Hom Srch Collctions Journls About Contct us My IOPscinc Positiv-nrgy D-br mthod for coustic tomogrphy: computtionl study This contnt hs bn downlodd from IOPscinc. Pls scroll down to s th full txt. 6 Invrs Problms 3 53 (http://iopscinc.iop.org/66-56/3//53) Viw th tbl of contnts for this issu, or go to th journl hompg for mor Downlod dtils: This contnt ws downlodd by: ssiltnn IP Addrss: 8.4.5.9 This contnt ws downlodd on 4/7/6 t 6:3 Pls not tht trms nd conditions pply.
Invrs Problms Invrs Problms 3 (6) 53 (35pp) doi:.88/66-56/3//53 Positiv-nrgy D-br mthod for coustic tomogrphy: computtionl study M V d Hoop, M Lsss, M Sntcsri,4, S Siltnn nd J P Tmminn 3 Dprtmnt of Computtionl nd Applid Mthmtics (CAAM), Ric Univrsity, 6 Min St. MS 34, Houston, TX, USA Dprtmnt of Mthmtics nd Sttistics, PO Box 68 (Gustf Hällströmin ktu b), FI-4 Univrsity of Hlsinki, Finlnd 3 Dprtmnt of Mthmtics, Tllinn Univrsity of Tchnology, Ehitjt t 5, 66 Tllinn, Estoni E-mil: mdhoop@ric.du, mtti.lsss@hlsinki.fi, mtto.sntcsri@hlsinki.fi, smuli.siltnn@hlsinki.fi nd jnn.tmminn@ttu. Rcivd July 5, rvisd 3 Novmbr 5 Accptd for publiction Dcmbr 5 Publishd Jnury 6 Abstrct A nw computtionl mthod for rconstructing potntil from th Dirichltto-Numnn (DN) mp t positiv nrgy is dvlopd. Th mthod is bsd on D-br tchniqus nd it works in bsnc of xcptionl points in prticulr, if th potntil is smll nough comprd to th nrgy. Numricl tsts rvl xcptionl points for prturbd, rdil potntils. Rconstructions for svrl potntils r computd using simultd DN mps with nd without ddd nois. Th nw rconstruction mthod is shown to work wll for nrgy vlus btwn 5 nd 5, smllr vlus giving bttr rsults. Kywords: coustic tomogrphy, gophysicl prospcting, rconstruction lgorithm, non-locl Rimnn Hilbrt problm, D-br qution, Dirichlt-to- Numnn mp, xcptionl points (Som figurs my ppr in colour only in th onlin journl). Introduction Indirct msurmnts cn oftn b ccurtly modlld using boundry vlu problms involving prtil diffrntil qutions (PDE). Th nd to intrprt such msurmnts lds to invrs problms whr on ims to rcovr sptilly vrying PDE cofficints from 4 Author to whom ny corrspondnc should b ddrssd. 66-56/6/53+35$33. 6 IOP Publishing Ltd Printd in th UK
Invrs Problms 3 (6) 53 boundry dt. Exmpls includ lctricl impdnc tomogrphy (EIT) nd coustic tomogrphy (AT). Th nonlinrity nd ill-posdnss of th bov kind of invrs problms cll for spcilid solution mthods. Idlly, th mthodology should provid computtionlly fsibl instructions for rconstructing th cofficint of intrst in nois-robust wy, nd rgulrition nlysis for th rsult. On promising pproch is th us of nonlinr Fourir trnsforms tilord to th problm t hnd. Known s th D-br mthod, it lrdy works for EIT in prctic [9,, 3, 4, 9, 3, 37], nd th thory for AT is wll-known [4, 3, 3]. A numricl mthod for AT ws rcntly prsntd in [4, 5] whr th clssicl nd gnrlid scttring mplitud r computd from th msurmnts s rconstruction stp. In this ppr w provid nw numricl implmnttion of th originl D-br mthod t positiv nrgy, pplicbl to AT, whr w us th nonlinr Fourir trnsform computd dirctly from msurmnts. Lt WÌ b th unit disk. Considr boundd function : W (,, ) which modls physicl dnsity nd stisfis () x c r >. Th prssur p stisfis th rducd coustic qution for tim-hrmonic wvs with frquncy ω, p + wkp = in W, ( ) whr k () x is th comprssibility nd th spd of sound is givn by c = ( k )-. Givn th boundry condition p = f on W, th invrs problm of AT is to rcovr ñ nd κ from th Dirichlt-to-Numnn (DN) mp L p,, : f wk. ( ) n W Assum for simplicity tht () x = h nd k () x = k for ll x nr W with som positiv constnts h nd k. Dfin not tht q = D - - ( wk - E ); ( 3) supp( q ) ÌWsinc w dfin th nrgy E by E = k h w >. ( 4) Th substitution u = - p trnsforms () into th Schrödingr qution (-D + qu ) = in W, ( 5) whr u = f on W nd q = q - E. This boundry-vlu-problm might b wll-posd; thn for ny f Î H ( W) it hs uniqu wk solution u Î H ( W ). In tht cs w dfin th DN mp u L : H q ( W) H- ( W), f, ( 6) n whr ν is th unit outr norml to th boundry. Mor prcisly for f, g Î H ( W), ( L f, g) = ( u v + quv) d x, ( 7) q W W whr u is th uniqu wk solution for th boundry vlu f, nd v Î H ( W) with v = g As mntiond in [7] w thn hv W W.
Invrs Problms 3 (6) 53 L q = Lwk,, - -, n so our ssumptions imply L q = h L wk,,. Assuming tht h, k nd ω r known, th invrs problm of AT thn tks th following form: givn Lq nd th nrgy E, rconstruct th potntil q. This is clld th Gl fnd Cldrón problm posd by Gl fnd [8] nd Cldrón [6]. W cn solv this problm using th D-br mthod bsd of xponntilly bhving complx gomtric optics (CGO) solutions first introducd by Fddv [7] nd ltr rdiscovrd in th contxt of invrs boundry-vlu problms by Sylvstr nd Uhlmnn [38]. Th D-br mthod is bsd on th boundry intgrl qution provd by Novikov [3], th D-br qution discovrd by Bls nd Coifmn [], nd th rltion of th CGO solution nd th potntil by Novikov [3]. S Nchmn [8] for discussion of th D-br mthod pplid to th AT problm. EIT nd AT r rltd to th Gl fnd Cldrón problm by trnsformtion rsulting to diffrnt nrgis: EIT is ro-nrgy problm with E = nd AT is positiv nrgy problm with E >. In th ro-nrgy cs in D, for conductivity-typ potntils, Nchmn [9] provd uniqunss nd rigorously justifid th D-br rconstruction. Th rsult ws ltr gnrlid by Bukhgim [3], who provd globl uniqunss for gnrl potntils t ny fixd nrgy. Th thr novltis of this ppr r: () W crt numricl lgorithm for Fddv Grn s function for positiv nrgy E >, significnt xtnsion of th ro-nrgy cs introducd in [36]. This is don in sction 3 ftr which th function will b usd throughout th numricl computtions. () W invstigt numriclly th xcptionl points which prvnt th strightforwrd us of th D-br mthod for rconstruction. This numriclly complmnts th rlir works [6, 7, 5, 6] focusing on th ro nd non-ro nrgy css. Th rsults cn b found in subsction 5.. (3) W propos nw numricl lgorithm for th D-br mthod t positiv nrgy nd tst it to rconstruct potntils using simultd DN-mps with nd without ddd nois. In contrst to othr mthods, our lgorithm is bl to do rconstructions t low nrgis. S sctions.3 nd 5.4 for comprisons with othr rconstruction schms ([3, 4,, 34]). Th lgorithm is dtild in sction 4 nd tstd in sction 5.3.. Prliminris.. CGOs solutions nd xcptionl points Lt q Î L ( W) b rl-vlud, nd E >. Rwrit (5) nd considr spcil xponntilly growing solutions y ( x, ) of (-D + q ) y(, ) = Ey(, ) in, ( 8) whr q is xtndd to th pln by ro, x = [ x, x T ] Î nd = [ T ] Î is spctrl prmtr with Im( ) ¹. Th xponntil bhviour is thn includd in th rquirmnt -i xy( x, ) -, s x +, ( 9) whr = + = E nd x = x + x. 3
