We study the problem of complex-valued independent subspace analysis (ISA). We introduce complex flag manifolds to tackle this problem, and, based on Riemannian geometry, propose the natural conjugate gradient method on this class of manifolds. Numerical experiments demonstrate that the natural conjugate gradient method yields better convergence compared to the natural gradient geodesic search ...

We still need to show that the directions p0, p1, . . . , pn−1 generated by Algorithms 1 and (2) are conjugate wrt A. If so, then by Theorem 11.3, this algorithm will terminate in n steps. The next theorem shows this property, along with two other important properties: (i) the residuals ri are mutually orthogonal, and (ii) each pk and rk is contained in the Krylov subspace of degree k for r0, d...

In this paper we present a fast iterative image superresolution algorithm using preconditioned conjugate gradient method. To avoid explicitly computing the tolerance in the inverse filter based preconditioner scheme, a new Wiener filter based preconditioner for the conjugate gradient method is proposed to speed up the convergence. The circulant-block structure of the preconditioner allows effic...

In four-dimensional variational data assimilation (4D-Var) an optimal estimate of the initial state of a dynamical system is obtained by solving a large-scale unconstrained minimization problem. The gradient of the cost functional may be efficiently computed using the adjoint modeling, at the expense equivalent to a few forward model integrations; for most practical applications, the evaluation...

Conjugate gradient methods are probably the most famous iterative methods for solving large scale optimization problems in scientific and engineering computation, characterized by the simplicity of their iteration and their low memory requirements. In this paper, we propose a new conjugate gradient method which is based on the MBFGS secant condition by modifying Perry’s method. Our proposed met...

On modern large-scale parallel computers, the performance of Krylov subspace iterative methods is limited by global synchronization. This has inspired the development of s-step (or communication-avoiding) Krylov subspace method variants, in which iterations are computed in blocks of s. This reformulation can reduce the number of global synchronizations per iteration by a factor of O(s), and has...

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the ...

Image reconstruction gives rise to some challenging large-scale constrained optimization problems. We consider a convex minimization problem with nonnegativity constraints that arises in astronomical imaging. To solve this problem, we use an efficient hybrid gradient projection-reduced Newton (active-set) method. By "reduced Newton," we mean that we take Newton steps only in the inactive variab...

In theory, the successive gradients generated by the conjugate gradient method applied to a quadratic should be orthogonal. However, for some ill-conditioned problems, orthogonality is quickly lost due to rounding errors, and convergence is much slower than expected. A limited memory version of the nonlinear conjugate gradient method is developed. The memory is used to both detect the loss of o...

We have implemented a software package for constructing and solving the sparse coefficient matrix linear systems arising from solving partial differential equations based on unstructured grids. The sparse symmetric complex linear system can be solved by either a precondi-tioned bi-conjugate gradient solver, or by a two-stage Cholesky factorization solver, or by a hybrid solver combining both. T...