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❆♥ ♦✉t✲♦❢✲❝♦r❡ ❡①t❡♥s✐♦♥ ♦❢ ❛ ♣❛r❛❧❧❡❧ s♣❛rs❡ ♠✉❧t✐❢r♦♥t❛❧ s♦❧✈❡r ❊✳ ❆❣✉❧❧♦ ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ❆✳ ●✉❡r♠♦✉❝❤❡ ❛♥❞ ❏✳✲❨✳ ▲✬❊①❝❡❧❧❡♥t ❈❙❈✵✺✱ ✷✶✲✷✸ ❏✉♥❡ ✷✵✵✺ ✶ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ❙t❛t❡✲♦❢✲t❤❡✲❛rt ✷ Pr❡❧✐♠✐♥❛r② ❙t✉❞② ✸ ❖✉t✲♦❢✲❝♦r❡ ❙t♦r❛❣❡ ♦❢ t❤❡ ❋❛❝t♦rs ❆❧❣♦r✐t❤♠s Pr❡❧✐♠✐♥❛r② ♣❡r❢♦r♠❛♥❝❡ ❛♥❛❧②s✐s ✹ ❖✉t✲♦❢✲❝♦r❡ ❆❝t✐✈❡ ▼❡♠♦r② ✭s❡q✉❡♥t✐❛❧ ❝❛s❡✮ ❆❝t✐✈❡ ♠❡♠♦r② ❜❡❤❛✈✐♦✉r ▼✐♥✐♠✐③✐♥❣ t❤❡ ✈♦❧✉♠❡ ♦❢ ■✴❖ ✺ ❚r❡❛t♠❡♥t ♦❢ ▲❛r❣❡ ❆❝t✐✈❡ ❋r♦♥t❛❧ ▼❛tr✐❝❡s ✻ ❋✉t✉r❡ ❲♦r❦

■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♠✉❧t✐❢r♦♥t❛❧ ♠❡t❤♦❞ ✭❉✉☛✱ ❘❡✐❞✬✽✸✮ 1 2 3 4 5 1 2 3 4 5 1 1 2 2 A= Fill−in L+U−I 3 3 5 5 4 4 5 5 4 4 5 Non Zero Zero Fill−in Factor 3 ▼❡♠♦r② ❞✐✈✐❞❡❞ ✐♥ t✇♦ ♣❛rts✿ 3 4 1 1 ❆❝t✐✈❡ ♠❡♠♦r② 4 2 2 5 3 Contribution block ❋❛❝t♦rs ❉❡♣❡♥❞❡♥❝② tr❡❡ Active Factors Stack memory matrix Active memory ▼❯▼P❙✿ ▼❯❧t✐❢r♦♥t❛❧ ▼❛ss✐✈❡❧② P❛r❛❧❧❡❧ ❙♦❧✈❡r ✭❆♠❡st♦②✱ ❉✉☛✱ ●✉❡r✲ ♠♦✉❝❤❡✱ ❑♦st❡r✱ ▲✬❊①❝❡❧❧❡♥t✱ Pr❛❧❡t✱ ✳✳✳✮ P0 P1 P0 2D static decomposition P2 P3 P2 P0 P0 P1 P0 P0 P3 P1 P0 P3 P0 TIME P1 P2 P0 P2 P1 P0 P0 P2 P3 P3 P3 P2 P2 P0 1D pipelined factorization P3 and P0 chosen at runtime P1 P2 P3 P3 P0 : STATIC : DYNAMIC SUBTREES s❡❡ ❤tt♣✿✴✴❣r❛❛❧✳❡♥s✲❧②♦♥✳❢r✴▼❯▼P❙ ♦r ❤tt♣✿✴✴✇✇✇✳❡♥s❡❡✐❤t✳❢r✴❛♣♦✴▼❯▼P❙

❙t❛t❡✲♦❢✲t❤❡✲❛rt ■✴❖ t♦♦❧s✿ ❈✴❋♦rtr❛♥ ❧✐❜r❛r✐❡s ✭✰ t❤r❡❛❞s✮ ❀ ❆■❖ ❀ ▼P■■❖ ❀ ❋● Pr❡✈✐♦✉s ♦✉t✲♦❢✲❝♦r❡ ❛♣♣r♦❛❝❤❡s ✭s❡q✉❡♥t✐❛❧✮✿ left−looking multifrontal ❋✐❣✉r❡✿ ❊✳ ❘♦t❤❜❡r❣ ❛♥❞ ❘✳ ❙❝❤r❡✐❜❡r✱ ✶✾✾✾ ❋✐❣✉r❡✿ ❱✳ ❘♦t❦✐♥ ❛♥❞ ❙✳❚♦❧❡❞♦✱ ✷✵✵✹