Invrs Problms 3 (6) 53 Th rquirmnt = E coms from th fct tht for lrg x w cn writ xp( i x) in plc of y ( x, ), nd th qution (8) still hs to hold, giving us -D i x = E i x = E. ( ) If Î, w hv stting of physicl scttring of prticl with momntum ζ. If Im( ) ¹, w cll th solutions y ( x, ) CGOs or Fddv-typ solutions (first introducd in [7]). In ordr to prcisly dfin th CGO solutions, w first writ m( x, ) = - i x y( x, ) nd cll it CGO solution s wll. This nw function stisfis nothr diffrntil qution; strting from (8) w gt whr (-D + q E x - ) i m( x, ) = i x( -D - i + + q - E) m( x, ) = ( L + q ) m( x, ) =, L -D - i. Th Grn function g ( x) of th oprtor L ζ is clld Fddv Grn s function nd it is xplicitly givn by th formul g () x = 4p iyx d, y y y y ( ) + for x Î nd Im( ) ¹. Thus w dfin m ( x, ) s th solution of th following Lippmnn Schwingr typ qution m( x, ) = - g ( x) *( q ( x) m( x, )). ( ) For givn potntil q, w cll Î n xcptionl point if nd only if intgrl qution () dos not dmit uniqu solution in L ( ). Whn Im( ) =, formul () nd qution () mk no sns; howvr, th following limits cn b dfind: y ( x, ) = y( x, + i g), g () x = g () x, ( 3) g g + ig m ( x, ) = m( x, + i g), ( 4) g whr, g Î, = E, g = nd f ( + ig) = lim f ( + ig) Following [4], w mk th chng of vribls +. E l i x i x,, l = + l = + + =. ( 5) E i E - l l W cll th nw prmtr λ lso th spctrl prmtr. Dpnding on whthr w us ( x, ) -or(, l) -nottion, in plc of y ( x, ), m ( x, ) nd g ( x) w writ y (, l ), m (, l) nd gl () rspctivly, vn if th nrgy is thn omittd. In th numricl prt of th ppr w clrly indict which nrgy lvl w r using in diffrnt numricl tsts. 4
Invrs Problms 3 (6) 53 Lt l = r xp( iq). Rwriting (5) givs E cos( ) sin( ) q q = r + i r. ( 6) r sin( ) + - r cos( ) q - q It is sy to s tht r implis Im( ), mning tht th CGO solution gos to th limit of physicl scttring. For l = (r = ) w thn dfin: y (, l) = y(, l( )), m (, l) = m(, l( )), ( 7) l g () = g (), ( 8) l ( ) whr f ( ) = lim + f ( )... Th D-br qution nd th boundry intgrl qution All of th following is includd in th pprs [3, 3] nd th survy [3] with diffrnt nottion. Rcll our ssumptions of q to b rl vlud nd E >. Dfin th diffrntil oprtors w = ( w - i w ), w = ( w + i w ), whr w = w + iw, nd th xponntil functions -l i E () xp - ( + )( l + l), ( 9) ll E xp i l () ( + )( l + l). ( ) ll For Î with = E, not n xcptionl point, w cn dfin, for th corrsponding λ, th non-physicl scttring trnsform by = l t ( l) () q () m(, l) drdim, ( ) hr dr dim stnds for th stndrd Lbsgu msur on th pln, i.. dx dx, sinc = x + ix. W hv th following symmtry tht w will us ltr: t( l) = t( l). ( ) Furthr, w dfin th functions h ±, i h ( l, l, E) = xp E( ) - l + l p q () y (, l) drdim, ( 3) for l = l =. It is lso usful to introduc th following uxiliry functions h, h, l l h( l, l ) = c - - h ( l, l + + ) i l l h (, ), i l l - c - l l + - l l ( 4) 5
Invrs Problms 3 (6) 53 l l h ( l, l ) = c - - h ( l, l + - ) i l l h (, ), ( 5) i l l - c - l l + + l l whr c + is th Hvisid stp function, nd ρ, solution of th following intgrl qutions, (, ) i (, ) i l l rl l + p rl l c - + l = l l h( l, l ) dl = - pi h( l, l ), ( 6) l l rl (, l ) + pi rl (, l ) c - - + l = i l l h ( l, l ) dl = - pi h ( l, l ), ( 6b) for l = l =. Hr nd throughout th ppr, dl (or ltr dl ) stnds for th surfc msur on { l Î : l = }. Th function μ stisfis th following non-locl Rimnn Hilbrt problm (NLRH) (s [4] nd [3]). W hv: ( t, ) sgn ( ) ( l ) lm l = l - -l () m (, l ), ( 7 ) 4 pl for λ not n xcptionl point nd l ¹, for l =, whr In ddition w hv l = m (, l) = m (, l) + r( l, l, ) m (, l ) d l, ( 8) + - r E,,, xp i ( l l ) = r ( l l ) ( ). ( 9) l - l + l - l lim m(, l ) =, ( 3) l m () - m(, l) = + +, for l, ( 3 ) l l q () = i E m - (). ( 3) Dfin th oprtors : f (, l) = - f (, w) d w, ( 33) p w - l t : f (, ) sgn( ) ( l l = l - ) -l () f (, l), ( 34) 4 pl dw : f (, l) = r( w, l, ) f (, l ) d l +, ( 35) pi w = w - l l = 6
Invrs Problms 3 (6) 53 whr f+ (, l) is th limit of f (, l) whn l s dfind in (7). By pplying th Cuchy Grn formul to (7) nd (8) th CGO solution m (, l) stisfis th intgrl qution m(, l) = - ( - ) m(, l), ( 36) whr for ch fixd ÎWw cn solv m (, l). W now rviw th rconstruction schm to obtin th scttring dt t( l) nd h ( l, l ) from th DN dt. Dfin th oprtors for Î W, l ¹ ( l + ) l ( f)( ) G ( - y) f( y) d s( y), G ( ) = l g ( ), ( 37) l l l W nd i E ( l + ) l l l l W ( f)( ) G ( - y) f( y) d s( y), G ( ) = l g ( ), ( 38) for Î W, l =. Th CGO solutions y (, l) nd y (, l) stisfy th boundry intgrl qutions [3] ( I + ( L - L )) y(, l) = l, for l ¹ ( non xcptionl ), ( 39) l q - E W l i E ( l+ ) W ( I + ( L - L )) y (, l) = l, for l =, ( 4) q -E W i E ( l+ ) W whr th DN-mp L - corrsponds to th potntil q =. In conjunction w hv i E E t( l) = ( l+ l ) ( L - L ) y(, l) d, for l ¹ ( non xcptionl ), ( 4) W q -E i E - h, ( ) ( l l l + l ) = ( q E) (, ) d, ( 4) ( p) L - L- y l W for l = l =, whr d stnds for th surfc msur on W. Thus w hv th ncssry stps to rconstruct th potntil q from th DN-mp L q, nmly: () Solv y (, l ) W nd y (, l) W from th boundry intgrl qutions (39) nd (4), rspctivly. () Comput th scttring trnsforms t( l) using (4) nd h ( l, l ) using (4). (3) Comput r ( l, l ) solving on of th qutions (6). (4) Choos rconstruction point nd solv m (, l) from (36). (5) Comput q ( ) from (3). W will rstrict th clss of potntils to thos with smll (clssicl) fixd-nrgy scttring mplitud, i.. smll ρ. A potntil such tht r º is sid to b trnsprnt nd it is wll known tht thr r no non-ro compctly supportd trnsprnt potntils [3]. Howvr, sinc th scttring trnsform t is rltd to ρ by nlytic continution tchniqus (s [3, sction 7]), its si (s wll s th si of th rltd potntil), roughly spking, cn b lrg vn for smll ρ. Thus, for potntils with smll ρ th lgorithm is simplifid by ssuming r º nd using only th trm t. W quntify th rror in lmm.. S [5] for mor discussions bout trnsprnt potntils nd [9 ] for th similr phnomnon of invisibility. Thus, w will us th following lgorithm: () Solv y (, l ) W from th boundry intgrl qutions (39). 7 i E
