465 research outputs found
Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?
A new, coercive formulation of the Helmholtz equation was introduced in
[Moiola, Spence, SIAM Rev. 2014]. In this paper we investigate -version
Galerkin discretisations of this formulation, and the iterative solution of the
resulting linear systems. We find that the coercive formulation behaves
similarly to the standard formulation in terms of the pollution effect (i.e. to
maintain accuracy as , must decrease with at the same rate
as for the standard formulation). We prove -explicit bounds on the number of
GMRES iterations required to solve the linear system of the new formulation
when it is preconditioned with a prescribed symmetric positive-definite matrix.
Even though the number of iterations grows with , these are the first such
rigorous bounds on the number of GMRES iterations for a preconditioned
formulation of the Helmholtz equation, where the preconditioner is a symmetric
positive-definite matrix.Comment: 27 pages, 7 figure
PDE-betinga optimering : prekondisjonerarar og metodar for diffuse domene
This thesis is mainly concerned with the efficient numerical solution of optimization problems subject to linear PDE-constraints, with particular focus on robust preconditioners and diffuse domain methods. Associated with such constrained optimization problems are the famous first-order KarushKuhn-Tucker (KKT) conditions. For certain minimization problems, the functions satisfying the KKT conditions are also optimal solutions of the original optimization problem, implying that we can solve the KKT system to obtain the optimum; the so-called âall-at-onceâ approach. We propose and analyze preconditioners for the different KKT systems we derive in this thesis.Denne avhandlinga ser i hovudsak pĂĽ effektive numeriske løysingar av PDE-betinga optimeringsproblem, med eit sĂŚrskilt fokus pĂĽ robuste prekondisjonerar og âdiffuse domainâ-metodar. Assosiert med slike optimeringsproblem er dei velkjende Karush-Kuhn-Tucker (KKT)-føresetnadane. For mange betinga optimeringsproblem, vil funksjonar som tilfredstillar KKT-vilkĂĽra samstundes vere ei optimal løysing pĂĽ det opprinnelege optimeringsproblemet. Dette impliserar at vi kan løyse KKT-likningane for ĂĽ finne optimum. Vi konstruerar og analyserar prekondisjonerar for dei forskjellige KKT-systema vi utleiar i denne avhandlinga
Block preconditioners for mixed-dimensional discretization of flow in fractured porous media
In this paper, we are interested in an efficient numerical method for the mixed-dimensional approach to modeling single-phase flow in fractured porous media. The model introduces fractures and their intersections as lower-dimensional structures, and the mortar variable is used for flow coupling between the matrix and fractures. We consider a stable mixed finite element discretization of the problem, which results in a parameter-dependent linear system. For this, we develop block preconditioners based on the well-posedness of the discretization choice. The preconditioned iterative method demonstrates robustness with regard to discretization and physical parameters. The analytical results are verified on several examples of fracture network configurations, and notable results in reduction of number of iterations and computational time are obtained.publishedVersio
CAUCHY-LIKE PRECONDITIONERS FOR 2-DIMENSIONAL ILL-POSED PROBLEMS
Ill-conditioned matrices with block Toeplitz, Toeplitz block (BTTB)
structure arise from the discretization of certain ill-posed problems in
signal and image processing. We use a preconditioned conjugate gradient
algorithm to compute a regularized solution to this linear system given
noisy data. Our preconditioner is a Cauchy-like block diagonal
approximation to an orthogonal transformation of the BTTB matrix.
We show the preconditioner has desirable properties when the kernel of the
ill-posed problem is smooth: the largest singular values of the
preconditioned matrix are clustered around one, the smallest singular
values remain small, and the subspaces corresponding to the largest and
smallest singular values, respectively, remain unmixed. For a system
involving variables, the preconditioned algorithm costs only
operations per iteration. We demonstrate the
effectiveness of the preconditioner on three examples
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