On Efficiently Updating Singular Value Decomposition Based Reduced Order Models
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1 On Efficiently dating Singula alue Decoosition Based Reduced Ode Models Ralf Zieann GAMM oksho Alied and Nueical Linea Algeba with Secial Ehasis on Model Reduction Been Se..-3.
2 he POD-based ROM aoach. Inut: CFD snahots Flow solutions at diffeent flow conditions
3 he POD-based ROM aoach. Inut: CFD snahots Flow solutions at diffeent flow conditions. POD Basis Othonoal basis odeed by infoation content sanning the sae sace 3
4 he POD-based ROM aoach. Inut: CFD snahots Flow solutions at diffeent flow conditions. POD Basis Othonoal basis odeed by infoation content sanning the sae sace 3. Ode Reduction Select ~ POD coonents with lagest infoation content 4
5 he POD-based ROM aoach. Inut: CFD snahots Flow solutions at diffeent flow conditions. POD Basis Othonoal basis odeed by infoation content sanning the sae sace 3. Ode Reduction Select ~ POD coonents with lagest infoation content 4. Pediction ste Deteine POD-ROM coefficients by inteolation / solving low-ode PDEs / least-squaes otiization 5
6 he POD-based ROM aoach. Inut: CFD snahots Flow solutions at diffeent flow conditions 5. Outut: aoxiated flow field. POD Basis Othonoal basis odeed by infoation content sanning the sae sace ntied flow condition 3. Ode Reduction Select ~ POD coonents with lagest infoation content 4. Pediction ste Deteine POD-ROM coefficients by inteolation / solving low-ode PDEs / least-squaes otiization 6
7 POD based educed ode odels ae a oweful tool Industial aicaft configuation ROM Seed-u facto > 3 Gid size ~9Mio subsonic Ma =. Snashot data at AoA = - Pediction at AoA = 7 (Extaolation!) 7
8 but teating lage snashots ay becoe a challenge! Incoing new snashots POD/SD eesentation known How to coute POD/SD of augented data set efficiently? 8
9 9 Noenclatue j j n A A A A R SD : )... ( : )... ( Centeing : Snashot aveage: )... ( Snashot atix : ) ( : Relative Infoation content ic i i i i
10 Discad coluns coesonding to sall singula values: Definition: he data set is called educed-ode odel of ode of he atio is called the coession ate. n R R n A. Reduced-ode eesentation
11 he SD basis udate oble n A n R. Given: ROM of new snashot obsevations... ask: Coute ROM of. A n Requieent: use only the evious stage ROM and the incoing snashots! n! Kee coutational costs deending on as low as ossible n
12 Objective Efficient SD basis udate date SD basis without having to stoe the initial snashots SNAPSHOS =SD POD MODES
13 Objective Efficient SD basis udate date SD basis without having to stoe the initial snashots SNAPSHOS =SD POD MODES SNAPSHOS + = + SD POD MODES 3
14 Objective Efficient SD basis udate date SD basis without having to stoe the initial snashots SNAPSHOS =SD POD MODES SNAPSHOS + SNAPSHOS =SD POD MODES 4
15 5 dating the snashot ean Shift vecto: Shift udate snashots to the evious-stage cente: o do: Decoose SD basis udate stategies i i A A A : i A i i... :!
16 6 dating the snashot ean Shift vecto: Shift udate snashots to the evious-stage cente: o do: Decoose SD basis udate stategies i i A A A : i A i i... :! stat hee
17 SD basis udate stategies (A): two-stes SD Setting X n B I it holds X B and X M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9) 7
18 SD basis udate stategies (A): two-stes SD Setting X n B I it holds X B and X ank- SD udate oble M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9) 8
19 SD basis udate stategies (A): two-stes SD Setting X n B I it holds X B and X Factoing out othogonal coonents: X B I ( I ) ( ) M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9) 9
20 SD basis udate stategies (A): two-stes SD Setting X n B I it holds X B and X Factoing out othogonal coonents: X B Poth P ( othp I ) ( ) whee P ( ) P oth( P) oth e.g. via Ga Schidt M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9)
21 SD basis udate stategies (A): two-stes SD Setting X n B I it holds X B and X othonoal Factoing out othogonal coonents: coluns X B Poth P ( othp I ) ( ) M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9)
22 SD basis udate stategies (A): two-stes SD Setting X n B I it holds X B and X Factoing out othogonal coonents: X B othonoal coluns Poth P ( othp I ) ( ) M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9)
23 SD basis udate stategies (A): two-stes SD Setting X n B I it holds X B and X Factoing out othogonal coonents: X B size ( ) ( ) Poth P ( othp I ) ( ) M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9) 3
24 SD basis udate stategies (A): two-stes SD Setting X n B I it holds X B and X Factoing out othogonal coonents: X B ~ ~ ~ Poth ( I ) ( ) M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9) 4
25 5 Setting it holds Factoing out othogonal coonents: SD basis udate stategies (A): two-stes SD n I B X X B X and M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9) ) ( ) ( ~ ~ ~ oth I P B X
26 6 Reeat fo shifting to new cente: SD basis udate stategies (A): two-stes SD M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9)! SD now known
27 SD basis udate stategies (A): two-stes SD Coents: e-othogonalization exensive additional (lage-scale) SD equied less obust via Ga-Schidt P ( ) P oth( P) oth Paallelization is oe involved M. Band: Fast low-ank odifications of the thin SD Lin. Alg. and its Al P. Hall et al.: Meging and slitting eigensace odels IEEE ans. Patten analysis and Machine Intelligence (9) 7
28 8 Objective: Fist ste: Coute SD of Reduce to syetic ED SD basis udate stategies (B): ED 3 +SD 3 Genealize ideas fo: J.R. Bunch C.P. Nielsen: dating the singula value decoosition Nue. Math !
