A Theoretical Study on the Rank of the Integral Operators for Large- Scale Electrodynamic Analysis

Transcription

1 Purdue Unversty Purdue e-pubs ECE Techncl Reports Electrcl nd Computer Engneerng A Theoretcl Study on the Rnk of the Integrl Opertors for Lrge- Scle Electrodynmc Anlyss Wenwen Ch Purdue Unversty, wch@purdue.edu Dn Jo Purdue Unversty, djo@purdue.edu Follow ths nd ddtonl works t: Ch, Wenwen nd Jo, Dn, "A Theoretcl Study on the Rnk of the Integrl Opertors for Lrge- Scle Electrodynmc Anlyss" (2). ECE Techncl Reports. Pper Ths document hs been mde vlble through Purdue e-pubs, servce of the Purdue Unversty Lbrres. Plese contct epubs@purdue.edu for ddtonl nformton.

2 A Theoretcl Study on the Rnk of the Integrl Opertors for Lrge- Scle Electrodynmc Anlyss Wenwen Ch Dn Jo TR-ECE--9 November 22, 2 School of Electrcl nd Computer Engneerng 285 Electrcl Engneerng Buldng Purdue Unversty West Lfyette, IN

3 A Theoretcl Study on the Rnk of the Integrl Opertors for Lrge-Scle Electrodynmc Anlyss Wenwen Ch nd Dn Jo School of Electrcl nd Computer Engneerng 465 Northwestern Ave. Purdue Unversty West Lfyette, IN , USA Ths work ws supported by NSF under wrd No nd No

4 Abstrct We theoretclly prove tht the mnml rnk of the ntercton between two seprted geometry blocks n n ntegrl-equton bsed nlyss of generl threedmensonl objects, for prescrbed error bound, scles lnerly wth the electrc sze of the block dmeter. We thus prove the exstence of the error-bounded lowrnk representton of both surfce nd volume bsed ntegrl opertors for electrodynmc nlyss, rrespectve of electrc sze nd sctterer shpe. The theoretcl nlyss developed n ths work permts n nlytcl study of the mnml rnk for prescrbed ccurcy, for rbtrrly shped objects wth rbtrry electrc szes. Numercl experments hve verfed ts vldty. Ths work provdes theoretcl proof on why the low-rnk mtrx lgebr cn be employed to ccelerte the computton of lrge-scle electrodynmc problems. The rnk studed n ths pper s bsed on sngulr vlue decomposton bsed mnml rnk pproxmton of ntegrl opertors, whch does not rely on the seprton of observton nd source coordntes. Methods tht do not generte mnml rnk pproxmton for prescrbed ccurcy cn result n rnk tht scles wth electrc sze t much hgher rte.