Invrs Problms 3 (6) 53 ( Comput th scttring trnsform t( l) using (4) (3) Choos rconstruction point nd solv m (, l) from (36) without th trm. (4) Comput n pproximtion q ( ) of q ( ) using (3). Lmm.. Lt WÌ b opn boundd domin with C boundry nd lt q Î W m, ( W ), rl-vlud, with supp( q ) ÌWnd m 3. Assum tht q m, N nd tht E > E( N, W) is sufficintly lrg (in prticulr thr r no xcptionl points). Lt t b th scttring trnsform dfind in () nd ρ b th function dfind in (6). Lt q b th potntil obtind solving th NLHM problm (7), (8) with scttring dt givn by t nd r º. Thn thr is constnt C = C( W, N, m) > such tht q - q C( W, N, m) E r, ( 43) whr T = { l Î : l = }. L ( W) L ( T ) Th proof of lmm. is givn t th nd of this sction. Rmrk.. W wnt to undrlin tht th ssumptions md on th potntil nd th nrgy r ndd for rigorous justifictions of our mthod. Numricl rsults prsntd in sction 5 strongly suggst tht our lgorithm prforms wll t low nrgis for ny L potntil (in bsnc of xcptionl points). S lso sction.3 for mor discussions bout th nrgy. W will now giv n intrprttion of our lgorithm in th Born pproximtion. Lt E > b fixd nd considr th clssicl scttring mplitud f ( k, l), k, l Î with k = l = E (s for instnc [3, (.3)] for dfinition). From [3, thorm 5., proposition 5.] f, h ± nd ρ r connctd through intgrl qutions nd, roughly spking, thy contin th sm informtion on potntil q. Assuming th Born pproximtion, i.. q L ( W) E, w hv tht f ( k, l)» [ q]( k - l), whr is th D Fourir trnsform. Thus th clssicl scttring mplitud f ( k, l) t fixd nrgy E >, or quivlntly r ( l, l ), dtrmins [ q]( p) for p E. Undr th sm ssumptions, th non-physicl scttring trnsform t( l) dtrmins [ q]( p) for p E. In prticulr, in sction 4.3 w will considr trunctd scttring trnsform t R, which is for l < R nd l > R, for R >, nd qul to t othrwis. In th Born pproximtion, t R dtrmins [ q ]( p) for ( ) E p E R +. Intuitivly, th proposd lgorithm llows th rconstruction of R th Fourir trnsform of potntil in th nnulus E p E( R + R) : thus it provids good rsults for dt cquird t fixd nrgy E fr from nd +. In sction 4 w giv mor dtild numricl lgorithm for this rconstruction procdur..3. Rltd work A rconstruction lgorithm for th Gl fnd Cldrón problm t fixd positiv nrgy ws proposd by Novikov nd on of th uthors in [34] (s [4, 5] for numricl rsults). This lgorithm nd th on prsntd in this ppr hv similr thorticl bckground but prsnt svrl diffrncs: In th lgorithm of [34], only th scttring function ρ (or h ± ) is rconstructd from th DN mp nd usd in th solution of th Rimnn Hilbrt problm. In th prsnt ppr w us only th scttring trnsform t. 8
Invrs Problms 3 (6) 53 Th lgorithm in [34] is Lipschit stbl with n rror trm dpnding on th nrgy. Our proposd lgorithm is logrithmic stbl, with n rror quntifid by lmms. nd 4.. Concrning spd, th most computtionlly xpnsiv stps in th lgorithm in [34] r two D linr intgrl qutions, whil in our mthod thy r D linr intgrl qution nd -qution (D linr intgrl qution). Th formr lgorithm is thn fstr thn th lttr. Both lgorithms, to b rigorously justifid, nd th nrgy to b sufficintly lrg with rspct to th L norm of th potntil. But numricl vidncs show tht thy prform wll in diffrnt nrgy rngs: th lgorithm in [34] t high nrgis, whil th prsnt on t low nrgis. In figur 6 w prsnt numricl comprison of th two lgorithms t diffrnt fixd nrgis: it is clr tht t low nrgis our mthod provids bttr rconstructions. Along with th rsults of figurs 6 nd 7 w conjctur tht for potntils q such tht q L ( W) = our mthod prforms wll, whn - 5 E 5. In conclusion, dspit our mthod prsnt svrl disdvntgs with rspct to th on proposd in [34], it is, s fr s w know, th only known lgorithm producing good rconstructions t low positiv nrgis. Othr rconstruction lgorithms for this problm hv bn proposd. Th fundmntl ppr [3] provids rconstruction mthod without ny ssumptions on th nrgy; this lgorithm is lss stbl thn th on prsntd in this ppr, sinc it is bsd on proprtis of gnrlid scttring dt for only lrg complx prmtrs. Vry rcntly, in th prprint [] nw globl rconstruction mthod ws proposd (without ny ssumptions on th nrgy). This mthod is bsd on gnrlition of th Rimnn Hilbrt problm introducd in sction., which is bl to dl with th prsnc of xcptionl points. To our knowldg, no numricl studis bsd on [3] or [] hv bn prsntd yt. Proof of lmm.. Thnks to th ssumptions on th potntil nd th nrgy, th NLHM problm (7), (8) is solvd with scttring dt ( t, r) nd ( t, ) (s [3, thorms 6. nd 6.] for proof). Estimt (43) is dirct consqunc of tchnicl rsults of [35] usd to prov stbility stimt for this problm. W cn rpt th rgumnts of [35, sction 4] in ordr to obtin th qulity [35, idntity (4.)] q() - q () = i E( A - A + B - B + C - C ), ( 44) whr A, A, B, B, C, C t r constructd s A, B, C in [35, sction 4] with sgn( ) ( l l - ) 4 pl instd of r ( l ), nd with ρ nd instd of ρ, rspctivly. In [35, sction 5] th right-hnd sid of (44) is stimtd in trms of scttring dt. Sinc both potntils r rconstructd from th sm non-physicl scttring trnsform t, w hv tht A - A º (this follows from th stimt ftr [35, stimt (5.6)]). For th sm rson, from th stimt ftr [35, stimt (5.8)] w hv nd for C B - B c( W, N, m) E r, ( 45) - C th sm rgumnt givs L ( T ) C - C c( W, N, m) E r. ( 46) L ( T ) Th proof follows from ths stimts nd idntity (44)., 9
Invrs Problms 3 (6) 53 3. Computtion of th Fddv Grn s function for positiv nrgy As in th prvious sction, w will idntify th pln s th complx pln by writing x = [ x x ] T Î, = x + i x Î. Rcll lso th trnsformtion (5) btwn λ nd ζ prmtrs. W nd numricl lgorithm for th Fddv Grn s function gl () for ny point nd ny λ with l >, s th symmtry (). W rmrk tht in th cs = E = nd Im( ) ¹ th numricl computtion of g () ws first prsntd in [36] nd thn usd in th contxt of th invrs conductivity problm in [37]. Th mthod ws ltr rfind in [8]. This pproch is bsd on implmnting pproprit numricl intgrtions dpnding on th loction of th vlution point. Th ro-nrgy computtion cn lso b simply implmntd using [, formul (3.)] nd Mtlb s built-in xponntil-intgrl function: g=xp(-i * ). * rl(xpint (-i * ))/( * pi). Th Fddv Grn s function gl () for positiv nrgy hs similr scling nd rottionl proprtis s in th ro nrgy cs. Howvr, our pproch is bsd on g (), = E > : th min rson is tht w cn us rsidu clculus to comput simplr formuls in similr wy to [36]. Whn using th ζ prmtrs, th scling nd rottionl proprtis will b mployd diffrntly from th ro-nrgy cs. Thus th formuls obtind in this sction cnnot b dirctly obtind from [8, 36, 37] nd this rgion-bsd mthod hs to b modifid. Rcll th formul (), g () = 4p Th following rltions cn b sn. iy d. y y + y y Lmm 3.. Lt Î { }, R rottionl mtrix with dt( R) = nd R = R( R( )) + ir( Im( )). Thn th Fddv Grn s function g () with = E > stisfis g ( ) = g ( ), ( 47) g R gr -, ( 48 x g ( x g, ) - = - () ( 49) x g ( x g, 5 ) - = () ( ) - g () = g (-). ( 5) W cn us th rottion rltion (48) to rduc ζ to th form k i, k k. ( 5) = + k > >
