Gauss-Quadrature for Fourier-Coefficients

Transcription

1 Copote Techology Guss-Qudtue o Foue-Coecets A. Glg. Hezog. Pth P. Retop d U. Weve Cotet: The tusve method The o-tusve method Adptve Guss qudtue Embeddg to optmzto Applcto u ü tee Gebuch / Copyght Semes AG 8. Alle Rechte vobehlte.

2 The tusve method o Fte Elemets Fte Elemet ethod: Expso o the stess-mtx d the soluto Ku K K u u Poected system: K < > u < >... Rems: The poected system s symmetc The equed stoge gows wth *+/ o the legth o the expso ely o chce to solve the system dectly Bloc-Guss-Sedel: Oly the dgol blocs must be decomposed Good tl vlues tems o the detemstc soluto Vey ew tetos o the Guss-Sedel 3-5 Complexty gows lmost le wth the legth o the ow u ü tee Gebuch Sete August 8 U. Weve CT PP Semes AG Copote Techology

3 Sete 3 August 8 Semes AG Copote Techology U. Weve CT PP u ü tee Gebuch Itusve ethod o Deetl Systems... > < > < t x t x t x x A Ax x & &... > >< < t x F t x t x x x F x & & Itusve ethod: umecl qudtue wth the lgothm Hgh soluto eot ole system Itusve ethod: o umecl qudtue Fst soluto method!! Le system

4 The Itusve ethod +/- Itusve ethod: Plus Complexty o computto gows lmost le wth the umbe o bss polyomls Fo le poblems o mult-dmesol Guss-Qudtue ecessy Bette hdlg o sttoy le poblems Itusve ethod: us Chges the souce code o the detemstc solve!! Hgh memoy usge *+/ wth the umbe o bss polyomls dcult to mplemet o some teestg vlues e.g voses stess u ü tee Gebuch Sete 4 August 8 U. Weve CT PP Semes AG Copote Techology

5 Polyoml chos s Hlbet spce Chos s descbed by Rdom spce Ω F P Ie poduct: < g > E * g gdp The Hlbet spce sped by othogol polyomls H wth sp... < > δ Ω Ech elemet o the Hlbet spce: α α < > umecl themtcs: Choose sutble bse whch coespods to the pobblty mesue Compute the Foue-Coecets whch s dcult o hgh dmesos o the dom spce u ü tee Gebuch Sete 5 August 8 U. Weve CT PP Semes AG Copote Techology

6 Sete 6 August 8 Semes AG Copote Techology U. Weve CT PP u ü tee Gebuch The Geel Guss-Qudtue R d P dp > < Ω Ω α S S S ω ω ω α S W W ω α Shot omulto: Expoetl gowth o ucto evlutos the dmeso o dom spce! Whee the weghts d the bscsse vlues coespod to the mesue P Deto o Foue coecets: ult-dmesol Geel Guss Qudtue:

7 Adptve Guss Qudtue Two methods o the computto o the momets: Geelzed Guss-Qudtue: E V S dp Ω W Whee W d belogs to the coespodg mesue P PC-Expso: E E E V Embeddg o the two methods d compe them u ü tee Gebuch Sete 7 August 8 U. Weve CT PP Semes AG Copote Techology

8 Sete 8 August 8 Semes AG Copote Techology U. Weve CT PP u ü tee Gebuch Adptve Guss Qudtue ℵ I + W W E m m m P P dp dp E E ℵ I ℵ I Appoxmto o by polyoml chos expso: Ide: Evlute the ucto oly t ew vlues d choose ppoxmto o the est } \ { } { } {... S S ℵ I Ω ℵ I Ω I I Ω ℵ Choose subset o exct ucto evluto: Estmte the ucto the complemety subset:: Guss-Qudtue:

9 Sete 9 August 8 Semes AG Copote Techology U. Weve CT PP u ü tee Gebuch Adptve Guss Qudtue 3 V V E E E E ε E E ε V V Eo cotol by compg the two methods: ε ε V V m E E Ou pocedue: Fd the mml umbe o ucto evlutos whch ulls the eo bouds

