491 research outputs found
Space-Time Discretizations Using Constrained First-Order System Least Squares (CFOSLS)
This paper studies finite element discretizations for three types of time-dependent PDEs, namely heat equation, scalar conservation law and wave equation, which we reformulate as first order systems in a least-squares setting subject to a space-time conservation constraint (coming from the original PDE). Available piece- wise polynomial finite element spaces in (n + 1)-dimensions for functional spaces from the (n + 1)-dimensional de Rham sequence for n = 3, 4 are used for the implementation of the method. Computational results illustrating the error behavior, iteration counts and performance of block-diagonal and monolithic geometric multi- grid preconditioners are presented for the discrete CFOSLS system. The results are obtained from a parallel implementation of the methods for which we report reasonable scalability
Algorithms and data structures for adaptive multigrid elliptic solvers
Adaptive refinement and the complicated data structures required to support it are discussed. These data structures must be carefully tuned, especially in three dimensions where the time and storage requirements of algorithms are crucial. Another major issue is grid generation. The options available seem to be curvilinear fitted grids, constructed on iterative graphics systems, and unfitted Cartesian grids, which can be constructed automatically. On several grounds, including storage requirements, the second option seems preferrable for the well behaved scalar elliptic problems considered here. A variety of techniques for treatment of boundary conditions on such grids are reviewed. A new approach, which may overcome some of the difficulties encountered with previous approaches, is also presented
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
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
Learning Preconditioner for Conjugate Gradient PDE Solvers
Efficient numerical solvers for partial differential equations empower
science and engineering. One of the commonly employed numerical solvers is the
preconditioned conjugate gradient (PCG) algorithm which can solve large systems
to a given precision level. One challenge in PCG solvers is the selection of
preconditioners, as different problem-dependent systems can benefit from
different preconditioners. We present a new method to introduce \emph{inductive
bias} in preconditioning conjugate gradient algorithm. Given a system matrix
and a set of solution vectors arise from an underlying distribution, we train a
graph neural network to obtain an approximate decomposition to the system
matrix to be used as a preconditioner in the context of PCG solvers. We conduct
extensive experiments to demonstrate the efficacy and generalizability of our
proposed approach in solving various 2D and 3D linear second-order PDEs
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