35 research outputs found
A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part I
We propose a class of preconditioners, which are also tailored for symmetric linear systems from linear algebra and nonconvex optimization. Our preconditioners are specifically suited for large linear systems and may be obtained as by-product of Krylov subspace solvers. Each preconditioner in our class is identified by setting the values of a pair of parameters and a scaling matrix, which are user-dependent, and may be chosen according with the structure of the problem in hand. We provide theoretical properties for our preconditioners. In particular, we show that our preconditioners both shift some eigenvalues of the system matrix to controlled values, and they tend to reduce the modulus of most of the other eigenvalues. In a companion paper we study some structural properties of our class of preconditioners, and report the results on a significant numerical experience.preconditioners; large indefinite linear systems; large scale nonconvex optimization; Krylov subspace methods
A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part II
In this paper we consider the parameter dependent class of preconditioners M(a,d,D) defined in the companion paper The latter was constructed by using information from a Krylov subspace method, adopted to solve the large symmetric linear system Ax = b. We first estimate the condition number of the preconditioned matrix M(a,d,D). Then our preconditioners, which are independent of the choice of the Krylov subspace method adopted, proved to be effective also when solving sequences of slowly changing linear systems, in unconstrained optimization and linear algebra frameworks. A numerical experience is provided to give evidence of the performance of M(a,d,D).preconditioners; large indefinite linear systems; large scale nonconvex optimization; Krylov subspace methods
The LBFGS Quasi-Newtonian Method for Molecular Modeling Prion AGAAAAGA Amyloid Fibrils
Experimental X-ray crystallography, NMR (Nuclear Magnetic Resonance)
spectroscopy, dual polarization interferometry, etc are indeed very powerful
tools to determine the 3-Dimensional structure of a protein (including the
membrane protein); theoretical mathematical and physical computational
approaches can also allow us to obtain a description of the protein 3D
structure at a submicroscopic level for some unstable, noncrystalline and
insoluble proteins. X-ray crystallography finds the X-ray final structure of a
protein, which usually need refinements using theoretical protocols in order to
produce a better structure. This means theoretical methods are also important
in determinations of protein structures. Optimization is always needed in the
computer-aided drug design, structure-based drug design, molecular dynamics,
and quantum and molecular mechanics. This paper introduces some optimization
algorithms used in these research fields and presents a new theoretical
computational method - an improved LBFGS Quasi-Newtonian mathematical
optimization method - to produce 3D structures of Prion AGAAAAGA amyloid
fibrils (which are unstable, noncrystalline and insoluble), from the potential
energy minimization point of view. Because the NMR or X-ray structure of the
hydrophobic region AGAAAAGA of prion proteins has not yet been determined, the
model constructed by this paper can be used as a reference for experimental
studies on this region, and may be useful in furthering the goals of medicinal
chemistry in this field
Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections
This work focuses on the iterative solution of sequences of KKT linear
systems arising in interior point methods applied to large convex quadratic
programming problems. This task is the computational core of the interior point
procedure and an efficient preconditioning strategy is crucial for the
efficiency of the overall method. Constraint preconditioners are very effective
in this context; nevertheless, their computation may be very expensive for
large-scale problems, and resorting to approximations of them may be
convenient. Here we propose a procedure for building inexact constraint
preconditioners by updating a "seed" constraint preconditioner computed for a
KKT matrix at a previous interior point iteration. These updates are obtained
through low-rank corrections of the Schur complement of the (1,1) block of the
seed preconditioner. The updated preconditioners are analyzed both
theoretically and computationally. The results obtained show that our updating
procedure, coupled with an adaptive strategy for determining whether to
reinitialize or update the preconditioner, can enhance the performance of
interior point methods on large problems.Comment: 22 page
Computational Potential Energy Minimization Studies on the Prion AGAAAAGA Amyloid Fibril Molecular Structures
X-ray crystallography, NMR (Nuclear Magnetic Resonance) spectroscopy, and
dual polarization interferometry, etc are indeed very powerful tools to
determine the 3D structures of proteins (including the membrane proteins),
though they are time-consuming and costly. However, for some proteins, due to
their unstable, noncrystalline and insoluble nature, these tools cannot work.
Under this condition, mathematical and physical theoretical methods and
computational approaches allow us to obtain a description of the protein 3D
structure at a submicroscopic level. This Chapter presents some practical and
useful mathematical optimization computational approaches to produce 3D
structures of the Prion AGAAAAGA Amyloid Fibrils, from a potential energy
minimization point of view.
X-ray crystallography finds the X-ray final structure of a protein, which
usually need refinements in order to produce a better structure. The
computational methods presented in this Chapter can be also acted as a tool for
the refinements.Comment: published in [Recent Advances in Crystallography, ISBN:
978-953-51-0754-5, Editor Jason B. Bendict, InTech Open Access Publisher, 19
Sept 2012, hardcover] Chapter 12, DOI: 10.5772/47733, pp.297-312:
http://www.intechopen.com/books/recent-advances-in-crystallography/computational-potential-energy-minimization-studies-on-the-prion-agaaaaga-amyloid-fibril-molecular-
A class of preconditioners for large indefinite linear systems, as by-product of Krylov subspace methods: Part I
We propose a class of preconditioners, which are also tailored for symmetric linear systems from linear algebra and nonconvex optimization. Our preconditioners are specifically suited for large linear systems and may be obtained as by-product of Krylov subspace solvers. Each preconditioner in our class is identified by setting the values of a pair of parameters and a scaling matrix, which are user-dependent, and may be chosen according with the structure of the problem in hand. We provide theoretical properties for our preconditioners. In particular, we show that our preconditioners both shift some eigenvalues of the system matrix to controlled values, and they tend to reduce the modulus of most of the other eigenvalues. In a companion paper we study some structural properties of our class of preconditioners, and report the results on a significant numerical experience