Hierarchical matrix: Difference between revisions

Revision as of 19:32, 12 May 2020

Revision as of 19:41, 12 May 2020

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<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=Numer. Math.|volume=95|pages=1&ndash;28|doi=10.1007/s00211-002-0445-6}}</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=Numer. Math.|volume=115|issue=2|pages=165&ndash;193|doi=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=Numer. Math.|volume=131|issue=4|pages=615&ndash;642|doi=10.1007/s00211-015-0706-9|arxiv=1308.0499}}</ref>

<ref name ="SWX16">{{cite journal|last=Shen|first=Jie|last2=Wang|first2=Yingwei|last3=Xia|first3=Jianlin|date=2016|title=Fast structured direct spectral methods for differential equations with variable coefficients|journal= SIAM Journal on Scientific Computing|volume=38|issue=1|pages=A28&ndash;A54|doi=https://epubs.siam.org/doi/abs/10.1137/140986815}}</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 approximate the entire matrix <math>G</math>, it is split into a family of submatrices.

Large submatrices are stored in factorized representation, while small submatrices are stored in standard representation

in order to improve the efficiency.

Low-rank matrices are closely related to degenerate expansions used in [[panel clustering]] and the [[fast multipole method]]