503 research outputs found
A Combined Preconditioning Strategy for Nonsymmetric Systems
We present and analyze a class of nonsymmetric preconditioners within a
normal (weighted least-squares) matrix form for use in GMRES to solve
nonsymmetric matrix problems that typically arise in finite element
discretizations. An example of the additive Schwarz method applied to
nonsymmetric but definite matrices is presented for which the abstract
assumptions are verified. A variable preconditioner, combining the original
nonsymmetric one and a weighted least-squares version of it, is shown to be
convergent and provides a viable strategy for using nonsymmetric
preconditioners in practice. Numerical results are included to assess the
theory and the performance of the proposed preconditioners.Comment: 26 pages, 3 figure
On the Convergence of the Laplace Approximation and Noise-Level-Robustness of Laplace-based Monte Carlo Methods for Bayesian Inverse Problems
The Bayesian approach to inverse problems provides a rigorous framework for
the incorporation and quantification of uncertainties in measurements,
parameters and models. We are interested in designing numerical methods which
are robust w.r.t. the size of the observational noise, i.e., methods which
behave well in case of concentrated posterior measures. The concentration of
the posterior is a highly desirable situation in practice, since it relates to
informative or large data. However, it can pose a computational challenge for
numerical methods based on the prior or reference measure. We propose to employ
the Laplace approximation of the posterior as the base measure for numerical
integration in this context. The Laplace approximation is a Gaussian measure
centered at the maximum a-posteriori estimate and with covariance matrix
depending on the logposterior density. We discuss convergence results of the
Laplace approximation in terms of the Hellinger distance and analyze the
efficiency of Monte Carlo methods based on it. In particular, we show that
Laplace-based importance sampling and Laplace-based quasi-Monte-Carlo methods
are robust w.r.t. the concentration of the posterior for large classes of
posterior distributions and integrands whereas prior-based importance sampling
and plain quasi-Monte Carlo are not. Numerical experiments are presented to
illustrate the theoretical findings.Comment: 50 pages, 11 figure
Natural preconditioning and iterative methods for saddle point systems
The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or the discrete setting, so saddle point systems arising from the discretization of partial differential equation problems, such as those describing electromagnetic problems or incompressible flow, lead to equations with this structure, as do, for example, interior point methods and the sequential quadratic programming approach to nonlinear optimization. This survey concerns iterative solution methods for these problems and, in particular, shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness---in terms of rapidity of convergence---is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends
Preconditioned NonSymmetric/Symmetric Discontinuous Galerkin Method for Elliptic Problem with Reconstructed Discontinuous Approximation
In this paper, we propose and analyze an efficient preconditioning method for
the elliptic problem based on the reconstructed discontinuous approximation
method. We reconstruct a high-order piecewise polynomial space that arbitrary
order can be achieved with one degree of freedom per element. This space can be
directly used with the symmetric/nonsymmetric interior penalty discontinuous
Galerkin method. Compared with the standard DG method, we can enjoy the
advantage on the efficiency of the approximation. Besides, we establish an norm
equivalence result between the reconstructed high-order space and the piecewise
constant space. This property further allows us to construct an optimal
preconditioner from the piecewise constant space. The upper bound of the
condition number to the preconditioned symmetric/nonsymmetric system is shown
to be independent of the mesh size. Numerical experiments are provided to
demonstrate the validity of the theory and the efficiency of the proposed
method
An asymptotically superlinearly convergent semismooth Newton augmented Lagrangian method for Linear Programming
Powerful interior-point methods (IPM) based commercial solvers, such as
Gurobi and Mosek, have been hugely successful in solving large-scale linear
programming (LP) problems. The high efficiency of these solvers depends
critically on the sparsity of the problem data and advanced matrix
factorization techniques. For a large scale LP problem with data matrix
that is dense (possibly structured) or whose corresponding normal matrix
has a dense Cholesky factor (even with re-ordering), these solvers may require
excessive computational cost and/or extremely heavy memory usage in each
interior-point iteration. Unfortunately, the natural remedy, i.e., the use of
iterative methods based IPM solvers, although can avoid the explicit
computation of the coefficient matrix and its factorization, is not practically
viable due to the inherent extreme ill-conditioning of the large scale normal
equation arising in each interior-point iteration. To provide a better
alternative choice for solving large scale LPs with dense data or requiring
expensive factorization of its normal equation, we propose a semismooth Newton
based inexact proximal augmented Lagrangian ({\sc Snipal}) method. Different
from classical IPMs, in each iteration of {\sc Snipal}, iterative methods can
efficiently be used to solve simpler yet better conditioned semismooth Newton
linear systems. Moreover, {\sc Snipal} not only enjoys a fast asymptotic
superlinear convergence but is also proven to enjoy a finite termination
property. Numerical comparisons with Gurobi have demonstrated encouraging
potential of {\sc Snipal} for handling large-scale LP problems where the
constraint matrix has a dense representation or has a dense
factorization even with an appropriate re-ordering.Comment: Due to the limitation "The abstract field cannot be longer than 1,920
characters", the abstract appearing here is slightly shorter than that in the
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