Hierarchical matrix: Difference between revisions

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<ref name="MB08">{{cite book|last=Bebendorf|first=Mario|date=2008|title=Hierarchical matrices: A means to efficiently solve elliptic boundary value problems|publisher=Springer}}</ref>

<ref name="HAKH00">{{cite journal|last=Hackbusch|first=Wolfgang|last2=Khoromskij|first2=Boris N.|date=2000|title=A sparse H-Matrix Arithmetic. Part II: Application to Multi-Dimensional Problems|journal=Computing|volume=64|pages=21&ndash;47}}</ref>

<ref name="MB00">{{cite journal|last=Bebendorf|first=Mario|title=Approximation of boundary element matrices|date=2000|journal=Num. Math.|volume=86|pages=565-589565–589}}</ref>

<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=http://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=http://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=http://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=http://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>.