Invrs Problms 3 (6) 53 For this rducd ζ, using rltions (5) nd (5) w hv x g ( ) g ( ) g( x, ) - = = - which rsults to switching rltion x x g ( x g x. 53 ) ( ) - = ( ) Th strtgy for computing gl () is now th following: () Us (5) to comput ζ from λ. () Find th rottionl mtrix R tht stisfis R ( Im( )) = [, k T ] for som k > ; thn writ g () = g - () = g ( R ), R R R whr R is in th rducd form (5). (3) Th smllr th mor computtionl problms w hv s will b sn ltr. Thus, for vry smll w us mthod of singl lyr potntil dscribd in sction 3.. Furthr, us rltion (47) to scl points outwrds nd rltion (53) to switch from x < to x. (4) Us computtionl domins to comput g () for rducd ζ nd with x. It tks som nlysis to find suitbl computtionl domins for th lst stp. Assum w hv th rducd ζ of (5) nd th switchd = x + ix with x. Writ t = y + k, = ( y + ki) - E nd subsquntly th dnomintor of th intgrnd in () s y y + y = y + y + y k y k i + = ( y + k) + ( y + ki) - E = t + = ( t + i )( t - i ). W dfin th squr root in th sm wy MATLAB clcults it by dfult, tht is for complx numbr = r xp( i q), q < p, r th squr root is = r iq, q p, r -i ( p-q), p < q < p. This wy th squr root hs th following proprtis: for ny Î w hv R ( ),, ( 54) =. ( 55) Th numrtor of th intgrnd in () bcoms iy = i( x( t- k) + xy) = i( xy -xk ) ixt. Th intgrl in () is thus trnsformd into iy y y y y d ixt i( xy xk) dt d y. = - + - - ( t + i )( t - i )
Invrs Problms 3 (6) 53 Th intgrl ovr th rl prmtr t is complxifid with w = w + iw Î nd w writ f ( w) = Th pols of th function f(w) r R I ixw ixw R-xw I =. ( 56) ( w + i )( w - i ) ( w + i )( w - i ) i, whr = ( y + ki) - E = y - E - k + y ki. It follows from our dfinition of tht whn y >, is in th uppr hlf pln, so i is in qudrnt nd - i in qudrnt 4 (not, tht k > ), nd whn y <, is in th lowr hlf pln, so i is in qudrnt nd - i in qudrnt 3. Whn wi, x w hv f ( w), s w. W choos th intgrtion pth G = [-R, R] È { Rxp( i q) : q p}, so whn R th ccpol w = i is insid th pth. Using rsidu clculus w gt Thus Bcus of (54) w hv p f ( w) dw = f ( w) dw = i Rs f ( w) Ì G w= i = pi lim ( w - i ) f ( w) = pi w i i -x = p. x g () = i xy -xk dy ( ) p + p - y 4 xk x y x = -i i d p + -ix y -x -x ix y = i x ( i ) + i - ( ) i( x y -x k ) - -x - ix y nd so th following formul is obtind. Lmm 3.. For x, g () =, p d y. -x dy R x t k i - ( + ) -E -ixk ixt d t k i E t. ( 57) p ( ) + - W wnt to numriclly comput th intgrl of (57). W cn only comput up to finit limit, sy from to T. Two problms might occur, th intgrnd ithr convrgs slowly, mning w hv to tk T vry lrg, or th intgrnd might oscillt fst, mning w hv
Invrs Problms 3 (6) 53 to tk grt numbr of intgrtion points in [, T]. W s tht problmtic situtions in using (57) ris whn x is lrg (oscilltion), x is smll (convrgnc) nd whn k is lrg (oscilltion). Whn x is lrg w hv oscilltion, but lso bttr convrgnc, mning (57) is usbl. Also worth noting is tht whn λ is clos to on, thn k is clos to ro nd th pols i r clos to th intgrtion pth Γ cusing numricl problms. Ths obsrvtions ld to dditionl vrsions of formul (57), usd by th diffrnt computtionl domins. Writ -x gw ( ) = ixw, = ( w + ki ) - E, w = w + i w Î, + nd considr th complxifid intgrl gw ( )dw of (57). W hv i xw= ixw -xw, so whn x, w or x, w < w hv gw ( ) s w. ( 58) This is bcus th numrtor xp( -x ) convrgs to ro, sinc x nd (54) holds. Th brnch points of g(w) r E - ki. Lmm 3.3. Lt x nd x, thn g t R x i t - + k + tk () = -ixk xt d -. ( 59) p t + k + tk Proof. W choos th intgrtion pth È G = [, R] { Rxp( i q) : q p } { ir( - t) : t }. W hv gw ( ) dw=, sinc g(w) is nlytic in th first qudrnt. Bcus of (58) w G thn hv È + - x i ( t+ k ) + E - gw d w g i t id t ( ) =- ( ) = xt d. t k E t ( + ) +, Th intgrnd in (59) convrgs to ro quickly for lrg x nd hs high oscilltion for lrg x. Lmm 3.4. Lt x nd x <, thn g I t R i -x b () = - ixk ( i E x xt - ( + ) d ), ( 6) p b E + whr I = gt () dt nd b = ( E + + ( k - t) i) - E. 3
Invrs Problms 3 (6) 53 Proof. Dfin th pths P = [, E + ], L = { E + - i Rt: t }, L E Rxp i : 3 = { + + ( q) p q p}, L3 = { E + + ( - t) R: t }. Th brnch point E - ki is voidd by intgrting long G = PÈ ( LÈ L È L3), whr g(w) is nlyticl insid th loop LÈ L È L3, nd gw ( ) on th circl L s R. Thus whr + gw ( ) dw= I+ g( E+ - it)( -i) d t, -x b g( E + - it)( - i) dt = i( E+ ) x x t (-id. ) t b Th intgrnd in (6) convrgs to ro quickly for lrg x nd hs high oscilltion for lrg x. 3.. Choosing uppr limits for th intgrls Writ g T,g T nd g T 3 for th finit intgrls for (57), (59) nd (6) rspctivly. W nd to choos th uppr limits T i, i =,, 3. Thr will b numricl rror cusd by th nglctd prt of th intgrl nd th numricl intgrtion mthod usd. It is dcidd to simply rquir T 8 i g - g < -. ( 6) Th rror inducd by th numricl intgrtion mthod is ssumd not to b dpndnt on λ or. For g T i, th intgrtion rng [, T i] is dividd into M i points (with g T 3 thr is lso th dditionl intgrl I ) nd th Gussin qudrtur is usd. Th intgrs M i r chosn lrg nough so tht for ny intgr M > M i th first ight digits r not chnging in th numricl T vlu of g i (). In this tst th choic of hs only minor ffct, it is don by choosing th T worst possibl point for ny givn computtionl domin; for xmpl, for g () using = [, ] th intgrnd hs mor oscilltion thn with th point = [, ], nd thus nds lrgr prmtr M. Finding T i is bit cumbrsom. Th following proposition gurnts us th rror rquirmnt (6). Proposition 3.. T Choos mx 4 4 =, k, ( 6) xc, T = 4, ( 63 ) x 4
Invrs Problms 3 (6) 53 4 T3 = cx-x whr c, c r constnts dpnding on k nd k. Thn + k, ( 64) T 8 i g ( ) - g ( ) < -, i =,,3. Proof. In (57) w hv th trm nd = ( t - k - E) + kti = (( t - k ) + 4k t ) t4 t = ( - k + k 4 + 4k t ) 4 t4 ( - k t ) 4 ( t4- t4 ) 4 t =, 4 whn t k. Thn, writing θ for th ngl = r xp( iq), R cos = t 4 t -x cos( q) t 4. t 4 -x -x ( ) -x ( q) 4 4 Th ngl gos to ro s t,socos( q). Sinc t k w writ c = cos( q), whr th ngl of t =k is q, nd so w hv for th intgrl x xct 4 - - ixt dt dt T T t 4 -s = 4 ds 4 = Ei ( xct xct 4 s 4). ( 65) Th xponntil intgrl function E i cn b computd in MATLAB with xpint.m. Bcus of (6) w rquir tht th rmindr (65) is of th ordr p 4-8» 7.47-8. W cn tst with MATLAB tht Ei ( 4) < 6-8,so w gt xct 4 = 4 from which (6) follows. From (59) w sily gt T t = E ( x T ). x i t - + k + tk -x t - xt dt T t + k + tk Thus th uppr limit (63) follows, s bfor, from xt = 4. i dt 5
