REGULARISATION TECHNIQUES FOR FIRST KIND INTEGRAL EQUATIONS

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1 AKTET ISSN Journal of Intitute Alb-Shkenca Revitë Shkencore e Intitutit Alb-Shkenca Copright Intitute Alb-Shkenca REGULARISATION TECHNIQUES FOR FIRST KIND INTEGRAL EQUATIONS BESIANA HAMZALLARI, FIORALBA CAKONI Department of mathematic, Facult of Natural Science, Univerit of Tirana, Albania cakoni@math.udel.edu Department of Mathematical Science, Univerit of Delaware, Newark, Delaware, USA beianahamzallari@hotmail.com AKTET V, 3: , 0 PËRMBLEDHJE Ne kete punim do te paraqeim dia rezultate te reja lidhur me analizen e ekuacioneve integrale te llojit te pare ne teorine e hperhapje invere. Me holleiht, do te trajtojme te ahtuquajturin operator Far field dhe ekuacionin integral korrepondue. Si operatori dhe ekuacioni Far field jane baze per ndertimin e algoritmeve ne teorine e hperhapje invere ic ehte metoda e modelimit linear.gjithahtu do te tudiojme ekuacionin integral te llojit te pare qe hfaqet ne metoden e dekompozimit per rikontruktimin e forme (konturit) ne zgjidhjen e problemit te hperhapje invere me pengea.do te zbatojme teknika te ndrhme i ajo e Tikhonov dhe e trungezimit per te arritur ne nje zgjidhje te qendruehme te ketre ekuacioneve. SUMMARY In thi paper we dicu ome new reult related to the anali of ill poed firt kind integral equation ariing in the olution of obtacle invere cattering theor. More pecificall, we invetigate the o-called far field operator and the correponding far field equation. Both the far field operator and the far field equation contitute theoretical bai of uniquene and recontruction algorithm in the invere cattering theor uch a the linear ampling method. We tud the firt kind integral equation aociated with the decompoition method and the linear ampling method for olving the invere obtacle cattering problem. We how how to appl variou regularization technique uch a cut-off and Tikhonov regularization to compute a table olution to thee equation. Ke word: integral equation, regularization technique, invere cattering INTRODUCTION The field of invere problem i a relativel new area of mathematical reearch having it origin in the fundamental paper of Tikhonov in mid The reaon the area i o oung i hitorical prejudice dating back to Hadamard who claimed that the onl problem of phical interet were thoe that had a unique olution depending continuoul on the given data. Such problem were called well-poed, and the problem that were not well poed were labeled ill-poed. The development of the mathematical theor of ill-poed problem, together with the rapid development of computing facilitie, et the tage for the ubequent mathematical invetigation in the invere problem [3], [6 ]. In thi paper we dicu ome new reult related to the anali of ill poed firt kind integral equation ariing in the olution of the obtacle invere cattering prolem. More pecificall, we invetigate the o called far field operator F:L [0,] L [0,] and the correponding far field equation. We alo invetigate an ill-poed firt kind integral equation which appear in the decompoition method for hape recontruction in the obtacle invere cattering problem. We how how variou regularization technique uch a pectral

2 cut-off, Tikhonov regularization and the dicrepanc principle are applied to regularize the far field equation and how it regularized olution i related to the invere cattering theor.. ILL- POSED EQUATIONS AND REGULARIZATION TECHNIQUES Let A :U V be an operator, from U X into V Y where X, Y are normed pace. The equation A f i called well-poed if A i bijective and A :VU i continuou. Otherwie A f i called ill-poed. Theorem. Let X and Y be normed pace and let A :U V be a compact operator. Than Af i ill-poed if X i not of finite dimenion. Proof. Aume A exit and i continuou. Then I A A:X X i compact hence X i finite dimenion, which end the proof. The dicontinuit of A - lead to the intabilit of the olution. Method for contructing a table approximate olution to an ill-poed problem are called regularization method. In particular, for A a bounded linear operator, we want to approximate the olution of A = f from a knowledge of a perturbed right hand ide with a known error level. When fa(x) then if A i injective there exit a unique olution of A f. However, in general we cannot except that f A(X). How do we contruct a reaonable approximation that depend continuoul on? Definition. Let X and Y be normed pace and A :U V be an injective linear bounded operator. Then a famil of bounded linear operator R :Y X, 0 uch that, i called a regularization cheme for A and α the regularization parameter. to We clearl have that R A f a 0. A regularization cheme approximate the olution of A f b : R f. Writing Rf RfRA, we etimate R R A. Since the operator R cannot be uniforml bounded with repect to α and R α A cannot be norm convergence a 0, the firt term on the right hand ide i large for α mall wherea the econd term on the right hand ide i large if α i not mall! So how do we chooe α? A reaonable trateg i to chooe () uch that a 0. Definition. A trateg for a regularization cheme R, 0 i called regular if for ever f A(X) and all f Y uch that we have that R( ) f A f a 0. A natural trateg for chooing () i the dicrepanc principle of Morozov, i.e the reidual A f hould not be maller than the accurac of the meaurement of f. From now on X and Y will be infinite dimenional and A:X Y,A 0 a compact operator. The operator A A:X X i compact and elf-adjoint hence there exit at mot a countable et of eigenvalue { n }, uch that A A nnn. Hence ( A A n, n) n which implie that n 0. The non negative quare root of the eigenvalue of AA are called the ingular value of A. Theorem. Let n be the equence of nonzero ingular value of the compact operator A:X Y ordered uch that: 3... Then there exit orthonormal equence n in X andg n in uch that : A nngn, A gnnn 30 AKTET Vol. V, Nr 3, 0

