by applying the discrepancy principle as stopping rule. According to the geometrical properties of Lp spaces, numerical experiments show that the method is fast, robust in terms of both restoration accuracy and stability, promotes sparsity and reduces the over-smoothness in reconstructing edges and abrupt intensity changes
Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
In this paper we combine concepts from Riemannian Optimization and the theory
of Sobolev gradients to derive a new conjugate gradient method for direct
minimization of the Gross-Pitaevskii energy functional with rotation. The
conservation of the number of particles constrains the minimizers to lie on a
manifold corresponding to the unit norm. The idea developed here is to
transform the original constrained optimization problem to an unconstrained
problem on this (spherical) Riemannian manifold, so that fast minimization
algorithms can be applied as alternatives to more standard constrained
formulations. First, we obtain Sobolev gradients using an equivalent definition
of an inner product which takes into account rotation. Then, the
Riemannian gradient (RG) steepest descent method is derived based on projected
gradients and retraction of an intermediate solution back to the constraint
manifold. Finally, we use the concept of the Riemannian vector transport to
propose a Riemannian conjugate gradient (RCG) method for this problem. It is
derived at the continuous level based on the "optimize-then-discretize"
paradigm instead of the usual "discretize-then-optimize" approach, as this
ensures robustness of the method when adaptive mesh refinement is performed in
computations. We evaluate various design choices inherent in the formulation of
the method and conclude with recommendations concerning selection of the best
options. Numerical tests demonstrate that the proposed RCG method outperforms
the simple gradient descent (RG) method in terms of rate of convergence. While
on simple problems a Newton-type method implemented in the {\tt Ipopt} library
exhibits a faster convergence than the (RCG) approach, the two methods perform
similarly on more complex problems requiring the use of mesh adaptation. At the
same time the (RCG) approach has far fewer tunable parameters.Comment: 28 pages, 13 figure
A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation
In this paper we improve traditional steepest descent methods for the direct
minimization of the Gross-Pitaevskii (GP) energy with rotation at two levels.
We first define a new inner product to equip the Sobolev space and derive
the corresponding gradient. Secondly, for the treatment of the mass
conservation constraint, we use a projection method that avoids more
complicated approaches based on modified energy functionals or traditional
normalization methods. The descent method with these two new ingredients is
studied theoretically in a Hilbert space setting and we give a proof of the
global existence and convergence in the asymptotic limit to a minimizer of the
GP energy. The new method is implemented in both finite difference and finite
element two-dimensional settings and used to compute various complex
configurations with vortices of rotating Bose-Einstein condensates. The new
Sobolev gradient method shows better numerical performances compared to
classical or gradient methods, especially when high rotation rates
are considered.Comment: to appear in SIAM J Sci Computin
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