Hierarchical matrix: Difference between revisions

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In [[numerical mathematics]], '''hierarchical matrices (H-matrices)'''

<ref name="HA99">{{cite journal|last=Hackbusch|first=Wolfgang|date=1999|title=A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices|journal=Computing|volume=62|pages=89&ndash;108}}</ref>

<ref name="GRHA02">{{cite journal|last=Grasedyck|first=Lars|last2=Hackbusch|first2=Wolfgang|date=2003|title=Construction and arithmetics of H-matrices|journal=Computing|volume=70|pages=295&ndash;334|url=httphttps://dx.doi.org/10.1007/s00607-003-0019-1}}</ref>

<ref name="HA09">{{cite book|last=Hackbusch|first=Wolfgang|date=2015|title=Hierarchical matrices: Algorithms and Analysis|publisher=Springer|url=httphttps://dx.doi.org/10.1007/978-3-662-47324-5}}</ref>

are used as data-sparse approximations of non-sparse matrices.

While a [[sparse matrix]] of dimension <math>n</math> can be represented efficiently in <math>O(n)</math> units of storage

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<ref name="BERJ03">{{cite journal|last=Bebendorf|first=Mario|last2=Rjasanow|first2=Sergej|date=2003|title=Adaptive low-rank approximation of collocation matrices|journal=Computing|volume=70|pages=1&ndash;24}}</ref>

<ref name="BOGR05">{{cite journal|last=Börm|first=Steffen|last2=Grasedyck|first2=Lars|date=2005|title=Hybrid cross approximation of integral operators|journal=Num. Math.|volume=101|pages=221&ndash;249}}</ref>

<ref name="BOCH16">{{cite journal|last=Börm|first=Steffen|last2=Christophersen|first2=Sven|date=2016|title=Approximation of integral operators by Green quadrature and nested cross approximation|journal=Num. Math.|volume=133|pages=409&ndash;442|url=httphttps://dx.doi.org/10.1007/s00211-015-0757-y}}</ref>,

preconditioning the resulting systems of linear equations

,<ref name="FAMEPR16">{{cite journal|last=Faustmann|first=Markus|last2=Melenk|first2=J.&nbsp;Markus|last3=Praetorius|first3=Dirk|date=2016|title=Existence of H-matrix approximants to the inverses of BEM matrices: The simple-layer operator|journal=Math. Comp.|volume=85|pages=119&ndash;152|url=httphttps://dx.doi.org/10.1090/mcom/2990}}</ref>

or solving elliptic partial differential equations

<ref name="BEHA03">{{cite journal|last=Bebendorf|first=Mario|last2=Hackbusch|first2=Wolfgang|date=2003|title=Existence of H-matrix approximants to the inverse FE-matrix of elliptic operators with <math>L^\infty</math>-coefficients|journal=Num. Math.|volume=95|pages=1&ndash;28}}</ref>

<ref name="BO10">{{cite journal|last=Börm|first=Steffen|date=2010|title=Approximation of solution operators of elliptic partial differential equations by H- and H²-matrices|journal=Num. Math.|volume=115|pages=165&ndash;193|url=httphttps://dx.doi.org/10.1007/s00211-009-0278-7}}</ref>

<ref name ="FAMEPR13">{{cite journal|last=Faustmann|first=Markus|last2=Melenk|first2=J.&nbsp;Markus|last3=Praetorius|first3=Dirk|date=2015|title=H-matrix approximability of the inverses of FEM matrices|journal=Num. Math.|volume=131|pages=615&ndash;642|url=httphttps://dx.doi.org/10.1007/s00211-015-0706-9}}</ref>,

a rank proportional to <math>\log(1/\epsilon)^\gamma</math> with a small constant <math>\gamma</math> is sufficient to ensure an

accuracy of <math>\epsilon</math>.

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In order to treat very large problems, the structure of hierarchical matrices can be improved:

H²-matrices

<ref name="HAKHSA02">{{cite journal|last=Hackbusch|first=Wolfgang|last2=Khoromskij|first2=Boris&nbsp;N.|last3=Sauter|first3=Stefan|date=2002|title=On H²-matrices|journal=Lectures on Applied Mathematics|pages=9&ndash;29|url=httphttps://link.springer.com/chapter/10.1007/978-3-642-59709-1_2}}</ref>

<ref name="BO10b">{{cite book|last=Börm|first=Steffen|date=2010|title=Efficient Numerical Methods for Non-local Operators: H²-Matrix Compression, Algorithms and Analysis|publisher=EMS Tracts in Mathematics|url=http://www.ems-ph.org/books/book.php?proj_nr=125}}</ref>

replace the general low-rank structure of the blocks by a hierarchical representation closely related to the

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In the context of boundary integral operators, replacing the fixed rank <math>k</math> by block-dependent ranks

leads to approximations that preserve the rate of convergence of the underlying boundary element method

at a complexity of <math>O(n).</math><ref name="SA00">{{cite journal|last=Sauter|first=Stefan|date=2000|title=Variable order panel clustering|journal=Computing|volume=64|pages=223&ndash;261|url=httphttps://link.springer.com/article/10.1007/s006070050045}}</ref>

<ref name="BOSA05">{{cite journal|last=Börm|first=Steffen|last2=Sauter|first2=Stefan|date=2005|title=BEM with linear complexity for the classical boundary integral operators|journal=Math. Comp.|volume=74|pages=1139&ndash;1177|url=http://www.ams.org/journals/mcom/2005-74-251/S0025-5718-04-01733-8}}</ref>