5 A Theoretcl Study on the Rnk of the Integrl Opertors for Lrge-Scle Electrodynmc Anlyss Wenwen Ch, Student Member, IEEE nd Dn Jo, Senor Member, IEEE Abstrct We theoretclly prove tht the mnml rnk of the ntercton between two seprted geometry blocks n n ntegrl-equton bsed nlyss of generl three-dmensonl objects, for prescrbed error bound, scles lnerly wth the electrc sze of the block dmeter. We thus prove the exstence of the error-bounded low-rnk representton of both surfce nd volume bsed ntegrl opertors for electrodynmc nlyss, rrespectve of electrc sze nd sctterer shpe. The theoretcl nlyss developed n ths work permts n nlytcl study of the mnml rnk for prescrbed ccurcy, for rbtrrly shped objects wth rbtrry electrc szes. Numercl experments hve verfed ts vldty. Ths work provdes theoretcl proof on why the low-rnk mtrx lgebr cn be employed to ccelerte the computton of lrge-scle electrodynmc problems. The rnk studed n ths pper s bsed on sngulr vlue decomposton bsed mnml rnk pproxmton of ntegrl opertors, whch does not rely on the seprton of observton nd source coordntes. Methods tht do not generte mnml rnk pproxmton for prescrbed ccurcy cn result n rnk tht scles wth electrc sze t much hgher rte. Index Terms Rnk, Integrl Opertors, Electrodynmc Anlyss, Three Dmensonl, Theoretcl Anlyss D I. INTRODUCTION RIVEN BY the desgn of dvnced engneerng systems, there exsts contnued need of reducng the complexty of computtonl electromgnetc methods. Recently, the - nd 2 -mtrx bsed mthemtcl frmework [-2] hs been ntroduced nd further developed to ccelerte both tertve nd drect solutons of the ntegrl equton bsed nlyss of electrodynmc problems [3-5]. The key technque n ths mthemtcl frmework s herrchcl low-rnk mtrx lgebr tht enbles compct representton nd effcent computton of dense mtrces. The drect ntegrl equton solver reported n [6] for solvng lrge-scle electrodynmc problems cn lso be vewed n ths mthemtcl frmework. It successfully solved electrclly lrge ntegrl equtons for problem szes up to Mnuscrpt receved Nov. 2, 2. Ths work ws supported by NSF under wrd No nd No W. Ch nd D. Jo re wth School of Electrcl nd Computer Engneerng, Purdue Unversty, West Lfyette, IN 4797 USA. (e-ml: djo@purdue.edu). M unknowns. In [4-5], the cost of n -mtrx bsed drect computton s reduced for electrodynmc nlyss. The resultnt drect ntegrl equton solver successfully solved electrodynmc problems of 96 wvelengths wth more thn mllon unknowns n fst CPU tme (less thn 2 hours n LU fctorzton, 85 seconds n LU soluton), modest memory consumpton, nd wth the prescrbed ccurcy stsfed, on sngle CPU runnng t 3 GHz. Why the low-rnk representton cn be employed to ccelerte the computton of electrodynmc problems? Does n error-bounded low-rnk pproxmton of ntegrl opertors exst, regrdless of electrc sze? In [5], through sngulr vlue decomposton (SVD) bsed nlyss, t s numerclly shown, for lrge electrc szes (over wvelengths) nd vrous sctterers, the rnk of mtrx block of sze N formed between two geometrclly seprted groups, rsng from the surfce ntegrl equton bsed electrodynmc nlyss, scles s O(N.5 ). As result, the block hs low rnk. However, no theoretcl proof hs been developed to support ths numercl fndng. The contrbuton of ths work s theoretcl proof to the fct tht the mnml rnk of the ntercton between two seprted geometry blocks n n ntegrl-equton bsed nlyss of generl three-dmensonl objects, for prescrbed error bound, scles lnerly wth the electrc sze of the block dmeter. Ths proof s pplcble to vrous ntegrl opertors encountered n electrodynmc nlyss such s electrc feld, mgnetc feld, combned feld, surfce-bsed, nd volumebsed ntegrl opertors. We hve lso derved n nlytcl error bound for the mnml rnk pproxmton of the ntegrl opertor for electrodynmc nlyss. Snce the rnk scles lnerly wth electrc sze of the block dmeter, whle the number of unknowns n surfce ntegrl equton bsed nlyss scles s electrc sze squre, nd tht n volume ntegrl equton bsed nlyss scles s electrc sze cube, we prove the exstence of the error-bounded low-rnk representton of both surfce nd volume ntegrl opertors for electrodynmc nlyss, rrespectve of electrc sze. It s worth mentonng tht the rnk studed n ths pper s the rnk of mnml rnk pproxmton of the ntegrl opertor. It hs been proven tht gven n ccurcy requrement, the low-rnk pproxmton generted from sngulr vlue decomposton (SVD) s the mnml rnk pproxmton [, pp. 63] for the gven ccurcy. Our numercl experments show tht methods tht do not generte

