Parallelization in time by diagonalization
arxiv(2023)
摘要
This is a review of preconditioning techniques based on fast-diagonalization
methods for space-time isogeometric discretization of the heat equation. Three
formulation are considered: the Galerkin approach, a discrete least-square and
a continuous least square. For each formulation the heat differential operator
is written as a sum of terms that are kronecker products of uni-variate
operators. These are used to speed-up the application of the operator in
iterative solvers and to construct a suitable preconditioner. Contrary to the
fast-diagonalization technique for the Laplace equation where all uni-variate
operators acting on the same direction can be simultaneously diagonalized in
the case of the heat equation this is not possible. Luckily this can be done up
to an additional term that has low rank allowing for the utilization of
arrow-head like factorization or inversion by Sherman-Morrison formula. The
proposed preconditioners work extremely well on the parametric domain and, when
the domain is parametrized or when the equation coefficients are not constant,
they can be adapted and retain good performance characteristics.
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