534 research outputs found
A class of nonsymmetric preconditioners for saddle point problems
For iterative solution of saddle point problems, a nonsymmetric preconditioning is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where the SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate efficiency of the new preconditioner, especially when the left-upper block is far from symmetric
Preconditioning and convergence in the right norm
The convergence of numerical approximations to the solutions of differential equations is a key aspect of Numerical Analysis and Scientific Computing. Iterative solution methods for the systems of linear(ised) equations which often result are also underpinned by analyses of convergence. In the function space setting, it is widely appreciated that there are appropriate ways in which to assess convergence and it is well-known that different norms are not equivalent. In the finite dimensional linear algebra setting, however, all norms are equivalent and little attention is often payed to the norms used. In this paper, we highlight this consideration in the context of preconditioning for minimum residual methods (MINRES and GMRES/GCR/ORTHOMIN) and argue that even in the linear algebra setting there is a ‘right’ norm in which to consider convergence: stopping an iteration which is rapidly converging in an irrelevant or highly scaled norm at some tolerance level may still give a poor answer
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
Time-parallel iterative solvers for parabolic evolution equations
We present original time-parallel algorithms for the solution of the implicit
Euler discretization of general linear parabolic evolution equations with
time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory
of parabolic problems, we show that the standard nonsymmetric time-global
system can be equivalently reformulated as an original symmetric saddle-point
system that remains inf-sup stable with respect to the same natural parabolic
norms. We then propose and analyse an efficient and readily implementable
parallel-in-time preconditioner to be used with an inexact Uzawa method. The
proposed preconditioner is non-intrusive and easy to implement in practice, and
also features the key theoretical advantages of robust spectral bounds, leading
to convergence rates that are independent of the number of time-steps, final
time, or spatial mesh sizes, and also a theoretical parallel complexity that
grows only logarithmically with respect to the number of time-steps. Numerical
experiments with large-scale parallel computations show the effectiveness of
the method, along with its good weak and strong scaling properties
Absolute value preconditioning for symmetric indefinite linear systems
We introduce a novel strategy for constructing symmetric positive definite
(SPD) preconditioners for linear systems with symmetric indefinite matrices.
The strategy, called absolute value preconditioning, is motivated by the
observation that the preconditioned minimal residual method with the inverse of
the absolute value of the matrix as a preconditioner converges to the exact
solution of the system in at most two steps. Neither the exact absolute value
of the matrix nor its exact inverse are computationally feasible to construct
in general. However, we provide a practical example of an SPD preconditioner
that is based on the suggested approach. In this example we consider a model
problem with a shifted discrete negative Laplacian, and suggest a geometric
multigrid (MG) preconditioner, where the inverse of the matrix absolute value
appears only on the coarse grid, while operations on finer grids are based on
the Laplacian. Our numerical tests demonstrate practical effectiveness of the
new MG preconditioner, which leads to a robust iterative scheme with minimalist
memory requirements
A Bramble-Pasciak conjugate gradient method for discrete Stokes equations with random viscosity
We study the iterative solution of linear systems of equations arising from
stochastic Galerkin finite element discretizations of saddle point problems. We
focus on the Stokes model with random data parametrized by uniformly
distributed random variables and discuss well-posedness of the variational
formulations. We introduce a Bramble-Pasciak conjugate gradient method as a
linear solver. It builds on a non-standard inner product associated with a
block triangular preconditioner. The block triangular structure enables more
sophisticated preconditioners than the block diagonal structure usually applied
in MINRES methods. We show how the existence requirements of a conjugate
gradient method can be met in our setting. We analyze the performance of the
solvers depending on relevant physical and numerical parameters by means of
eigenvalue estimates. For this purpose, we derive bounds for the eigenvalues of
the relevant preconditioned sub-matrices. We illustrate our findings using the
flow in a driven cavity as a numerical test case, where the viscosity is given
by a truncated Karhunen-Lo\`eve expansion of a random field. In this example, a
Bramble-Pasciak conjugate gradient method with block triangular preconditioner
outperforms a MINRES method with block diagonal preconditioner in terms of
iteration numbers.Comment: 19 pages, 1 figure, submitted to SIAM JU
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