EFFICIENT DOMAIN DECOMPOSITION METHOD FOR ACOUSTIC SCATTERING IN MULTI-LAYERED MEDIA

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1 European Conference on Computatonal Flud Dynamcs ECCOMAS CFD 2006 P. Wesselng, E. Oñate and J. Péraux (Eds) c TU Delft, The Netherlands, 2006 EFFICIENT DOMAIN DECOMPOSITION METHOD FOR ACOUSTIC SCATTERING IN MULTI-LAYERED MEDIA Kazufum Ito and Jar Tovanen Center for Research n Scentfc Computng, Box 8205 North Carolna State Unversty Ralegh, NC , USA e-mal: kto@ncsu.edu, jatovan@ncsu.edu Key words: Acoustc Scatterng, Geologcal survey, Helmholtz equaton, Doman Decomposton Method, Precondtoned Iteratve Method Abstract. An effcent doman decomposton method s developed for dscretzed Helmholtz equatons resultng from acoustcal geologcal survey problems. A Schwarz-type multplcatve precondtoner s constructed usng a fast drect method n subdomans by embeddng them to rectangular domans. The GMRES teratons are reduced to a subspace correspondng to nterfaces. Numercal experments wth several two-dmensonal problems demonstrate the effectveness and scalablty of the proposed method. 1 INTRODUCTION Acoustcal geologcal survey problems are often modeled usng the Helmholtz equaton wth varyng wave number. The dscretzaton of ths equatons requres suffcently many nodes per wavelength 7 and due to ths the resultng system of lnear equatons s very large. It s usual that grds have from 500 to a few thousands grd ponts n each drecton. Thus, the resultng problem can have several mllon unknowns for a two-dmensonal problem and several bllon unknowns for a three-dmensonal problem. The system matrx s ndefnte and complex-valued. Thus, many effcent soluton procedures avalable for postve defnte problems are not applcable. Iteratve methods wthout a good precondtoner converge slowly for Helmholtz problems whch leads to too hgh computatonal cost. Furthermore, whle drect methods lke LU decomposton mght be applcable for two-dmensonal problems they are computatonally too expensve for three-dmensonal problems. Thus, the research on effcent solvers for the Helmholtz equaton wth varyng wave number has concentrated on developng good precondtoners. The man technques employed to construct these precondtoners are doman decomposton methods, multgrd methods, and fast drect solvers. Plessx and Mulder 13 studed tensor product form precondtoners based on fast drect solvers. Whle for low frequency problems they were very effectve ther performance 1

2 deterorated severely when the frequency approached practcally nterestng level. The authors 6,8 used a smlar approach wth good success for almost perfectly vertcally layered meda. Larson and Holmgren 12 developed a doman decomposton precondtoner employng a fast drect solver and tested t on a parallel computer. Erlangga, Oosterlee, and Vuk 3 developed a multgrd precondtoner whch s farly effectve for geologcal survey problems. In ths paper, we develop a Schwarz-type multplcatve doman decomposton precondtoner. The subdoman precondtoners employ a fast drect solver va embeddng the subdomans nto rectangles. By fast drect methods, we mean drect methods based on FFT or cyclc reducton, for example, whch requre order of m log m or m(log m) 2 operatons to solve a system wth m unknowns. Wth a pecewse constant wave number, the precondtoner can be made to match the system matrx n the nteror of the subdomans. Ths has two mportant consequences: The resdual vanshed n the nteror of the subdomans and due to ths the teratons can be reduced on the nterfaces and ther near-by grd ponts. Thus, the soluton procedure can be consdered as a precondtoned teratve method on a sparse subspace 9,10. Due to ths the memory requrement of the GMRES method 15 s greatly reduced. Secondly, the condtonng of the precondtoned system s mproved due to the matchng n the nteror of the subdomans. The numercal examples demonstrate the effectveness of the developed approach. 2 MODEL PROBLEMS In the rectangular computatonal doman Π, the pressure feld p satsfes the Helmholtz partal dfferental equaton p k 2 p = g, (1) where k s the wave number gven by k = ω, ω s the angular frequency, c s the speed of c sound, and g corresponds to a sound source. We assume the speed of sound and, thus, also the wave number to be pecewse constant functons. On the boundares Π, we mpose a second-order absorbng boundary condton 1,8 p n = kp p on Π 2 s k s (2) p n = 3k 4 p at C, where n and s denote the unt outward normal and tangent vectors, respectvely, and C denotes the set of the corner ponts of Π. 3 NUMERICAL METHOD 3.1 Dscretzaton We dscretze the Helmholtz equaton on a unform grd wth the grd step sze h. The dscretzaton stencls for nteror grd ponts, boundary grd ponts, and corner grd ponts 2

