18,266 research outputs found
Preconditioning complex symmetric linear systems
A new polynomial preconditioner for symmetric complex linear systems based on
Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear
systems is herein presented. It applies to Conjugate Orthogonal Conjugate
Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative
solvers and does not require any estimation of the spectrum of the coefficient
matrix. An upper bound of the condition number of the preconditioned linear
system is provided. Moreover, to reduce the computational cost, an inexact
variant based on incomplete Cholesky decomposition or orthogonal polynomials is
proposed. Numerical results show that the present preconditioner and its
inexact variant are efficient and robust solvers for this class of linear
systems. A stability analysis of the method completes the description of the
preconditioner.Comment: 26 pages, 4 figures, 4 table
The solution of linear systems of equations with a structural analysis code on the NAS CRAY-2
Two methods for solving linear systems of equations on the NAS Cray-2 are described. One is a direct method; the other is an iterative method. Both methods exploit the architecture of the Cray-2, particularly the vectorization, and are aimed at structural analysis applications. To demonstrate and evaluate the methods, they were installed in a finite element structural analysis code denoted the Computational Structural Mechanics (CSM) Testbed. A description of the techniques used to integrate the two solvers into the Testbed is given. Storage schemes, memory requirements, operation counts, and reformatting procedures are discussed. Finally, results from the new methods are compared with results from the initial Testbed sparse Choleski equation solver for three structural analysis problems. The new direct solvers described achieve the highest computational rates of the methods compared. The new iterative methods are not able to achieve as high computation rates as the vectorized direct solvers but are best for well conditioned problems which require fewer iterations to converge to the solution
Preconditioned iterative methods for implicit equations
This paper was written for The European Centre for Medium Range Weather Forecasting Workshop on Developments in Numerical Methods for Very High Resolution Global Models, June 2000
Second order adjoints for solving PDE-constrained optimization problems
Inverse problems are of utmost importance in many fields of science and engineering. In the
variational approach inverse problems are formulated as PDE-constrained optimization problems,
where the optimal estimate of the uncertain parameters is the minimizer of a certain cost
functional subject to the constraints posed by the model equations. The numerical solution
of such optimization problems requires the computation of derivatives of the model output
with respect to model parameters. The first order derivatives of a cost functional (defined
on the model output) with respect to a large number of model parameters can be calculated
efficiently through first order adjoint sensitivity analysis. Second order adjoint models
give second derivative information in the form of matrix-vector products between the Hessian
of the cost functional and user defined vectors. Traditionally, the construction of second
order derivatives for large scale models has been considered too costly. Consequently, data
assimilation applications employ optimization algorithms that use only first order derivative
information, like nonlinear conjugate gradients and quasi-Newton methods.
In this paper we discuss the mathematical foundations of second order adjoint sensitivity
analysis and show that it provides an efficient approach to obtain Hessian-vector products. We
study the benefits of using of second order information in the numerical optimization process
for data assimilation applications. The numerical studies are performed in a twin experiment
setting with a two-dimensional shallow water model. Different scenarios are considered with
different discretization approaches, observation sets, and noise levels. Optimization algorithms
that employ second order derivatives are tested against widely used methods that require
only first order derivatives. Conclusions are drawn regarding the potential benefits and the
limitations of using high-order information in large scale data assimilation problems
Some Preconditioning Techniques for Saddle Point Problems
Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud
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The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336
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