Invrs Problms 3 (6) 53 Strting from (6) w hv b = (( k - t) 4 - ( E + )( k - t) + ( E + ) + 4( E + ) ( k - t) ) (( k - t) 4 ) 4= t - k. 4 Using th sm rgumnt s prcding (65), w writ c b is q. Thn for T3 > k w hv t = k nd (64) follows from T - x b xt-xc( t-k) x t 3 b dt ( cx- x)( T- k) = 4. 3 T t - k dt = cos( q ), whr th ngl of -s = xk ds ( cx -x)( T3-k) s = xk Ei (( cx- x)( T3- k)) E (( c x - x )( T - k )), i 3 3, 3.. Us of singl-lyr potntil for smll For smll vlus of thr is problm of slow convrgnc. W will vd this problm by th us of th singl-lyr potntil for function tht stisfis th Hlmholt qution. Writ E = k, G () = xp ( i ) g (), G() = i H ( k ) 4, whr H is Hnkl s function of th first typ. W hv (- - k) G ( ) = d (- - k) G( ) = d, so (- - k)( G - G) =. Writ H G - G. For ny rdius R thr xists singl-lyr potntil p(), which givs th vlu of H ζ by th intgrl i H ( ) = H ( k - y ) p( y) d m( y) S( p( ))( ). ( 66) D(, R) 4 Assum w know H () on th circl D(, R), whr R is lrg nough so tht w do not hv th problms of slow convrgnc. Th potntil cn b rcovrd by th invrs of th intgrl oprtor, p = S- ( H ( )). Thn H () cn b clcultd using (66) for ny < R. Finlly w hv g () = -i ( H () + G()). ( 67) Th numricl implmnttion of this submthod is strightforwrd with th dditionl trick tht th potntil p() is computd on circl of rdius R + > R so tht w void singulritis in th oprtor S. 6
Invrs Problms 3 (6) 53 3.3. Computtionl domins nd th computtion of g ζ ðþ For th rducd ζ w now hv th qutions T g T g g t R T - x t + tki-k () = -ixk ixt d, ( 68) p t tki k + - T t x R T - x i t + tk+ k () = -ixk -xt d,, 69 t tk k ( ) p + + t R E x t + - + tki-k () = -ixk i x t d p t + tki - k T3 -x b - ii( E+ ) x xt d t, x <, ( 7) b 3 whr b = ( E + + ( k - t) i) - E. S figur. In gnrl th point lis in on of th computtionl domins: In domin, w us th singl-lyr potntil nd th qution (67). In domin b, w scl th point to th nnulus D D using (47), g () = g ( 5 5x) = g ( 5x) 5 4 3 (not tht th nrgy E chngs vi this scling trnsformtion). In domin c w do th sm s bov with th scling fctor. In domin w us (68), sinc x is smll nd x is lrg. Th uppr limit T is computd from (6). In domin 3 w us (69), sinc x is smll nd x > is lrg. Th uppr limit T is computd from (63). In domin 7 w us (7), sinc x is smll, x < nd x is lrg. Th uppr limit T 3 is computd from (64). In domins 4, 5, 6 w us (53) to switch thm to domins 3,, 7 rspctivly.a smpl of th function gl () is picturd in, in 4 4 -grid of points, l = + i, E =. 4. Numricl implmnttion of th D-br mthod 4.. Simultion of msurmnt dt W us trunctd Fourir bsis to pproximt oprtors on th boundry W of th unit disk W= D (, ) by finit mtrics. Choos n intgr N > nd dfin th following bsis functions: f ( n )( q ) = n N N i nq, = -, ¼ p,. ( 7) Th dt of th invrs problm is th DN-mp (6). Solv th problm (-D + qu ) ( n) = in W, u ( n) = f ( n) on W ( 7) 7
Invrs Problms 3 (6) 53 Figur. Computtionl domins. Domin is th disk D, domin b is th nnulus D D, domin c is th nnulus D3 D. Domins, 3, 4, 5, 6 nd 7 form th nnulus D D. 4 3 Figur. Th rl nd imginry prts of gl () in 4 4 grid of points, l = + i, E =. for u ( n) using finit lmnt mthod. Dfin th mtrix Lq = [ u ( l, n)] by u( n) u ( l, n) = () l f d s. ( 73) W n 8
Invrs Problms 3 (6) 53 Hr l is th row indx nd n is th column indx. Th intgrtion cn b computd whn th st [, p) is dividd into discrt points. Th mtrix L q rprsnts th oprtor Lq pproximtly. W dd simultd msurmnt nois by dfining q L L + c G, ( 74) q whr G is ( N + ) ( N + ) mtrix with rndom ntris indpndntly distributd ccording to th Gussin norml dnsity (, ). Th constnt c > cn b djustd for diffrnt rltiv rrors Lq - Lq Lq, whr is th stndrd mtrix norm. Th DN-mp L - E is rprsntd by th mtrix L -E in similr wy, in th boundry vlu problm (7) w thn hv q =- E. 4.. Solving th boundry intgrl qution For ny λ with l ¹ w cn comput th mtrix rprsnttion Sl = [ s ( l, n)] of th oprtor l (37) dfind by s ( l, n) = s( n) f( l) d s, s( n) = G ( - y) f( n) ( y) d s( y). ( 75) W W Hr l is th row indx nd n is th column indx, nd th st [, p) is dividd into discrt points. Th functions y (, l), xp( i E ( l + l)), W W vc vc cn b xprssd s vctors y l, l rspctivly using th bsis functions f ( n). Thn, th boundry intgrl qution (39) is pproximtd by th qution vc vc l q -E l l ( I + S ( L - L )) y =, ( 76) vc whr I is th corrct sid unit mtrix. Ths r sily solvd for th vctors y l,by invrting th mtrix I + S ( L - L ) l q - E. 4.3. Trunction of th scttring trnsform Th computtion of th CGO solutions for l clos to nd + is computtionlly unstbl. For this rson w will clcult th vlus of th scttring trnsform only whn R < l < R, for som R > fixd. Th following lmm rigorously justifis th us of such trunction nd givs n xplicit stimt, ssuming som smoothnss of th potntil. As corollry w obtin tht th low frquncy prt of th potntil is symptoticlly clos to its non-linr low frquncy prt. Mor prcisly, th potntil rconstructd from th trunctd scttring trnsform on th nnulus { R < l < R} nd th on obtind from th trunctd Fourir trnsform on bll of rdius E R, coincid up to O (( ER) -( m - )) whr m is rltd to th rgulrity of th potntil. S corollry 4. for clrity. Lmm 4.. Lt WÌ b opn boundd domin with C boundry nd lt q Î W m, ( W ), rl-vlud, with supp( q ) ÌWnd m 3. Assum tht q m, N nd tht E > E( N, W) is sufficintly lrg, so tht thr r no xcptionl points. Fix R R ( N, m) > nd dfin tr( l) = t( l) cr ( l), whr t( l) is dfind in () nd c R is th chrctristic function of th nnulus R = { l Î : R l R}. Lt ρ b th function dfind in (6). Lt q R b th potntil obtind solving th NLHR problm (7), (8) with scttring dt givn by t R nd ρ. Thn thr is constnt C = C( W, N, m) > such tht 9 l
Invrs Problms 3 (6) 53 q q CE-( m-) R -( m- ). ( 77) - R L ( W) Proof. First w must vrify tht th potntil q R is wll dfind, tht is tht w cn solv th NLRH problm with scttring dt t R nd ρ. This is consqunc of rsults of [3]. Th NLRH problm cn b solvd with th formuls nd qutions of [3, thorm 6.]. Sinc tr( l) t( l) for vry l Î, th trunctd scttring trnsform stisfis th stimts rquird in [3, thorms 6. nd 6.] (which r lrdy stisfid by t, thnks to our ssumptions) in ordr to solv th NLRH problm vi intgrl qutions by itrtion. Estimt (77) will b consqunc of tchnicl rsults in [35], originlly obtind to prov stbility stimts for this problm. Sinc th NLRH problm cn b solvd for q nd q R, w cn rpt th rgumnts of [35, sction 4] in ordr to obtin th qulity [35, idntity (4.)] q () - q () = i E( A - A + B - B + C - C ), ( 78) R R R R whr A, AR, B, BR, C, CR r constructd s A, B, C in [35, sction 4] with t sgn( ) ( l l ) t - nd sgn R( l) ( l - ) instd of r ( l ).In[35, sction 5] th right 4 pl 4 pl hnd sid of (78) is stimtd in trms of scttring dt. First, th stimt ftr [35, stimt (5.6)] in th prsnt nottion rds t A AR c(, N, m) E ( l) - W + l + L l l R t( l) l L p ( ) ( R), ( 79) for som p Î ], [, whr R is th nnulus dfind in th sttmnt. For B B R w us th stimt ftr [35, stimt (5.8)]. Sinc q nd q R corrspond to scttring dt ( t, r) nd ( t R, r ), th first trm in this stimt vnishs; but lso th third on corrsponding to d r, dfind in th sttmnt of [35, proposition 4.] vnishs if w fix, sinc w chos R > (so t º t R in th nnulus ). Thus w obtin th stimt t B BR c(, N, m) ( l) - W + l l l for som < s < < s <+, whr to C C R nd w gt L ss, ( ) ss, s L L s L t C CR c(, N, m) ( l) - W + l l l R, ( 8) = +. Th sm rgumnt pplis L ss, ( ) Finlly, w us [35, lmm 3.], which givs L p stimts of corrsponds to r ( l) in tht lmm) nr nd. W hv j t( l) l l L p( ) R R. ( 8) t sgn( ) ( l l - ) (this 4pl cm (, NE ) -m R- m+, forj=-,,, ( 8) for R R ( N, m) nd p. This combind with (78) (8) yilds th min stimt (77).,