3 1 4 (k(x + h/2, y + h/2) h) (3) Kazufum Ito and Jar Tovanen are gven by Fgure 1, where the dagonal weghts are defned by d = 4 1 (k(x ± h/2, y ± h/2) h) 2, 4 b = ( ) k(x, y ± h/2) h 1 (k(x + h/2, y ± h/2) h) 2, 2 k(x, y ± h/2) h 4 a = ( ) k(x, y + h/2) h + k(x + h/2, y) h 2 k(x, y + h/2) h k(x + h/2, y) h (1 + k(x,y+h/2) h ) 1 2 (1 + k(x,y+h/2) h ) 1 d 1 b 1 a 1 2 (1 + k(x+h/2,y) h ) (1 + k(x,y h/2) h ) Fgure 1: The dscretzaton stencl at (x, y) for an nteror grd pont (left), a grd pont on the left boundary (mddle), and the left lower corner pont (rght). The weghts d, b, and a are gven by (3). The dscretzaton leads to a system of lnear equatons 3.2 Iteratve soluton procedure Ap = b. (4) Instead of solvng the system of lnear equatons (4), we solve teratvely usng the GMRES method 15 a left precondtoned system AB 1 q = b, (5) where B s the precondtoner. Once we have obtaned q the soluton of the orgnal problem s gven by p = B 1 q. In order to descrbe our doman decomposton precondtoner B, we defne each subdoman as the unon of cells n whch the speed of sound s a gven constant. More precsely, let c j, j = 1,..., n, denote the dfferent speeds of sound. The closure of the jth subdoman Ω j s defned by Ω j = [x h/2, x + h/2] [y h/2, y + h/2], (6) c(x, y) = c j (x, y) M 3

4 where M s the set of cell mdponts. Let the rectangular matrx R j correspond to the restrcton operator nto the subdoman Ω j. Multplyng a vector by t results a vector contanng the components assocated to the subdoman Ω j and ts boundary. We denote the precondtoner n Ω j by B j and t wll be defned n Secton 3.3. We employ a Schwarz-type multplcatve doman decomposton precondtoner B = P n whch s defned recursvely as P 1 j = P 1 j 1 + RT j B 1 j R j (I AP 1 j 1 ), j = 2,..., n, (7) where P 1 1 = R T 1 B 1 1 R 1. Thus, we obtan y = y (n) = B 1 x by performng the followng sequence of operatons: 3.3 Subdoman precondtoners y (1) = R T 1 B 1 1 R 1 x, y (2) = y (1) + R T 2 B 1 2 R 2 (x Ay (1) ),. y (n) = y (n 1) + R T n B 1 n R n (x Ay (n 1) ). Systems of lnear equatons correspondng to general shaped subdomans cannot be solved usng a fast drect solver. For ths reason, we embed each subdoman nto a larger rectangular doman and we defned the subdoman precondtoner as a Schur complement matrx. In order to descrbe ths more precsely, we defne the mnmum and maxmum values of the coordnates for each subdoman Ω j as x j mn = argmn (x,y) Ω j x, x j max = argmax (x,y) Ω j x, y j mn = argmn (x,y) Ω j y, and y j max = argmax (x,y) Ω j y. Smlarly, we defne x mn, x max, y mn, and y max for the computatonal doman Π. Then the extended rectangular subdomans are defned by Ω j = [ max{x j mn lh, x mn}, mn{x j max + lh, x max } ] [ max{y j mn lh, y mn}, mn{y j max + lh, y max } ], where l s a nonnegatve nteger. The jth subdoman precondtoner s based on the Helmholtz equaton (8) (9) (10) p k 2 j p = g n Ω j, (11) where k j = ω/c j and the absorbng boundary condtons (2) are posed on the boundares Ω j. By usng the same grd and dscretzaton for the problem (11), we obtan a matrx C j whch has a block form ( ) Cj,dd C C j = j,de (12) C j,ed 4 C j,ee

