66 research outputs found
Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization
Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations
Enhancing structure relaxations for first-principles codes: an approximate Hessian approach
We present a method for improving the speed of geometry relaxation by using a
harmonic approximation for the interaction potential between nearest neighbor
atoms to construct an initial Hessian estimate. The model is quite robust, and
yields approximately a 30% or better reduction in the number of calculations
compared to an optimized diagonal initialization. Convergence with this
initializer approaches the speed of a converged BFGS Hessian, therefore it is
close to the best that can be achieved. Hessian preconditioning is discussed,
and it is found that a compromise between an average condition number and a
narrow distribution in eigenvalues produces the best optimization.Comment: 9 pages, 3 figures, added references, expanded optimization sectio
Prospective, randomized, doubleâblind assessment of topical bakuchiol and retinol for facial photoageing
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/147746/1/bjd16918_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/147746/2/bjd16918.pd
On Block Triangular Preconditioners for the Interior Point Solution of PDE-Constrained Optimization Problems
We consider the numerical solution of saddle point systems of equations resulting from the discretization of PDE-constrained optimization problems, with additional bound constraints on the state and control variables, using an interior point method. In particular, we derive a Bramble-Pasciak Conjugate Gradient method and a tailored block triangular preconditioner which may be applied within it. Crucial to the usage of the preconditioner are carefully chosen approximations of the (1,1)-block and Schur complement of the saddle point system. To apply the inverse of the Schur complement approximation, which is computationally the most expensive part of the preconditioner, one may then utilize methods such as multigrid or domain decomposition to handle individual sub-blocks of the matrix system
On Convergence of the Inexact Rayleigh Quotient Iteration with the Lanczos Method Used for Solving Linear Systems
For the Hermitian inexact Rayleigh quotient iteration (RQI), the author has
established new local general convergence results, independent of iterative
solvers for inner linear systems. The theory shows that the method locally
converges quadratically under a new condition, called the uniform positiveness
condition. In this paper we first consider the local convergence of the inexact
RQI with the unpreconditioned Lanczos method for the linear systems. Some
attractive properties are derived for the residuals, whose norms are
's, of the linear systems obtained by the Lanczos method. Based on
them and the new general convergence results, we make a refined analysis and
establish new local convergence results. It is proved that the inexact RQI with
Lanczos converges quadratically provided that with a
constant . The method is guaranteed to converge linearly provided
that is bounded by a small multiple of the reciprocal of the
residual norm of the current approximate eigenpair. The results are
fundamentally different from the existing convergence results that always
require , and they have a strong impact on effective
implementations of the method. We extend the new theory to the inexact RQI with
a tuned preconditioned Lanczos for the linear systems. Based on the new theory,
we can design practical criteria to control to achieve quadratic
convergence and implement the method more effectively than ever before.
Numerical experiments confirm our theory.Comment: 20 pages, 8 figures. arXiv admin note: text overlap with
arXiv:0906.223
ParIC : A Family of Parallel Incomplete Cholesky Preconditioners
A class of parallel incomplete factorization preconditionings
for the solution of large linear systems is investigated. The approach may
be regarded as a generalized domain decomposition method. Adjacent
subdomains have to communicate during the setting up of the preconÂ
ditioner, and during the application of the preconditioner. Overlap is
not necessary to achieve high performance. FillÂin levels are considered
in a global way. If necessary, the technique may be implemented as a
global reÂordering of the unknowns. Experimental results are reported
for twoÂdimensional problems
On Inner Iterations in the Shift-Invert Residual Arnoldi Method and the Jacobi--Davidson Method
Using a new analysis approach, we establish a general convergence theory of
the Shift-Invert Residual Arnoldi (SIRA) method for computing a simple
eigenvalue nearest to a given target and the associated eigenvector.
In SIRA, a subspace expansion vector at each step is obtained by solving a
certain inner linear system. We prove that the inexact SIRA method mimics the
exact SIRA well, that is, the former uses almost the same outer iterations to
achieve the convergence as the latter does if all the inner linear systems are
iteratively solved with {\em low} or {\em modest} accuracy during outer
iterations. Based on the theory, we design practical stopping criteria for
inner solves. Our analysis is on one step expansion of subspace and the
approach applies to the Jacobi--Davidson (JD) method with the fixed target
as well, and a similar general convergence theory is obtained for it.
Numerical experiments confirm our theory and demonstrate that the inexact SIRA
and JD are similarly effective and are considerably superior to the inexact
SIA.Comment: 20 pages, 8 figure
Eigenvalue bounds of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices
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