Modified memoryless spectral-scaling Broyden family on Riemannian manifolds

This paper presents modified memoryless quasi-Newton methods based on the spectral-scaling Broyden family on Riemannian manifolds. The method involves adding one parameter to the search direction of the memoryless self-scaling Broyden family on the manifold. Moreover, it uses a general map instead of vector transport. This idea has already been proposed within a general framework of Riemannian conjugate gradient methods where one can use vector transport, scaled vector transport, or an inverse retraction. We show that the search direction satisfies the sufficient descent condition under some assumptions on the parameters. In addition, we show global convergence of the proposed method under the Wolfe conditions. We numerically compare it with existing methods, including Riemannian conjugate gradient methods and the memoryless spectral-scaling Broyden family. The numerical results indicate that the proposed method with the BFGS formula is suitable for solving an off-diagonal cost function minimization problem on an oblique manifold.Comment: 20 pages, 8 figure

The RHMC algorithm for theories with unknown spectral bounds

The Rational Hybrid Monte Carlo (RHMC) algorithm extends the Hybrid Monte Carlo algorithm for lattice QCD simulations to situations involving fractional powers of the determinant of the quadratic Dirac operator. This avoids the updating increment ($dt$) dependence of observables which plagues the Hybrid Molecular-dynamics (HMD) method. The RHMC algorithm uses rational approximations to fractional powers of the quadratic Dirac operator. Such approximations are only available when positive upper and lower bounds to the operator's spectrum are known. We apply the RHMC algorithm to simulations of 2 theories for which a positive lower spectral bound is unknown: lattice QCD with staggered quarks at finite isospin chemical potential and lattice QCD with massless staggered quarks and chiral 4-fermion interactions ($\chi$QCD). A choice of lower bound is made in each case, and the properties of the RHMC simulations these define are studied. Justification of our choices of lower bounds is made by comparing measurements with those from HMD simulations, and by comparing different choices of lower bounds.Comment: Latex(Revtex 4) 25 pages, 8 postscript figure

Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations

A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup discretization analysis; and application of the method to high-order variational discretization

Parallel eigensolvers in plane-wave Density Functional Theory

We consider the problem of parallelizing electronic structure computations in plane-wave Density Functional Theory. Because of the limited scalability of Fourier transforms, parallelism has to be found at the eigensolver level. We show how a recently proposed algorithm based on Chebyshev polynomials can scale into the tens of thousands of processors, outperforming block conjugate gradient algorithms for large computations

Exploiting spatial sparsity for multi-wavelength imaging in optical interferometry

Optical interferometers provide multiple wavelength measurements. In order to fully exploit the spectral and spatial resolution of these instruments, new algorithms for image reconstruction have to be developed. Early attempts to deal with multi-chromatic interferometric data have consisted in recovering a gray image of the object or independent monochromatic images in some spectral bandwidths. The main challenge is now to recover the full 3-D (spatio-spectral) brightness distribution of the astronomical target given all the available data. We describe a new approach to implement multi-wavelength image reconstruction in the case where the observed scene is a collection of point-like sources. We show the gain in image quality (both spatially and spectrally) achieved by globally taking into account all the data instead of dealing with independent spectral slices. This is achieved thanks to a regularization which favors spatial sparsity and spectral grouping of the sources. Since the objective function is not differentiable, we had to develop a specialized optimization algorithm which also accounts for non-negativity of the brightness distribution.Comment: This version has been accepted for publication in J. Opt. Soc. Am.

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 $L^2$ 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 $H^1$ 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