5 once the unknowns correspondng to the subdoman Ω j are numbered frst (subscrpt d) and the unknowns correspondng to the extenson Ω j \ Ω j are numbered second (subscrpt e). The precondtoner B j for the jth subdoman s gven by a Schur complement matrx A system of lnear equatons B j y j = x j can solved as B j = C j,dd C j,de C 1 j,ee C j,ed. (13) y j = ( 1 0 ) C 1 j ( ) 1 x 0 j (14) usng a fast drect solver lke the cyclc reducton method 5,14. Ths soluton requres order of m j log m j floatng pont operatons, where m j s the sze of C j. 3.4 Sparse subspace teratons The GMRES method 15 forms and stores bass vectors for a Krylov subspace whch after l teratons s K l (v) = { v, AB 1 v, (AB 1 ) 2 v,..., (AB 1 ) l v }, (15) where v s the ntal resdual. For example, f the ntal guess for soluton n the GMRES method s zero then v = b. In general case, storng the bass vector requres a vast amount of memory for large problems unless the teratons converge already after a few teratons. The vectors v j, j = 1,..., l, defnng K l (v) are gven by v j = AB 1 v j 1 = [(A B) + B] B 1 v j 1 = (A B)B 1 v j 1 + v j 1, (16) where we have denoted v 0 = v. It s easy to see from (16) usng nducton that v j+1 X {v}, where X = range(a B). (17) The subspace X corresponds to the rows of the matrces A and B whch dffer. Due to the constructon of the precondtoner, these rows are assocated to the nterfaces between subdomans and ther near-by grd ponts 4,9,10. Thus, the subspace X s very sparse. For a two-dmensonal problem, the number of nonzero components n vectors on X s of order m, where m s the sze of the system of lnear equatons. If the ntal guess s chosen to be b then the ntal resdual v belongs to X and, thus, K l (v) X, that s, the Krylov subspaces are sparse. Alternatvely, X {v} s sparse when b s sparse and the ntal guess s zero. These observatons reduce the memory usage by a large factor and allows us to use the GMRES method wthout restarts. Furthermore, the so-called partal soluton technque 2,11 can be used to solve the systems of lnear equatons wth B on the sparse subspace 4,9. 5

6 4 NUMERICAL RESULTS In all experments, the stoppng crteron for the GMRES method was that the norm of the resdual s reduced by the factor In all embeddngs, we extended the subdomans by one mesh step, that s, l = 1 n (10). The sound source s always a pont source at the mdpont of the upper boundary. The ntal guess for the soluton n the GMRES method s zero for all problems. All experments are performed wthout GMRES restarts. 4.1 Perfectly vertcally layered medum These test problems are defned n the unt square and we use a grd to solve these problems. In the frst test problem, the speed of sound s constant c = 1 n the whole doman. However, we dvde the unt square nto two and four equally szed subdomans whch are extended accordng to (10) for precondtonng. The number of GMRES teratons for the resultng precondtoned systems are gven by the thrd and fourth column of Table 1. The next two problems have perfectly vertcally layered meda shown n Fgure 2. The number of GMRES teratons for these problems are gven by the two last columns of Table 1. The real part of the soluton at the frequency f = 25 s shown by Fgure 2. c = 1 c = 1 c = 2 c = 2 c = 3 c = 4 Fgure 2: Layered test problems n the unt square and the soluton for the four layer problem at f = 25. homog. homog. layered layered subdomans f grd ter. ter. ter. ter Table 1: The number of teratons for dfferent test problems and frequences f. 6