29 9 Objective: Fist ste: Coute SD of Reduce to syetic ED SD basis udate stategies (B): ED 3 +SD 3 Genealize ideas fo: J.R. Bunch C.P. Nielsen: dating the singula value decoosition Nue. Math ! Exloit evious stage SD
30 3 Objective: Fist ste: Coute SD of Reduce to syetic ED SD basis udate stategies (B): ED 3 +SD I I 3 Genealize ideas fo: J.R. Bunch C.P. Nielsen: dating the singula value decoosition Nue. Math ! facto out
31 3 Objective: Fist ste: Coute SD of Reduce to syetic ED SD basis udate stategies (B): ED 3 +SD I I 3 Genealize ideas fo: J.R. Bunch C.P. Nielsen: dating the singula value decoosition Nue. Math ! size ) ( ) (
32 3 Objective: Fist ste: Coute SD of Reduce to syetic ED SD basis udate stategies (B): ED 3 +SD I Q Q I ~~ ~ 3 Genealize ideas fo: J.R. Bunch C.P. Nielsen: dating the singula value decoosition Nue. Math !
33 33 Objective: Fist ste: Coute SD of Reduce to syetic ED Coute left singula vectos via SD basis udate stategies (B): ED 3 +SD I Q Q I ~~ ~ 3 Genealize ideas fo: J.R. Bunch C.P. Nielsen: dating the singula value decoosition Nue. Math ! ~ ~ ˆ Q
34 34 se Band s ethod fo shifting to new cente: SD basis udate stategies (B): ED 3 + SD! SD now known 3 Genealize ideas fo: J.R. Bunch C.P. Nielsen: dating the singula value decoosition Nue. Math
35 35 Objective: Let Reduce to syetic ED exloit evious stage SD in atix oducts Coute left singula vectos via SD basis udate stategies (C): one-ste ED ) ( ) ( ~ ~ ~ R Q Q X X! ) ( : n R X ~ ~ Q X
36 SD basis udate stategies (B) and (C) Coents: Staight fowad aallelization! (only standad atix oducts equied in aallel) 36
37 Analysis of Coutational costs n Count flos fo atix oduct AB A n R B R Stategy (B) is oe efficient than stategy (A): flos(a) flos(b) O(( ( ) n) ) n O( oth( P)) 37
38 Analysis of Coutational costs he anking of Stategy (B) vs. (C) deends on the coession ate! Assution: x x (coession ate) hen the coutational costs diffe by flos(c) flos(b) n ( x x ) ( 3x)( ) 38
39 Analysis of Coutational costs he anking of Stategy (B) vs. (C) deends on the coession ate! Solving the quadatic equation shows that Stategy (B) is oe efficient than Stategy (C) if x 3( ) ( ) 7 4 In actical ileentations use switch to select the ost efficient udate stategy. 39
40 Analysis of Coutational costs Rules of thub: Fo highly coessed odels Stategy (B) is oe efficient than Stategy (C) Fo weakly coessed odels the oosite holds tue Moe ecisely Stategy (B) is oe efficient than Stategy (C) if and. x Stategy (C) is oe efficient than Stategy (B) if x 3. 4
41 Exale: dating SDs of Rando atices R n n R n 5 Method Reconstuction eo tie (sec) Stategy (A) SD-oth.54e Stategy (A) QR-oth.333e- 8.5 Stategy (B).34e- 6.3 Stategy (C).79e- 4.4 ncoessed odels
42 Exale: dating SDs of Rando atices R n n R n 5 Method Reconstuction eo tie (sec) Stategy (A) SD-oth Stategy (A) QR-oth Stategy (B) Stategy (C) Coession level x.4 4
43 Suay he udate stategies following the syetic ED aoach ae oe efficient than the SD udate known fo liteatue (Band Hall et al.) (Assution: n>>) Industial oint of view: SD/ED efoed by black box function Most efficient: choose ethod deending on the coession ate he exales suggest that all ethods shae a siila level of accuacy (SD-aoach ay suffe fo othogonal out-factoing ED-aoaches ay suffe fo squaing the condition nube) 43
44 Suay Fo details and additional efeences see: A coehensive coaison of vaious algoiths fo efficiently udating singula value decoosition based educed ode odels DLR IB 4-/3 Feely available at DLR s electonic libay: htt://elib.dl.de/75 44
45 hank you fo you attention! 45
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