25,821 research outputs found
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
All-at-Once Solution if Time-Dependent PDE-Constrained Optimisation Problems
Time-dependent partial differential equations (PDEs) play an important role in applied mathematics and many other areas of science. One-shot methods try to compute the solution to these problems in a single iteration that solves for all time-steps at the same time. In this paper, we look at one-shot approaches for the optimal control of time-dependent PDEs and focus on the fast solution of these problems. The use of Krylov subspace solvers together with an efficient preconditioner allows for minimal storage requirements. We solve only approximate time-evolutions for both forward and adjoint problem and compute accurate solutions of a given control problem only at convergence of the overall Krylov subspace iteration. We show that our approach can give competitive results for a variety of problem formulations
Refined saddle-point preconditioners for discretized Stokes problems
This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying the preconditioned MINRES method. The codes used in the numerical studies are available online
All-at-once solution of time-dependent PDE-constrained optimization problems
Time-dependent partial differential equations (PDEs) play an important role in applied mathematics and many other areas of science. One-shot methods try to compute the solution to these problems in a single iteration that solves for all time-steps at the same time. In this paper, we look at one-shot approaches for the optimal control of time-dependent PDEs and focus on the fast solution of these problems. The use of Krylov subspace solvers together with an efficient preconditioner allows for minimal storage requirements. We solve only approximate time-evolutions for both forward and adjoint problem and compute accurate solutions of a given control problem only at convergence of the overall Krylov subspace iteration. We show that our approach can give competitive results for a variety of problem formulations
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