Invrs Problms 3 (6) 53 Corollry 4.. Lt q, W, m, N, R, E nd q R b s in lmm 4.. Lt c R b th chrctristic function of th bll of rdius E R cntrd in th origin nd dfin q R = - [ c R q ], whr is th D Fourir trnsform. Thn thr is constnt C = C( W, N, m) > such tht q q CE-( m- ) R -( m- ). ( 83) R - R L ( W) Proof. Sinc q Î W m, ( W) w hv q w c, N, m w m ( ) ( W ) - for w. Thn q ( ) - q () = - [( - c ) q ] R c( W, N, m) R w ER drwdim w c( W, N, m), w m ( ER) m- for vry ÎW. This combind with lmm 4. givs th corollry., Now, choos n intgr Nl > nd rdii < R < R. For spctrl prmtrs R < l < R dfin Nl Nl grid. For ths vlus w prcomput th mtrics S l in ordr to solv th boundry intgrl qution. Th rdius R > is tking out vlus of λ clos to th unit circl, sinc w hv problms in computing th Fddv Grn s function for ths vlus. Th us of R ws justifid in th bov Lmm nd is nlogous to th trunction rdius of th ro-nrgy cs cting s rgultion prmtr. Dpnding on th cs th vlu of R, outsid of which th computtionl problms ris, chngs. Th computtionl problms cn b sn from th computd scttring trnsform. W dnot by - th trnsformtion from th Fourir sris domin to th function domin nd simply us (4) to gt t( l ): i E t( l ) = ( l+ l ) -(( L - L ) y ) d s. W Rcll th symmtry () for ny non-xcptionl λ. Using this w cn construct th scttring trnsform insid th unit circl. Dpnding on th scttring trnsform, choos th rdius R insid of which th numricl computtion is usbl. Thn us th trunctd scttring trnsform, l R t( l), R l < R tr( l) =, R l R ( 84) t( l), R< l < R, l R. q -E l Rmrk 4.3. Although th bov-dfind trunctd scttring trnsform diffrs from th on in lmm 4., rgulrition stimt similr to (77)(but lss shrp) cn b provd using th sm ids. Undr th hypothsis of lmm 4., by[33, stimt (.8c)] w hv t( l) C( N)( + E( l + l - ) )-m, l Î,
Invrs Problms 3 (6) 53 which givs j t( l) l l Lp( R l R) OE ( -m ( R- )), forj= -,,, whr p. This, combind with th proof of lmm 4., yilds rconstruction rror of th ordr E- m -( m- ) mx ( OR ( - ), OR ( )). 4.4. Solving th D-br qution W cn solv th priodic vrsion of th intgrl qution (36), without th trm, using t R nd th nlog of th solvr fully dtild in []. W will dl with th following intgrl qution m = - ( ) m, ( 85) R R R whr R is th oprtor of (34) with t R ( l) instd of t( l ). Eqution (85) is solvd by priodition nd using mtrix-fr implmnttion of GMRES. S [4, sction 5.4] for dtils. 4.5. Rconstructing th potntil Lt r b th rconstruction point of our choosing. Lt d b th finit diffrnc nd dfin th points = r + d, = r - d, 3 = r + i d, 4 = r - i d. Using th rlir i dscribd mthods w cn solv th corrsponding CGO solutions mrs = mr (, i l), i =,, 3, 4. W combin th qutions (3) nd (3), omit th trm ( l ), us finit λ nd finit diffrnc mthod for th diffrntition to gt th pproximt rconstruction qution 3 4 m - m m - m R R R R q ( r)» l E i +. ( 86) d d Not tht th rsult is computd in grid of prmtrs λ (sinc (85) is). W comput n vrg of q ( r ) ovr vlus corrsponding to l = R. 5. Numricl rsults In sction 5. w tst th lgorithm for gl () by computing th CGO solutions m (, l) with l > for som potntils. This is don by solving th Lippmnn Schwingr typ qution () using th numricl solution mthod dscribd in [4, sction 4.3]. Evluting numriclly th D-br qution (7) llows us to ssss th ccurcy of th CGO solutions. W comput in sction 5. th scttring trnsform for vrious rdilly symmtric potntils nd obsrv th mrgnc of xcptionl points. This is nlogous to th ronrgy study [5]. In sction 5.3 w tst th full D-br lgorithm for rconstructing svrl tst potntils from thir pproximt DN mps. In sction 5.4 w tst our lgorithm ginst th Novikov Sntcsri lgorithm of [34] with diffrnt nrgis.
Invrs Problms 3 (6) 53 Figur 3. Msh plot nd profil plot of th rottionlly symmtric conductivity s () = s ( ). 5.. Vlidtion of th numricl Fddv Grn s function 5... Dfinition of potntils. W us xctly th sm potntils s in th numricl prt of [5]. Tk rdii < r < r < nd polynomil pt ( )= - t3 + 5t4-6t5. St for r t r p t p t - r () = ( ) r r. - Thn, th pproximt tst function for r j ( ) = p( ) for r < < r ( 87) for r, is in C. Th vlus r =.8 nd r =.9 wr usd. Considr th rdilly symmtric potntils ( ) q = j, ( 88) ( ) s q = D + j, ( 89) s whr Î nd s Î C ( W) with s c >. Th notion of conductivity typ potntils is rlvnt lso t positiv nrgis s th problm of AT will includ such trm in corrsponding potntil of th Gl fnd Cldrón problm. Th msh plot nd th profil plot ( of th conductivity σ r picturd in figur 3. Th conductivity-typ potntil q ) nd th pproximt tst function j r picturd in figurs 4 nd 5 rspctivly. 5... Vrifiction of th computd CGO solutions. In this sction w fix E =. To vrify tht th CGO solutions (nd subsquntly th Fddv Grn s function) r corrct, w tst th -qution (7) using th fiv-point stncil mthod with th finit diffrnc dl =.. 3
Invrs Problms 3 (6) 53 Figur 4. Msh plot nd profil plot of th conductivity-typ potntil q () = q ( ) ( ) ( ). Figur 5. Msh plot nd profil plot of th tst function j() = j( ). ( ) Tk prmtrs λ from. to 4.5. Tk th potntils q diffrntly sid potntils. For ch l = l + li, comput ( ) nd q 35 to tst two vry () Th CGO solution m in th -grid, corrsponding to th prmtr λ. () Th CGO solutions m, m, m3, m4, m5, m6, m7 nd m 8 using l + dl, l + dl, l - dl, l - dl, l + dli, l + dli, l - dli nd l - dli rspctivly. (3) Th functions l ( ) nd - l ( ). ( ) ( ) (4) Th scttring trnsform t( l) of () with m = m, q = q nd q = q. 4 35