Pr❡❧✐♠✐♥❛r② ❙t✉❞②✿ ❊①♣❡r✐♠❡♥t❛❧ ❊♥✈✐r♦♥♠❡♥t ▼❯▼P❙✿ ▼✉❧t✐❢r♦♥t❛❧ P❛r❛❧❧❡❧ ❙♦❧✈❡r ❢♦r ❜♦t❤ ▲❯ ❛♥❞ ▲❉▲ ❚ ✳ ❘❡♦r❞❡r✐♥❣ t❡❝❤♥✐q✉❡s✿ ▼❊❚■❙ ✱ P❖❘❉ ✳ ❚❡st ♣❧❛t❢♦r♠✿ ■❇▼ ♣❧❛t❢♦r♠ ❛t ■❉❘■❙ ✭❖rs❛②✱ ❋r❛♥❝❡✮ ❝♦♠✲ ♣♦s❡❞ ♦❢ ✹✲✇❛② ❛♥❞ ✸✷✲✇❛② P♦✇❡r✹✰ ♣r♦❝❡ss♦rs✳ ▼❡♠♦r② ❧✐♠✐ts ♣❡r ♣r♦❝❡ss♦r✿ ◆✉♠❜❡r ♦❢ ♣r♦❝s ✶ ✷✲✶✻ ✶✼✲✻✹ ✻✺✲ ▼❛① ♠❡♠♦r② ✶✻ ●❇ ✹●❇ ✸✳✺●❇ ✶✳✸●❇ ❚❡st ♣r♦❜❧❡♠s✿ r❛♥❣❡ ♦❢ ❧❛r❣❡ ♠❛tr✐❝❡s ❡①tr❛❝t❡❞ ❢r♦♠ st❛♥❞❛r❞ ❝♦❧❧❡❝t✐♦♥s ♦r ♣r♦✈✐❞❡❞ ❜② ▼❯▼P❙ ✉s❡rs✳ ❙✐♠✉❧❛t✐♦♥ ♦❢ ❛♥ ♦✉t✲♦❢✲❝♦r❡ ❜❡❤❛✈✐♦✉r✿ ❋r❡❡ ❢❛❝t♦rs ❛s s♦♦♥ ❛s t❤❡② ❛r❡ ❝♦♠♣✉t❡❞ ❖♥❧② ❢❛❝t♦r✐③❛t✐♦♥ st❡♣ ✐s ♣♦ss✐❜❧❡ ✭❢❛❝t♦rs ❛r❡ ❧♦st✮ ❙❡❧❡❝t❡❞ ✈❛❧✉❡s✿ t❤❡ ❜✐❣❣❡r ♦✈❡r ❛❧❧ ♣r♦❝❡ss♦rs ❢♦r ✿ ❚❤❡ s✐③❡ ♦❢ ❢❛❝t♦rs ❚❤❡ ♣❡❛❦ ♦❢ ❛❝t✐✈❡ ♠❡♠♦r② ❚❤❡ ♣❡❛❦ ♦❢ t♦t❛❧ ♠❡♠♦r②

Pr❡❧✐♠✐♥❛r② ❙t✉❞②✿ ❊①♣❡r✐♠❡♥t❛❧ ❘❡s✉❧ts ❚②♣✐❝❛❧ ♠❡♠♦r② ❜❡❤❛✈✐♦✉r ✭ ❆❯❉■❑❲ ✶ ♠❛tr✐①✮ 1600 1600 Active memory Active memory Factors zone Factors zone Total memory Total memory 1400 1400 1200 1200 1000 1000 Memory peak Memory peak 800 800 600 600 400 400 200 200 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Nb processors Nb processors ▼❊❚■❙ P❖❘❉ ❆❝t✐✈❡ ♠❡♠♦r② ✴ t♦t❛❧ ♠❡♠♦r② r❛t✐♦ 0.8 Maximum peak of active memory / maximum peak of total memory (ratio) AUDIKW_1 CONESHL_MOD CONESHL2 CONV3D 0.7 ULTRASOUND80 0.6 0.5 0.4 0.3 0.2 0.1 0 10 20 30 40 50 60 70 Number of processors ❈♦♥s❡q✉❡♥❝❡ ❋✐rst st❡♣✿ ❢❛❝t♦rs ♦✉t✲♦❢✲❝♦r❡ ✭✇❡❧❧ ❛❞❛♣t❡❞ ❢♦r ❢❡✇ ♣r♦❝❡ss♦rs✮ ❙❡❝♦♥❞ st❡♣✿ ❢❛❝t♦rs ❛♥❞ st❛❝❦ ♦✉t✲♦❢✲❝♦r❡ ✭❧❛r❣❡st ♣r♦❜❧❡♠s ♦r ♠❛♥② ♣r♦❝❡ss♦rs✮