6 2 mnml rnk pproxmton such s nterpolton [3], Tylor seres expnson, nd plne-wve expnson bsed methods cn result n rnk tht s much hgher thn the mnml rnk requred by ccurcy. The rnk lso scles wth electrc sze t much hgher rte. II. THEORETICAL STUDY A. Problem Descrpton The ntegrl equton bsed nlyss of electrodynmc problems results n dense lner system of equtons Z I = V. () Consder Z,t,s, n rbtrry m n off-dgonl block of the system mtrx Z, whch descrbes the ntercton between two seprted groups (t nd s) of the sctterer beng nlyzed. The objectve of ths work s to theoretclly study whether there exsts n error-bounded low-rnk representton of Z,t,s rrespectve of electrc sze nd sctterer shpe, nd f such representton exsts, how the rnk scles wth electrc sze, nd hence the number of unknowns N. Gven n ccurcy requrement ε, s shown n [, pp. 63], the rnk-r representton (R) generted from SVD s mnml rnk pproxmton of the orgnl mtrx M tht fulfls M R 2 ε. However, n SVD nlyss s numercl, whch mkes t not fesble to fnd the ctul rnk requred by ccurcy for rbtrrly lrge electrc szes. As result, n nlytcl pproch, whch s not restrcted by computtonl resources nd s vld for rbtrry shpe, becomes necessry to develop theoretcl understndng on the rnk s dependence wth electrc sze. Ths pper provdes such n nlytcl pproch. In ths pproch, we re ble to mke connecton between n SVD nlyss nd Fourer nlyss. By utlzng the reltonshp between the two nlyses n lner nd shft-nvrnt system, we succeed n nlytclly revelng the rnk of the ntegrl opertors nd ts dependence wth electrc sze. mtrx. Thus, (5) cn be vewed s representng the response b n the V bss ( b V U ), the nput f n the U bss ( f ), nd reltng these two projectons by dgonl mtrx ( Σ ). When the opertor H s both lner nd shft nvrnt (LSIV), SVD turns to Fourer nlyss [7]. More specfclly, the sngulr vectors of n LSIV system re weghted Fourer bss functons (complex exponentls) nd the sngulr vlues re the bsolute vlues of the Fourer trnsform of the system s pont spred functon (mpulse response functon) [7, 8]. To see ths more clerly, let s consder n LSIV system. Becuse n LSIV system opertor s convoluton opertor [7], the response b n spce domn s convoluton of the nput f wth n mpulse response h br ( ) = f( r)* hr ( ), (7) n whch r denotes n rbtrry pont n spce. The bove convoluton cn be converted to smple multplcton by Fourer nlyss. Thus we hve br ( ) = hr ( ) f( r), (8) ( ) ( ) ( ) where ( ) denotes Fourer trnsform. We cn rewrte (8) s FT FT br () = ( hr ()) f() r, (9) where br () FT s the representton of br () n the Fourer bss, nd f () r FT s the representton of f () r n the Fourer bss. In other words, we represent the nput n untry bss (Fourer bss), we lso represent the response n untry bss (Fourer bss), nd relte the two by ( hr ( )). From (5) nd (9), the reltonshp between SVD nd Fourer nlyss cn be clerly seen. The Fourer bses my be dfferent from the SVD-generted bses. However, f the system s lner nd shft-nvrnt, the two bses re both Fourer bses [7]. Therefore, the Fourer nlyss ccomplshes the SVD nlyss of lner shft-nvrnt system. B. Reltonshp between SVD nd Fourer Anlyss n Lner Shft-Invrnt System A lner system cn be modeled by: b= H f, (2) where f nd b re vectors, nd H s lner opertor. We cn perform SVD on H to obtn H b = VΣU f, (3) where superscrpt H denotes complex conjugte trnspose, Σ s the dgonl mtrx comprsng sngulr vlues, nd V nd U re mtrces comprsng sngulr vectors. Snce V nd U re both untry, we hve H H V b = Σ( U f ). (4) whch cn be wrtten compctly s V U b = Σ f, (5) where V H U H b = V b; f = U f. (6) Multplyng untry mtrx by vector cn be thought of s projectng ths vector onto the orthonorml set defned by the C. Rnk Revelng v n Anlytcl Fourer Anlyss of the Integrl Opertor There exst mny ntegrl equton bsed formultons for nlyzng 3-D electrodynmc problems. Exmples re electrc feld ntegrl equton, mgnetc feld ntegrl equton, combned feld ntegrl equton, ech of whch cn be formulted n surfce-bsed or volume-bsed form. The underlyng ntegrl opertors re ll lner nd shft nvrnt. Therefore, we cn use Fourer nlyss to nlytclly study the rnk of the ntegrl equton bsed system mtrx. The pont-spred functon n ntegrl equton bsed opertors s Green s functon nd ts vrnts. The Green s functon for 3-D problem cn be wrtten s: jk r jkr e e gr ( ) = = 4 π r 4πR. () Wthout loss of generlty, n ntegrl equton bsed opertor cn be expressed s the convoluton of certn source f wth Green s functon s the followng: br ( ) = g( r r ' ) f( r ') dr ', ()