7 4.2 Wedge problem The speed of sound for the wedge test problem 3,13 s shown n Fgure 3. The doman decomposton s gven by Fgure 4 together wth the one grd step extensons for the subdomans. Table 2 gves the number of GMRES teratons and CPU tmes n seconds at dfferent frequences on a PC wth 3.8 GHz Xeon processor. The real parts of the solutons at f = 30 Hz and 50 Hz are plotted n Fgure m c = 2000 m/s 500 m c = 1500 m/s 400 m 100 m m m c = 3000 m/s 600 m Fgure 3: The wedge test problem and the solutons at f = 30 Hz and 50 Hz. subdoman Ω 1 subdoman Ω 2 subdoman Ω 3 Fgure 4: The subdomans Ω j for the wedge test problem (red and black), ther extensons Ω j \ Ω j (green), and the assocated parts of the sparse subspace X {v} (black). 4.3 Salt depost problem Ths test problem mmcs a smple salt depost. The length of the sde of the square computatonal doman s chosen to be 600 meters. The profle for the speed of sound s shown n Fgure 5. It s perfectly vertcally layered except for the salt dsk wth the radus 7

8 f [Hz] grd ter. tme Table 2: The number of teratons and CPU tmes n seconds for dfferent frequences f for the wedge problem. of 150 meters and the center at (300 m, 250 m) when the lower left corner s at orgn. We consder three frequences f = 30 Hz, 60 Hz, and 120 Hz and three grds , , and Table 3 gves the number of GMRES teratons for all possble combnatons of frequences and grds. Fgure 6 shows the real parts of the solutons at the consdered frequences. The doman decomposton s shown n Fgure m/s 2500 m/s 3500 m/s 4500 m/s 5500 m/s Fgure 5: The speed of sound for the salt depost problem. Fgure 6: The solutons for the salt depost problem at f = 30 Hz, 60 Hz, and 120 Hz. 8

9 subdoman Ω 1 subdoman Ω 2 subdoman Ω 3 subdoman Ω 4 subdoman Ω 5 Fgure 7: The subdomans Ω j for the salt depost problem (red and black), ther extensons Ω j \ Ω j (green), and the assocated parts of the sparse subspace X {v} (black). grd f [Hz] Table 3: The number of teratons for dfferent frequences f and grds for the salt depost problem. 5 CONCLUSIONS We proposed a Schwarz-type multplcatve doman decomposton precondtoner for the Helmholtz equaton wth a pecewse constant wave number. The computatonal effcency results from the use of fast drect solvers for subdoman precondtonng and the good condtonng of the precondtoned system. Furthermore, teratons can be reduced on the subdoman boundares and ther neghborng grd ponts. The number of GMRES teratons requred to reduce the resdual by a gven factor depends only mldly on the grd step sze and frequency. The numercal experments demonstrated the method to be effcent for two-dmensonal problems wth realstc frequences and jumps of the speed of sounds encountered n acoustcal geologcal survey problems. The proposed method can be generalzed n a straghtforward manner for three-dmensonal problems. ACKNOWLEDGEMENTS The research was supported by the Academy of Fnland grant # and the Offce of Naval Research grant N REFERENCES [1] A. Bamberger, P. Joly, and J E. Roberts, Second-order absorbng boundary condtons for the wave equaton: a soluton for the corner problem. SIAM J. Numer. Anal., 27, , (1990). [2] A. Banegas, Fast Posson solvers for problems wth sparsty, Math. Comp., 32, , (1978). 9

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