Invrs Problms 3 (6) 53 Figur 6. Errors in th -qution for two diffrnt potntils nd diffrnt ccurcis of th LS-solvr; horiontl xis is th spctrl prmtr λ, vrticl xis is th norm m - t( ) ( ) 4pl l m ( - l. On th lft w usd q ) ( nd on th right q ) 35. In rd L using circls is M = 7, in blck using crosss M = 8 nd in blu using squrs is M = 9. Two smllst vlus for λ wr omittd, for l =. th mgnitud of th rror ws btwn 3 nd 3, for th scond smllst it ws btwn.3 nd.. (4) Th drivtivs nd th -oprtion by (6) Th rror m 8m 8m m m = - + - + l dl m 8m 8m m m = - + - + l dl m= i. ( l + l ) m 3 4 6 5 7 8 m - t( ) ( ). ( 9) 4pl l -l m L ( D(,)) Th bov computtions of CGO solutions r don with th solution lgorithm dscribd in [4, sction 4.3]; it is clld LS-solvr blow. Th -grid hs M M points. In figur 6 w ( s th rror (9) s function of λ using q ) ( on th lft, q ) 35 on th right. Th prmtr M is incrsd from 7 to 9. As xpctd, th rror dcrss s M incrss s it incrss th ccurcy of th LS-solvr. Th smllst vlus of λ wr omittd in th picturs, for l =. th mgnitud of th rror ws btwn 3 nd 3, for th scond smllst λ it ws btwn.3 nd.. For vlus of λ nr l = th numricl mthod of gl () hs grt rror du to vry smll vlu of k. In th rconstruction of th potntil w us vlus s lrg s l» 5. Not picturd hr, this tst ws don lso for lrgr vlus of λ, th rror sms to b of similr mgnitud for ny. < l < 5. In conclusion, th mthod for computing gl () is vlid nd ccurt nough for our purposs. 5
Invrs Problms 3 (6) 53 Figur 7. Scttring trnsform for th potntil q ( ) = j on th lft, for th potntil q ( ) =D s s + j on th right. Th x-xis is =-35¼ 35, y-xis is l =.¼ 4.5. Compr to figurs 3 nd 9 in [5]. 5.. Numricl invstigtion of xcptionl points In this sction w fix E =. For givn potntil thr my b vlus of prmtr λ for which thr xists no uniqu CGO solution. Such λ vlus r clld xcptionl points. W follow hr th ro-nrgy study [5] nd comput numriclly CGO solutions t posivit nrgy. Excptionl points will show up s singulritis in computtion. Rcll th rottionlly symmtric potntils q ( ) nd q ( ) from (88) nd (89). S figurs 3 5. W us 5 discrt points of λ nd 7 discrt points of α, l =., ¼,4.5, = -35, ¼,35. W us M = 8 for th LS-solvr (s [4, sction 4.3]) lding to M M sid -grid. In figur 7 w plot th rdilly symmtric nd rl-vlud scttring trnsform t( l) = t( l ) for th potntil q ( ) = j on th lft, for th potntil q ( ) =D s s + j on th right. Th x-xis is th prmtr α nd th y-xis is th modulus l of th spctrl prmtr. Blck colour rprsnts vry smll ngtiv vlus, nd whit vry lrg positiv vlus of t( l ). Th lins whr it bruptly chngs btwn ths colours r xcptionl circls tht mov s th prmtr α chngs. In figur 8 w plot th profil of th scttring trnsform t( l) s function of λ, using th potntil q ( ) = j, with th vlus =-5, -5, - 3. Th xcptionl circls cn b sn s singulritis in th profils. 5.3. Rconstructions of q In this sction w fix E = - 3. W rconstruct two non-symmtric potntils. Th first is picturd in figur 9. Th scond is of conductivity typ, th conductivity σ is picturd in figur 3 nd th corrsponding potntil q = s - Ds in figur. 5.3.. Choic of prmtrs. For th DN- nd S l -mtrics w us N = 6, s (73) nd (75). W dd gussin nois to ch lmnt with (74) so tht th rltiv mtrix norm btwn th 6
Invrs Problms 3 (6) 53 Figur 8. On th right: th profil of t( l) s function of λ using th potntil q ( ) = j with thr diffrnt vlus of α. On th lft: th pln t( l) for ll prmtrs α with n indiction of th loction of th profil on th right. Compr to figur 4 in [5]. originl DN-mtrix nd th noisy DN-mtrix is.5%. In th msh for th FEM w hv 48 576 tringls. Dpnding on th cs, w cut off non-usbl prts of th scttring trnsform. As n xmpl, in figur w hv th rl nd imginry prts of th scttring trnsform of cs potntil computd using th non-noisy DN-mtrix L q nd th noisy DN-mtrix L q. In th whit rs th computtion brks down du to nois nd/or lrg vlus of λ. Th blck lin indicts th trunction rdius R usd in t R. 7
Invrs Problms 3 (6) 53 Figur 9. Msh plot nd D plot of th cs potntil q (). Figur. Msh plot nd D plot of th cs conductivity s (). W choos R =.37 to tk cr of problms with l clos to. W hv Nl = 56 s th grid prmtr of sction 4.3. W solv th priodid intgrl qution (85) in Md Md -sid λ-grid with M d = 7. 5.3.. Effct of trunction. Using th cs potntil w tst diffrnt circulr trunction rdii R for th scttring trnsform t R. Th rsult is in figur 3, whr w s how th rconstruction improvs by using lrgr trunction rdius. 5.3.3. Rconstructions. Bsd on figur nd similrly for th othr css, w choos for R = 4 nd R = 3 for cs non-noisy nd noisy rconstructions from th DN-mtrix. For cs w choos R = 435 nd R = 3 rspctivly. In figurs 4 nd 5 w pictur th originl potntils on th lft, th rconstruction using (86) without nois in th middl nd th rconstruction using (86) with ddd nois on th right. Rltiv rrors 8
Invrs Problms 3 (6) 53 Figur. Msh plot nd D plot of th cs potntil q () = s() Ds() -. Figur. Th scttring trnsform t( l) of cs, th non-symmtric potntil of figur 9. Rl prt on th lft, imginry prt on th right, in λ-grid [- 6, 6] [- 6, 6] i. On th top row: th non-noisy DN-mtrix L q ws usd. On th bottom row: th noisy DN-mtrix L q ws usd. In th whit rs th computtion brks down. Th blck lin indicts th lrgst usbl circl for th trunction t R ( l ). Enrgy lvl E =.. 9
Invrs Problms 3 (6) 53 Figur 3. Cs rconstructions using thr diffrnt trunction rdii, non-noisy rconstructions on th lft nd noisy rconstructions on th right. Attchd r th rltiv L rrors comprd to th originl potntil. S figur 4 for th originl potntil. Enrgy lvl E =.. Figur 4. On th lft: th originl cs potntil, s figur 9. In th middl: rconstruction using th non-noisy DN-mtrix L q. On th right: rconstruction using th noisy DN-mtrix L q. Rltiv rrors q - qrc L ( W) q L ( W) r givn. Enrgy lvl E =.. Th colormp is diffrnt from th on usd in figur 3. 3
Invrs Problms 3 (6) 53 Figur 5. On th lft: th originl cs potntil, s figur. In th middl: rconstruction using th non-noisy DN-mtrix L q. On th right: rconstruction using th noisy DN-mtrix L q. Rltiv rrors q - qrc L ( W) q L ( W) r givn. Enrgy lvl E =.. q q q, - rc L ( W) L ( W) whr q rc is th rconstruction, r givn. 5.4. Comprison of lgorithms W usd nothr tst potntil for th comprison of our lgorithm ginst th Novikov Sntcsri lgorithm [34]. Th rsult is picturd in 6: on top w hv th originl potntil hving vlus btwn nd, blow w hv th rconstructions of both lgorithms using non-noisy DN-mps for nrgis E =.,, 5, nd E = 3. Our mthod is on th lft, th othr mthod on th right. Rltiv rrors nd trunction rdii for our mthod r givn. For smllr nrgis thn E =., w gt pproximtly th sm rconstruction. In figur 7 w us nrgis -5, - 3 nd.. Th smll diffrncs cn b ttributd to diffrncs in th trunction tht w choos bsd on th plottd scttring trnsform s in figur. 