7 3 where the ntegrl could be one-, two-, or three-dmensonl. The Fourer nlyss of the bove results n: Bk ( ) = G ( kfk ) ( ), (2) whch, n dscrete form, cn be wrtten s: B G F B G F B = G F 2 2 2, (3) B G F n n n where G, F, nd B re, respectvely, G ( k ), Fk ( ), nd Bk ( ) t dscrete k ( =,, ). Fg.. Illustrton of two seprted groups t nd s. Consder two seprted groups t nd s of the entre unknown set of generl 3-D object, s llustrted n Fg.. The correspondng domns re Ω t nd Ω s. The mnmum dstnce between the two groups s denoted by R, nd the mxmum dstnce s R 2 : R = mn( r r' ),. (4) R2 = mx( r r' ) where, r Ωt, r' Ωs. The strong η-dmssblty condton s usully used n the low-rnk mtrx lgebr to quntfy the seprton between two groups [-2]: = mx{ dm( Ωt), dm( Ωs)} ηdst( Ωt, Ω s), (5) n whch dm() s the Euclden dmeter of group, dst(,) s the Euclden dstnce between two groups, s the mxmum group dmeter, nd η s postve prmeter. It s not dffcult to fnd the reltonshp between R nd R 2 n (4) nd η nd the block dmeter n (5) s: R = η. (6) R2 = R+ c, < c 2 where c s constnt coeffcent between nd 2. In n ntegrl equton bsed system mtrx Z, the ntercton between two seprted groups t nd s s chrcterzed by mtrx block Z t,s. Gven prescrbed ccurcy, ts mnml rnk cn be numerclly determned by SVD. From the nlyss gven n Secton II.B, ths rnk s the sme s the rnk of the dgonl mtrx G n (3), the entres of whch re the Fourer trnsform of Green s functon t dscrete frequency ponts. Thus, next, we perform Fourer nlyss of the Green s functon to nlytclly determne the rnk of Z t,s. The Green s functon, whch s the kernel functon of (), pprently, depends on both observton pont r nd source pont r '. However, the Green s functon does not depend on the shpe of the sctterer. Insted, t s only determned by the dstnce between the observton nd source ponts, r r', thus defned over fnte one-dmensonl rnge of ( R, R ). 2 Therefore, we perform Fourer nlyss of () n the sme rnge s the followng j2 π R 2 ( R R ) gr ( ) = G e, R ( R, R), (7) 2 where dscrete G = G ( k ) cn be evluted from j2π R R2 R ( ),,, R 2 G = gre dr = R R 2 R. (8) Substtutng () nto (8), we cn nlytclly evlute (8) s: ( + ) ( + 2 ) n e = ( ) jk k R e jk k R G j ( n )!, (9) n n n 4π n = ( k + k ) R R 2 where k = 2 π, =,,.... (2) R2 R Substtutng (6) nto (2), we obtn k = 2 π = 2 π. (2) cη R c Usng (2) nd (6), (9) becomes G = 4π ( ck+ 2π) ( ck+ 2π ) j( + c η) j n n η η η c c n c e e ( j) ( n )! n n n = ( ck+ 2π ) ( + c η) (22) The mxmum of G ( =,,...) s G, wth correspondng ndex beng. Gven n ccurcy requrement ε, the rnk of the dgonl mtrx G n (3) cn be determned from the number of G s tht stsfy the followng condton G > ε. (23) G Before genertng quntttve plot of (22) to exmne the rnk, we cn conduct quck nlytcl nlyss. For electrclly lrge problems, (22) cn be pproxmted by c η G ~ (24) 4 π ( ck+ π 2 ) s the other hgher order terms decy much fster. Substtutng (24) nto (23), we obtn c < k. (25) 2π ε

