1,177 research outputs found
A Parallel-in-Time Preconditioner for the Schur Complement of Parabolic Optimal Control Problems
For optimal control problems constrained by a initial-valued parabolic PDE,
we have to solve a large scale saddle point algebraic system consisting of
considering the discrete space and time points all together. A popular strategy
to handle such a system is the Krylov subspace method, for which an efficient
preconditioner plays a crucial role. The matching-Schur-complement
preconditioner has been extensively studied in literature and the
implementation of this preconditioner lies in solving the underlying PDEs
twice, sequentially in time. In this paper, we propose a new preconditioner for
the Schur complement, which can be used parallel-in-time (PinT) via the so
called diagonalization technique. We show that the eigenvalues of the
preconditioned matrix are low and upper bounded by positive constants
independent of matrix size and the regularization parameter. The uniform
boundedness of the eigenvalues leads to an optimal linear convergence rate of
conjugate gradient solver for the preconditioned Schur complement system. To
the best of our knowledge, it is the first time to have an optimal convergence
analysis for a PinT preconditioning technique of the optimal control problem.
Numerical results are reported to show that the performance of the proposed
preconditioner is robust with respect to the discretization step-sizes and the
regularization parameter
Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods
Use of the stochastic Galerkin finite element methods leads to large systems
of linear equations obtained by the discretization of tensor product solution
spaces along their spatial and stochastic dimensions. These systems are
typically solved iteratively by a Krylov subspace method. We propose a
preconditioner which takes an advantage of the recursive hierarchy in the
structure of the global matrices. In particular, the matrices posses a
recursive hierarchical two-by-two structure, with one of the submatrices block
diagonal. Each one of the diagonal blocks in this submatrix is closely related
to the deterministic mean-value problem, and the action of its inverse is in
the implementation approximated by inner loops of Krylov iterations. Thus our
hierarchical Schur complement preconditioner combines, on each level in the
approximation of the hierarchical structure of the global matrix, the idea of
Schur complement with loops for a number of mutually independent inner Krylov
iterations, and several matrix-vector multiplications for the off-diagonal
blocks. Neither the global matrix, nor the matrix of the preconditioner need to
be formed explicitly. The ingredients include only the number of stiffness
matrices from the truncated Karhunen-Lo\`{e}ve expansion and a good
preconditioned for the mean-value deterministic problem. We provide a condition
number bound for a model elliptic problem and the performance of the method is
illustrated by numerical experiments.Comment: 15 pages, 2 figures, 9 tables, (updated numerical experiments
A constrained pressure-temperature residual (CPTR) method for non-isothermal multiphase flow in porous media
For both isothermal and thermal petroleum reservoir simulation, the
Constrained Pressure Residual (CPR) method is the industry-standard
preconditioner. This method is a two-stage process involving the solution of a
restricted pressure system. While initially designed for the isothermal case,
CPR is also the standard for thermal cases. However, its treatment of the
energy conservation equation does not incorporate heat diffusion, which is
often dominant in thermal cases. In this paper, we present an extension of CPR:
the Constrained Pressure-Temperature Residual (CPTR) method, where a restricted
pressure-temperature system is solved in the first stage. In previous work, we
introduced a block preconditioner with an efficient Schur complement
approximation for a pressure-temperature system. Here, we extend this method
for multiphase flow as the first stage of CPTR. The algorithmic performance of
different two-stage preconditioners is evaluated for reservoir simulation test
cases.Comment: 28 pages, 2 figures. Sources/sinks description in arXiv:1902.0009
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