6. Conclusions W dvlopd nw numricl mthod for rconstructing th potntil from boundry msurmnts in th Gl fnd Cldrón problm. Th mthod sms to work s vidncd by th rconstructions, vn if th thory is still missing dtils: th oprtor is not usd in th rconstructions. S th rconstructions of css nd, picturd in figurs 4 nd 5. Also s th comprison rconstructions of figurs 6 nd 7. Th rconstructions of th potntil in th comprisons hs significntly lowr numricl rltiv rror, which w ttribut to th lowr contrst of th potntil. W conclud tht our mthod works bttr for lowr contrst potntils, vn if th rconstructions of css nd r visully stisfctory nd do rvl importnt fturs of th originl potntil. Th numricl mthod for g λ is not ccurt nr l = which rsultd in high rrors in th vrifiction tst of figur 6 using th -qution. For othr vlus of λ this numricl mthod is ccurt nough for good qulity rconstructions. Rgrding th rdil potntils, th numricl vidnc show no xcptionl points for smll α nor for lrg λ which is to b xpctd ccording to th thory for smll potntils. Also ccording to our tsts thr r no xcptionl points for positiv α. For ngtiv α, thr r ithr on or two xcptionl circls in th rng of prmtrs invstigtd. Th two 3
Invrs Problms 3 (6) 53 Figur 6. Numricl comprison of th two lgorithms. Top row: th originl potntil q. Th nxt rows show th rconstruction using incrsing nrgy E, our mthod on th lft, th mthod of Novikov Sntcsri on th right. Th trunction rdii R for out mthod nd rltiv L rrors r lso shown. 3
Invrs Problms 3 (6) 53 Figur 7. Rconstruction using thr diffrnt, smll nrgy lvls. Rltiv rrors q - q W q W nd trunction rdii r givn. rc L ( ) L ( ) typs of potntils q ( ) nd q ( ) hv littl diffrnc in thir xcptionl points, minly in th scond xcptionl circl forming t =- s α is dcrsd from ro. Th comprison rsult of figur 6 shows tht currntly our mthod works bttr with smllr nrgis nd bcoms unffctiv t lrgr nrgis s is xpctd from th stbility rsults. Incrsing th nrgy scls th usbl r of th trunction down, s indictd by th trunction rdii usd in th comprison. Using nrgis - 5 E. rsult to virtully th sm rconstruction, nd s of now this is th optiml rng for our mthod furthr fintuning is lft for futur works. Acknowldgmnts JPT ws supportd in prt by th Finnish Culturl Foundtion nd Europn Rsrch Council (ERC). ML nd SS wr supportd by th Finnish Cntr of Excllnc in Invrs Problms Rsrch 7 (Acdmy of Finlnd CoE-projct 55). MS ws supportd by FiDiPro projct of Acdmy of Finlnd, numbr 6335. Rfrncs [] Richrd B nd Coifmn R R 98 Scttring, trnsformtions spctrls t équtions d évolution non linéirs Goulouic Myr Schwrt Sminr, 98 98 (Plisu: Écol Polytch) p Exp. No. XXII [] Boiti M, Lon J P, Mnn M nd Pmpinlli F 987 On spctrl trnsform of KdV-lik qution rltd to th Schrödingr oprtor in th pln Invrs Problms 3 5 36 [3] Bukhgim A 8 Rcovring th potntil from Cuchy dt in two-dimnsions J. Invrs Ill- Posd Problms 6 9 34 [4] Burov V A, Srgv S N, Shurup A S nd Rumyntsv O D Appliction of functionlnlyticl Novikov lgorithm for th purposs of ocn tomogrphy Proc. Mtings Acoust. 7 765 [5] Burov V A, Shurup A S, Zotov D I nd Rumyntsv O D 3 Simultion of functionl solution to th coustic tomogrphy problm for dt from qusi-point trnsducrs Acoust. Signl Procss. Comput. Simul. Acoust. Phys. 59 39 47 33
Invrs Problms 3 (6) 53 [6] Cldrón A-P 98 On n invrs boundry vlu problm Sminr on Numricl Anlysis nd its Applictions to Continuum Physics (Rio d Jniro, 98) (Rio d Jniro: Soc. Brsil. Mt.) pp 65 73 [7] Fddv L D 966 Incrsing solutions of th Schrödingr qution Sov. Phys. Dokl. 33 5 [8] Glfnd I M 96 Som problms of functionl nlysis nd lgbr Intrntionl Mthmticl Congrss in Amstrdm (in Russin) (Moscow: Nuk) pp 49 74 [9] Grnlf A, Kurylv Y, Lsss M nd Uhlmnn G 7 Full-wv invisibility of ctiv dvics t ll frquncis Commun. Mth. Phys. 75 749 89 [] Grnlf A, Kurylv Y, Lsss M nd Uhlmnn G 9 Cloking dvics, lctromgntic wormhols, nd trnsformtion optics SIAM Rv. 5 3 3 [] Grnlf A, Lsss M nd Uhlmnn G 3 On nonuniqunss for Cldrón s invrs problm Mth. Rs. Ltt. 685 94 [] Grnlf A, Kurylv Y, Lsss M nd Uhlmnn G 9 Invisibility nd invrs problms Bull. Am. Mth. Soc. 46 55 79 [3] Grinvich P G Scttring trnsformtion t fixd non-ro nrgy for th two-dimnsionl Schrödingr oprtor with potntil dcying t infinity Russ. Mth. Surv. 55 5 83 [4] Grinvich P G nd Mnkov S V 986 Invrs scttring problm for th two-dimnsionl Schrödingr oprtor, th -mthod nd nonlinr qutions Funct. Anl. Appl. 94 3 [5] Grinvich P G nd Novikov R G 995 Trnsprnt potntils t fixd nrgy in dimnsion two. Fixd-nrgy disprsion rltions for th fst dcying potntils Commun. Mth. Phys. 74 49 46 [6] Grinvich P G nd Novikov R G Fddv ignfunctions for point potntils in twodimnsions Phys. Ltt. A 376 6 [7] Grinvich P G nd Novikov R G 3 Fddv ignfunctions for multipoint potntils Eursin J. Mth. Comput. Appl. 76 9 [8] Ikht M nd Siltnn S 4 Numricl solution of th Cuchy problm for th sttionry Schrödingr qution using Fddvʼs grn function SIAM J. Appl. Mth. 64 97 3 [9] Iscson D, Mullr J L, Nwll J C nd Siltnn S 6 Imging crdic ctivity by th D-br mthod for lctricl impdnc tomogrphy Physiol. Ms. 7 S43 5 [] Knudsn K, Lsss M, Mullr J L nd Siltnn S 9 Rgulrid D-br mthod for th invrs conductivity problm Invrs Problms Imging 3 599 64 [] Knudsn K, Mullr J L nd Siltnn S 4 Numricl solution mthod for th dbr-qution in th pln J. Comput. Phys. 98 5 7 [] Lkshtnov E L, Novikov R G nd Vinbrg B R 5 A globl Rimnn Hilbrt problm for two-dimnsionl invrs scttring t fixd nrgy rxiv:59.6495 [3] Mullr J L nd Siltnn S 3 Dirct rconstructions of conductivitis from boundry msurmnts SIAM J. Sci. Comput. 4 3 66 [4] Mullr J L nd Siltnn S Linr nd Nonlinr Invrs Problms with Prcticl Applictions (Phildlphi: SIAM) [5] Music M, Prry P nd Siltnn S 3 Excptionl circls of rdil potntils Invrs Problms 9 454 [6] Music M 4 Th nonlinr Fourir trnsform for two-dimnsionl subcriticl potntils Invrs Problms Imging 8 5 67 [7] Nchmn A, Sylvstr J nd Uhlmnn G 988 An n-dimnsionl Borg Lvinson thorm Commun. Mth. Phys. 5 595 65 [8] Nchmn A I 988 Rconstructions from boundry msurmnts Ann. Mth. 8 53 76 [9] Nchmn A I 996 Globl uniqunss for two-dimnsionl invrs boundry vlu problm Ann. Mth. 43 7 96 [3] Novikov R G 986 Rconstruction of two-dimnsionl Schrödingr oprtor from th scttring mplitud for fixd nrgy Funct. Anl. Appl. 46 8 [3] Novikov R G 988 A multidimnsionl invrs spctrl problm for th qution - dy + ( vx ( ) - ux ( )) y = Funct. Anl. Appl. 63 7 [3] Novikov R G 99 Th invrs scttring problm on fixd nrgy lvl for th two-dimnsionl Schrödingr oprtor J. Funct. Anl. 3 49 63 [33] Novikov R G 999 Approximt invrs quntum scttring t fixd nrgy in dimnsion Proc. Stklov Inst. Mth. 5 85 3 34