8 4 As result, we hve rnk = Ok ( ). (26) Hence, we prove tht the rnk of G, nd thereby the rnk of Z t,s, whch s the ntercton between two seprted geometry blocks, s lnerly proportonl to the electrc sze of the mxmum block dmeter. In surfce ntegrl equton bsed solver, the row nd column dmenson of Z t,s s proportonl to the squre of the electrc sze of the mxmum block dmeter. In volume ntegrl equton bsed solver, the row nd column dmenson of Z t,s s proportonl to the cube of the electrc sze. In ether cse, Z t,s s low rnk rrespectve of electrc sze. Moreover, t cn be seen from the forementoned theoretcl nlyss, s long s R (mnmum dstnce between the two groups) s greter thn zero,.e. the two groups re geometrclly seprted (non-overlppng), the error bounded low-rnk representton exsts, regrdless of electrc sze. It cn lso be seen tht f the two groups overlp, thus the correspondng mtrx block contns dgonl elements; the error-controlled low-rnk representton does not exst. In Fg. 2() nd (b), bsed on (22), we plot G / G wth G sorted n descendng order, for electrc sze k=, 2, 4, 8, 6, 32, 64, nd 28 respectvely. Wthout the theoretcl pproch developed n ths work, t would not be fesble to obtn the rnk for such lrge electrc szes. From the reltonshp between n SVD nlyss nd Fourer nlyss n n LSIV system, G / G s nothng but normlzed sngulr vlues. From Fg. 2, t s cler tht gven n ccurcy requrement, the rnk of G ncreses wth electrc sze. The quntttve reltonshp cn be seen from Tble I, where we lst the rnk wth respect to electrc sze requred for chevng ccurcy ε=.5, nd ε=. respectvely. It s cler tht the rnk scles lnerly wth the electrc sze of the block dmeter. When genertng Fg. 2 nd Tble I, η=.5 nd c =2 re used n the clculton of (22). D. Anlytcl Error Bound of the SVD-Bsed Mnml Rnk Approxmton of Integrl Opertors for Lrge-Scle Electrodynmc Anlyss The Fourer coeffcent of the Green s functon shown n (22) lso revels the error bound of the mnml-rnk representton of the ntegrl equton bsed opertor for lrge-scle electrodynmc nlyss. Wth the rnk of the ntercton between two seprted geometry blocks chosen s r, the error s bounded by ts, ts, Z ( Z ) G G G rnk r rnk r r =, (27) ts, Z G G where G s the r-th dgonl entry of G, ssumng the r dgonl elements re sorted n descendng order from G to G. The equlty n (27) s due to the reltonshp between n n SVD nlyss nd Fourer nlyss for lner nd shftnvrnt opertor s nlyzed n Secton II.B. The nequlty n (27) s becuse G s dgonl. As cn be seen from (27) nd (8), the center-pont bsed pproxmton of Green s functon only cptures G, thus beng rnk- pproxmton. For electrclly lrge problems, from (22), we obtn lrge k G r =. (28) G π π 2 r 2 r + + c kηr c k It s cler tht ny desred order of ccurcy cn be cheved v rnk r rrespectve of electrc sze k. In ddton, gven requred order of ccurcy, when electrc sze k ncreses by certn rto, the error of the rnk-r pproxmton cn be kept to the desred order by ncresng rnk r by the sme rto. Tble I. Rnk s dependence wth electrc sze for chevng requred ccurcy for 8 dfferent electrc szes from k= to k=28. k ε =.5 Rnk ε =. Rnk k ε =.5 Rnk ε =. Rnk G/G G/G k = k = 2 k = 4 k = () k = 6 k = 32 k = 64 k = (b) Fg. 2. Normlzed Fourer expnson coeffcent of Green s functon n descendng order for eght dfferent electrc szes from k= to k=28.

