Hierarchical Matrices
News
H2Lib released.
The first release of the new H2Lib package has been released to
the public.
It is closely related to HLib,
but differs in a number of important points:
 H2Lib is published as free and open source software under
the GNU lesser public license.
It can be obtained free of charge at
GitHub.
 The library offers matrix arithmetic operations
(multiplication, inversion, factorization) both for
hierarchical matrices and H²matrices.
 Integral equations are supported by a wide range of
optimized approximation algorithms.
 Dense matrices and vectors are represented by separate
data types, allowing the library to perform, e.g., bounds
checking.
 Although the library's performance benefits significantly
from additional packages like
LAPACK, these
packages are not required.
 Online documentation is available at the
project website.
Bug in HLib: Incorrect array operations when
writing to a file.
The function write_supermatrix used to write a hierarchical
matrix to a file computed an index with the wrong leading dimension.
The problem is solved by a patch available at the
HLib patches page.
Winterschool on hierarchical matrices.
The next winterschool on hierarchical matrices will take place at the
Max Planck Institute for Mathematics
in the Sciences from the 11th to the 15th of March, 2013. The
deadline for registrations is the 29th of February, 2013.
The winterschool will focus on the theoretical foundation of hierarchical
matrix techniques and on the practical implementation of the corresponding
algorithms and data structures in the context of the
HLib package. Details can be found on the
winterschool homepage.
New Book on H²Matrices.
The first book on H²matrix
methods is now available.
It offers a comprehensive introduction into the relevant algorithms,
the techniques for the corresponding complexity and error analysis,
and a collection of numerical examples illustrating the properties
of the H²matrix method.
All algorithms presented in the book will be included in the
next release HLib.
Big Book on Hierarchical Matrices.
The big book on
hierarchical matrix techniques has been published.
If offers an overview of the current state of the art in the field
of hierarchical matrix techniques and covers topics like algebraic
and analytic lowrank approximations, matrix arithmetic operations
and applications to integral operators, matrix equations and
preconditioning of elliptic partial differential equations.
Bug in HLib: Incorrect quadrature for twodimensional
boundary elements.
The singular quadrature routines in quadrature1d used for
discretizing integral operators in curvebem relied on invalid
symmetry assumptions, and the logarithmic kernel function used in
curvebem was treated incorrectly in the computation of the
diagonal entries of the single layer potential matrix.
Both problems are solved by a patch available at the
HLib patches page.
New Paper: Adaptive variablerank approximation
of general dense matrices.
The paper [Börm2005b]
introduces a modification of the standard H^{2}matrix
recompression algorithm presented in
[Börm/Hackbusch2002]
that allows it to reduce the storage requirements by using techniques
introduced in the context of variableorder approximation schemes
[Sauter2000].
Lecture notes of the winterschool on hierarchical
matrices.
The lecture notes
[Börm/Grasedyck/Hackbusch2004]
of the winterschool on hierarchical matrices have been updated.
The current version covers the application of hierarchical matrices to
integral and partial differential equations, approximative arithmetics,
complexity estimates and H^{2}matrices.
HLib 1.3 released.
The new version 1.3 of the HLib
package is now available.
Important new features include hybrid cross approximation of boundary
element operators, improved orthogonalization and recompression
routines for H^{2}matrices, new (and hopefully more
accessible) example programs, and H^{2}matrix
arithmetics in linear complexity.
New paper: Hybrid cross approximation of
integral operators.
A popular approach to constructing lowrank approximations of discretized
integral operators is based on the application of cross approximation
techniques
[Tyrtyshnikov2000],
[Bebendorf/Rjasanow2003]
to entries of the discrete matrix. This has the advantage that a complete
Hmatrix approximation can be created based only on a routine for
evaluating integrals and some geometric information.
Unfortunately, this approach can be shown to fail in a number of important
situations, e.g., for the classical double layer operator on polygonal
domains. This is due to the fact that too much information is lost in the
transition from the continuous to the discrete problem.
The paper
[Börm/Grasedyck2004] proposes a different approach: the cross
approximation is applied to the original kernel function, and the resulting
kernel expansion is then discretized. Numerical experiments indicate that
this hybrid technique is both faster and more reliable than previous methods.
