In mathematics, Hiptmair–Xu (HX) preconditioners[1] are preconditioners for solving
and
problems based on the auxiliary space preconditioning framework.[2] An important ingredient in the derivation of HX preconditioners in two and three dimensions is the so-called regular decomposition, which decomposes a Sobolev space function into a component of higher regularity and a scalar or vector potential. The key to the success of HX preconditioners is the discrete version of this decomposition, which is also known as HX decomposition. The discrete decomposition decomposes a discrete Sobolev space function into a discrete component of higher regularity, a discrete scale or vector potential, and a high-frequency component.
HX preconditioners have been used for accelerating a wide variety of solution techniques, thanks to their highly scalable parallel implementations, and are known as AMS[3] and ADS[4] precondition. HX preconditioner was identified by the U.S. Department of Energy as one of the top ten breakthroughs in computational science[5] in recent years. Researchers from Sandia, Los Alamos, and Lawrence Livermore National Labs use this algorithm for modeling fusion with magnetohydrodynamic equations.[6] Moreover, this approach will also be instrumental in developing optimal iterative methods in structural mechanics, electrodynamics, and modeling of complex flows.
HX preconditioner for ![{\displaystyle H(\operatorname {curl} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9334e4faff3dd2251ff0634a0688f78d1fad506)
[edit]
Consider the following
problem: Find
such that
with
.
The corresponding matrix form is
The HX preconditioner for
problem is defined as
where
is a smoother (e.g., Jacobi smoother, Gauss–Seidel smoother),
is the canonical interpolation operator for
space,
is the matrix representation of discrete vector Laplacian defined on
,
is the discrete gradient operator, and
is the matrix representation of the discrete scalar Laplacian defined on
. Based on auxiliary space preconditioning framework, one can show that
where
denotes the condition number of matrix
.
In practice, inverting
and
might be expensive, especially for large scale problems. Therefore, we can replace their inversion by spectrally equivalent approximations,
and
, respectively. And the HX preconditioner for
becomes
HX Preconditioner for ![{\displaystyle H(\operatorname {div} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da7d26ac45e05287d1e4caa24fcc03028314b030)
[edit]
Consider the following
problem: Find
with
.
The corresponding matrix form is
The HX preconditioner for
problem is defined as
where
is a smoother (e.g., Jacobi smoother, Gauss–Seidel smoother),
is the canonical interpolation operator for
space,
is the matrix representation of discrete vector Laplacian defined on
, and
is the discrete curl operator.
Based on the auxiliary space preconditioning framework, one can show that
For
in the definition of
, we can replace it by the HX preconditioner for
problem, e.g.,
, since they are spectrally equivalent. Moreover, inverting
might be expensive and we can replace it by a spectrally equivalent approximations
. These leads to the following practical HX preconditioner for
problem,
The derivation of HX preconditioners is based on the discrete regular decompositions for
and
, for the completeness, let us briefly recall them.
Theorem:[Discrete regular decomposition for
]
Let
be a simply connected bounded domain. For any function
, there exists a vector
,
,
, such that
and
Theorem:[Discrete regular decomposition for
]
Let
be a simply connected bounded domain. For any function
, there exists a vector
,
such that
and
Based on the above discrete regular decompositions, together with the auxiliary space preconditioning framework, we can derive the HX preconditioners for
and
problems as shown before.
- ^ Hiptmair, Ralf; Xu, Jinchao (2007-01-01). "{Nodal auxiliary space preconditioning in $backslash$bf H($backslash$bf curl) and $backslash$bf H($backslash$rm div)} spaces". SIAM J. Numer. Anal. ResearchGate: 2483. doi:10.1137/060660588. Retrieved 2020-07-06.
- ^ J.Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing. 1996;56(3):215–35.
- ^ T. V. Kolev, P. S. Vassilevski, Parallel auxiliary space AMG for H (curl) problems. Journal of Computational Mathematics. 2009 Sep 1:604–23.
- ^ T.V. Kolev, P.S. Vassilevski. Parallel auxiliary space AMG solver for H(div) problems. SIAM Journal on Scientific Computing. 2012;34(6):A3079–98.
- ^ Report of The Panel on Recent Significant Advancements in Computational Science, https://science.osti.gov/-/media/ascr/pdf/program-documents/docs/Breakthroughs_2008.pdf
- ^ E.G. Phillips, J. N. Shadid, E.C. Cyr, S.T. Miller, Enabling Scalable Multifluid Plasma Simulations Through Block Preconditioning. In: van Brummelen H., Corsini A., Perotto S., Rozza G. (eds) Numerical Methods for Flows. Lecture Notes in Computational Science and Engineering, vol 132. Springer, Cham 2020.