9 5 6 Plte 25 Cylnder 5 2 k mx 4 3 k mx κ () κ (b) Open cone 5 k mx 5 4 Cone sphere κ (c) 25 Sphere 3 2 k mx κ /9 k mx κ (d) (e) Fg. 3. Rnk generted by ACA+ [9, ] nd SVD wth respect to electrc sze for vrety of sctterer shpes. () Plte. (b) Cylnder. (c) Open cone. (d) Cone sphere. (e) Sphere. III. NUMERICAL VERIFICATION To further verfy the proposed theoretcl nlyss, we numerclly determned the rnk of plte, cylnder, open cone, cone sphere, nd sphere, resultng from surfce-bsed electrc feld ntegrl opertor, by ACA+ [9, ] nd SVD from smll to very lrge electrc szes. In Fg. 3, we plot the mxml rnk k mx mong ll the off-dgonl blocks of the system mtrx versus electrc sze for ll of the fve dfferent sctterers. It s cler tht k mx s O(k). Thus t verfed the proposed theoretcl nlyss. IV. CONCLUSION A theoretcl study s conducted n ths work to nlyze the mnml rnk of ntegrl opertors encountered n electrodynmc nlyss nd ts dependence wth electrc sze for prescrbed error bound. We show tht the rnk generted by sngulr vlue decomposton s the mnml rnk requred

10 6 by ccurcy becuse the rnk-r pproxmton produced by SVD s proven to be the mnml rnk pproxmton for prescrbed ccurcy. The SVD-bsed low-rnk pproxmton does not rely on the seprton of observton nd source coordntes for seprted geometry blocks, whle methods tht seprte observton nd source coordntes such s nterpolton nd plne wve expnson bsed methods do not led to mnml rnk pproxmton of the electrodynmc kernel. As result, the rnk obtned from these methods s observed to scle wth electrc sze t much hgher rte. The SVD nlyss s numercl, whch prevents n nlytcl study. By recognzng the reltonshp between n SVD nlyss nd Fourer nlyss n lner nd shftnvrnt system, we successfully derved n nlytcl error bound of the SVD-bsed mnml rnk pproxmton of the ntegrl opertor, nd reveled the reltonshp between the rnk nd the electrc sze for stsfyng prescrbed ccurcy. The rnk of the ntercton between two seprted geometry blocks s shown to scle lnerly wth the electrc sze of the block dmeter. We thus theoretclly proved the exstence of n error bounded low-rnk representton of electrodynmc ntegrl opertors rrespectve of electrc sze nd sctterer shpe. Numercl experments further verfed ths theoretcl fndng. The theoretcl proof developed n ths work provdes theoretcl bss for employng nd further developng lowrnk mtrx lgebr for ccelertng the ntegrl equton bsed computton of electrodynmc problems. REFERENCES [] S. Börm, L. Grsedyck, nd W. Hckbusch, Herrchcl mtrces, Lecture note 2 of the Mx Plnck Insttute for Mthemtcs n the Scences, 23. [2] S. Börm. 2 -mtrces multlevel methods for the pproxmton of ntegrl opertors, Comput. Vsul. Sc., 7: 73-8, 24. [3] W. Ch nd D. Jo, An 2 -Mtrx-Bsed Integrl-Equton Solver of Reduced Complexty nd Controlled Accurcy for Solvng Electrodynmc Problems, IEEE Trns. Antenns Propg., vol. 57, no., pp , Oct. 29. [4] W. Ch nd D. Jo, A Complexty-Reduced -Mtrx Bsed Drect Integrl Equton Solver wth Prescrbed Accurcy for Lrge-Scle Electrodynmc Anlyss, the 2 IEEE Interntonl Symposum on Antenns nd Propgton, 4 pges, July 2. [5] W. Ch nd D. Jo, A Fst -Mtrx Bsed Drect Integrl Equton Solver wth Prescrbed Accurcy for Lrge-Scle Electrodynmc Anlyss, under revew, IEEE Trns. Antenns Propg., 2. (See lso Purdue ECE techncl report of ths work tht cn be downloded t [6] J. Sheffer, Drect Solve of Electrclly Lrge Integrl Equtons for Problem Szes to M unknowns, IEEE Trns. Antenns Propg., vol. 56, no. 8, pp , Aug. 28. [7] H. H. Brrett nd K. J. Myers, Foundtons of mge scence, John Wley nd Sons, New York (24). [8] S. Prk, J. M. Wtten, nd K. J. Myers, Sngulr vectors of lner mgng system s effcent chnnels for the Byesn del observer, IEEE Trns. Med. Imgng 28, (29). [9] M. Bebendorf, Approxmton of boundry element mtrces, Numer. Mth., vol. 86, no. 4, pp , 2.