❖✉t✲♦❢✲❝♦r❡ ❙t♦r❛❣❡ ♦❢ t❤❡ ❋❛❝t♦rs ❙②♥❝❤r♦♥♦✉s ❱❡rs✐♦♥✿ ❯s❡ st❛♥❞❛r❞ ✇r✐t❡ ♦♣❡r❛t✐♦♥s ❋❛❝t♦rs ❛r❡ ✇r✐tt❡♥ t♦ ❞✐s❦ ✭♣♦ss✐❜❧② ✇✐t❤ ❧♦✇✲❧❡✈❡❧ s②st❡♠ ❜✉☛❡r✐♥❣✮ ❛s s♦♦♥ ❛s t❤❡② ❛r❡ ❝♦♠♣✉t❡❞ ❙♦❧✉t✐♦♥ st❡♣✿ ✶✳ ❘❡❛❞ ❛ ❢❛❝t♦r ❜❧♦❝❦ ✷✳ ❲♦r❦ ✇✐t❤ t❤❡ ❢❛❝t♦r ✮ ❋❛❝t♦rs ♠❛② ❜❡ r❡❛❞ t✇✐❝❡ ✭❢♦r✇❛r❞ ❡❧✐♠✐♥❛t✐♦♥ ❛♥❞ ❜❛❝❦✇❛r❞ s✉❜st✐t✉t✐♦♥✮ ❆s②♥❝❤r♦♥♦✉s ❱❡rs✐♦♥✿ I/O Request ❚❤r❡❛❞❡❞ ✈❡rs✐♦♥ ✇✐t❤ ❜✉☛❡rs I/O ❙♦❧✉t✐♦♥ st❡♣✿ st✐❧❧ s②♥❝❤r♦♥♦✉s Compute I/O thread thread ❘❡s✉❧ts✿ ✇❡ ❝❛♥ s♦❧✈❡ ❜✐❣❣❡r ♣r♦❜❧❡♠s s❛♠❡ ♣r♦❜❧❡♠s ✇✐t❤ ❧❡ss ♠❡♠♦r② ✭❝❢ ♣r❡❧✐♠✐♥❛r② st✉❞②✮ ❡①❛♠♣❧❡✿ ❯▲❚❘❆❙❖❯◆❉✽✵ t♦t❛❧ ♠❡♠ ♣❡r ♣r♦❝ ❛❝t✐✈❡ ♠❡♠ ♣❡r ♣r♦❝ ✶ ♣r♦❝ ✭✶✻●❇ ♠❡♠♦r②✮ ✶✶✵✶ ♠✐❧❧✐♦♥ r❡❛❧s ✷✶✽ ♠✐❧❧✐♦♥ r❡❛❧s ✹ ♣r♦❝s ✸✻✵ ♠✐❧❧✐♦♥ r❡❛❧s ✶✺✹ ♠✐❧❧✐♦♥ r❡❛❧s s❛♠❡ ♣r♦❜❧❡♠s ✇✐t❤ ❧❡ss ♣r♦❝❡ss♦rs ▼❛tr✐① ❙tr❛t❡❣② ♠✐♥ ♣r♦❝s ❯▲❚❘❆❙❖❯◆❉✽✵ ✐♥✲❝♦r❡ ✽ ♦✉t✲♦❢✲❝♦r❡ ✷ ❈❖◆❱✸❉ ✐♥✲❝♦r❡ ✸✷ ♦✉t✲♦❢✲❝♦r❡ ✶✻ ❈❖◆❱✸❉ ♦♥ ✶ ♣r♦❝ ✇✐t❤ ✶✻ ●❇ ♠❡♠♦r②✿ ♦✉t✲♦❢✲❝♦r❡ ✈❡rs✐♦♥ ♦❦✱ ✐♥✲ ❝♦r❡ ✈❡rs✐♦♥ r✉♥s ♦✉t ♦❢ ♠❡♠♦r②

Pr❡❧✐♠✐♥❛r② P❡r❢♦r♠❛♥❝❡ ❆♥❛❧②s✐s ❙❛♠❡ ❡♥✈✐r♦♥♠❡♥t✿ ■❉❘■❙ ♣❧❛t❢♦r♠ ❖r❞❡r✐♥❣ s❡❧❡❝t❡❞✿ ▼❊❚■❙ ❉✐☛❡r❡♥t str❛t❡❣✐❡s✿ ✲ s②♥❝❤r♦♥♦✉s ■✴❖ ✲ ❛s②♥❝❤r♦♥♦✉s ■✴❖ ✇✐t❤ ❛ ❜✉☛❡r ✲ ✐♥ ❝♦r❡ ❚✐♠❡ ❢♦r ❢❛❝t♦r✐③❛t✐♦♥✿ t②♣✐❝❛❧ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ st❡♣ 1200 IC Synchronous OOC Asynchronous OOC 1000 Elapsed time for factorization (seconds) 800 600 400 200 0 0 10 20 30 40 50 60 70 Number of processors ❋✐❣✉r❡✿ ❈❖◆❊❙❍▲ ▼❖❉ ♠❛tr✐①