10 Adptve Guss Qudtue 4 Algothm:. Choose sutble d the subspces I d ℵ. Estmte the ucto complemety subset ℵ 3. Compute E d V by the Geelzed Guss Qudtue 4. Compute the Polyoml Chos Expso d the coespodg momets E d 5. Compute the eos E E ε d 6. I ot covegece Set + choose ew dex wth I I Goto 3 Ed V V V ε ℵ + Ω \ I + Eo cotol by compg two embedded methods such s umecl tegto u ü tee Gebuch Sete August 8 U. Weve CT PP Semes AG Copote Techology

11 Test-Exmple: Ctleve bem Degee p #Evl p+ 5 Rel. eo e e e e e-5 Tol.e-5 Degee p #Evl p+ 5 Rel. eo e e e e d p I E b l p bl 8E I 4 Tol.e-3 u ü tee Gebuch Sete August 8 U. Weve CT PP Semes AG Copote Techology

12 The o-itusve ethod +/- o-itusve ethod: Plus Usge o detemstc solve s blc box!! Esly pplcble to deet quttes Idepedet coecets o PC expso Lttle ddtol memoy usge o-itusve ethod: us umbe o ucto evlutos ceses pdly wth depedet stochstc puts umbe o bscsse vlues must be chose dvce u ü tee Gebuch Sete August 8 U. Weve CT PP Semes AG Copote Techology

13 Embeddg to optmzto m x R x g x Detemstc & stochstc optmzto m E x μ R x D μ σ P g x 99.9% Respose Suce Optmzto. Choose tl box o evluto. Evlute the ucto t cet pots 3. Compute espose suce ppoxmto o the bse o the evluted pots 4. mze the espose suce 5. Reduce the gve box oud the optml vlue Respose suce gee o the bse o dscete evlutos ed Gdet Optmzto:. Compute the detemstc gdet e.g. by te deeces. Compute the stochstc gdet 3. Compute the ext pot o bse o the gdet u ü tee Gebuch Sete 3 August 8 U. Weve CT PP Semes AG Copote Techology

14 Robust Desg Optmzto Optmzto the ce o ucettes The opetg pot chges compso to the detemstc oe The opetg pot moe obust wth espect to the sctteg put vlues Computto o lue pobbltes: Computg the exct s o lue Computg the opetg pot wth gve s o the lue xmzto o toleces: xml muctug toleces llow chepe poducto Keepg the s o lue t gve vlue Developmet o the stochstc optmze RoDeO u ü tee Gebuch Sete 4 August 8 U. Weve CT PP Semes AG Copote Techology

15 Optmzto o hed up dsply o c Poducto toleces o the hed up dsply A hed-up dsply cossts o cbet whch lght s emtted d elected oto the wdsheld v thee cocve d oe ple mo. Poducto toleces led to mge dstotos o the wd sheld d to hgh eect te. Gol: Reducto o the poducto eect te o hed-up dsply. u ü tee Gebuch Sete 5 August 8 U. Weve CT PP Semes AG Copote Techology

16 Pobblstc Optmzto o Hedup Dsply Smpled model o the mo system Actos: Developmet o sotwe whch computes the dstoto o the mge t the wd sheld. The vbles e the dmesos o the house Detemstc optmzto: mzto o the dstoto o the mge by vyg the dmesos o the house. Pobblstc optmzto: xmzto o the llowed toleces ce o the gve pobblty tht the dstoto does ot theshold pescbed vlue. u ü tee Gebuch Sete 6 August 8 U. Weve CT PP Semes AG Copote Techology

17 Results o Optmzto Desty o the scee qulty te detemstc optmzto Desty o the scee qulty te pobblstc optmzto Reect te: Wthout optmzto: 87% Wth det. optmzto: 65% Wth pob. optmzto: 46% u ü tee Gebuch Sete 7 August 8 U. Weve CT PP Semes AG Copote Techology

18 Th you o you tteto u ü tee Gebuch Sete 8 August 8 U. Weve CT PP Semes AG Copote Techology