3 For ever X we have the ingular value decompoition: (, n ) n P Where P:X N(A) i the orthogonal projection operator of X onto N(A) and A n(, n) gn The tem ( n ngn) i called a ingular tem of A. The following theorem known a Picard Theorem provide a ufficient condition for the exitence of a olution to A f and reveal the ill-poed nature of thi equation. Theorem.3 (Picard Theorem) Let A : X Y be a compact operator with ingular tem ( n ngn). The the equation A f i olvable if and onl if f N(A ) and (f,gn). n In thi cae a olution to A f i given b (f,gn) n n Note that Picard Theorem illutrate the ill-poed nature of the equation A f. In particular, etting f fgn, we obtain a olution of A f given b n / n, f f n Since b Hilbert-Schmidt Theorem we have that n 0. We a that A f i mildl ill-poed if the ingular value deca lowl to zero and everel ill-poed if the deca ver rapidl (for example exponentiall). All of the invere cattering problem conidered in thi book are everel ill-poed. There are two well-known regular regularization cheme, namel the pectral cutoff method and the Tikhonov regularization. Spectral cut-off. Let A:X Y be an injective compact operator with ingular tem ( n ngn). Then the pectral cut-off: R m f : (f,gn) n n m n Decribe a regularization cheme with regularization parameter m and Rm / m (ee [3] for detail) Tikhonov Regularization. Let A:X Y be a compact operator. Then for ever α > 0 the operator I A A:X X i bijective and ha a bounded invere. Furthermore, if A i injective than; R : ( I A A) A Decribe a regularization cheme, known a Tikhonov regularization with R. The Tikhonov regularization cheme ha an equivalent formulation which i formulated in the following theorem (ee [3] for the proof). Theorem.4 Let A:X Y be a compact operator and let α >0. Then for ever fy there exit a unique X uch that: A f inf A f X The minimize i the unique olution of A A A f. We finih thi ection b conidering a cla of compact integral operator that will appear in the following tud of invere problem. Let m G R be a meaurable et. Definition.3 The linear operator A:L (G) L (G) defined b ( A)(x): GK(x,) () d where K:Gx G C i a given function known i call an integral operator with kernel K. If K:G x G C i a continuou function the operator A i called integral operator with continuou kernel. The following theorem will be of great importance to u in the following and the proof can be found in [7] and [9]. Theorem.5 The integral operator with continuou kernel i compact in L (G). AKTET Vol. V, Nr 3, 0 3

4 3. Invere Scattering and Ill-poed Equation Let u conider the cattering of acoutic plane i ikx.d wave u : e in the direction d b a ound hard obtacle D (called Scatterer) which for ake of implicit we aume i a connected bounded region of R at a given fixe frekuenc ω where k c, c being the ound peed (note that here it aumed that field i time harmonic i.e i t e the time dependent term). The cattered field u atifie u k u 0 in R \ D (3) ikxd u e 0 ond (4) u lim r( iku ) 0 (5) r r where the Sommerfeld radiation (5) i aumed to hold uniforml in θ with (r, θ) are polar coordinate. Thi exterior boundar value problem i well-poed, i.e. a unique olution exit in appropriate pace [5]. It i known tha the (radiating) fundamental olution to the Helmholtz equation i given b; (x,) : where H ( 0 ) i the Hankel fuction of the firt kind, and note that Φ(x,) atifie the Sommerfeld radiation condition with repect to both x and. The cattered field u atifie the amptotic the amptotic behavior [6], [3] ikr e 3 u (x) u(, ) O(r ) r Where d (co,in), k i fixed and u u v e (, ) 8k e The function i ( ) H (k x ) 4 0 i 4 D ikr Co( ) u ) (u )d() (6) v e ikr Co( ) (8) ) (7) i called the far field pattern correponding to the cattering problem (3) (5). Pleae note that the far field pattern of the fundamental olution i 4 e ikr Co( ) (,) e, where 8k (r, ) The invere obtacle cattering problem now i: given the (meaured) far field pattern u (, ), for, [0,] find D. A we will ee bellow thi problem i everel ill-poed and non-linear ince the far field pattern doe not depend linearl on D. We note that thi invere problem arie from man application in medical imaging, nondetructive teting, etc. We alo note that the exact far field pattern, for, [0,] uniquel determine D [6], []. We will preent two method for doing thi invere problem, namel the decompoition method [4] and the linear ampling method [8]. Both method lead to olving an ill-poed integral equation of the firt kind for which we are going to ue regularization technique a developed in Section. 3. The decompoition method The main idea of the decompoition method i to break the invere obtacle cattering problem into two part: the firt part deal with the illpoede b contructing the cattered wave u from the far field pattern u and the econd part deal with the non-linearit b determining the unknown boundar D of the catterer a the location where the boundar condition for the ikxd total field u e u i atified in a leat quare ene. We aume that the unknown catter D i bounded and impl onnected and enough a priori information on the unknown catterer i aumed o that one can place a cloed urface inide D. Then the cattered field u i ought a a ingle laer potential [9], [3] u (x) () (x,) d() D xr \D where fundamental olution (, ) i given b (6) and L ( ) (the pace of quare integrable function in ) i a function to be determined. In (9) 3 AKTET Vol. V, Nr 3, 0

5 thi cae the far field pattern repreentation u ( ) e D ikr Co( ) u ()d() ha the And i now determined b olving the integral equation (0). The kernel of the integral operator on the right-hand ide of (0) i analtic, whence thi i a compact operator according to Theorem.5. The later mean that (0) i an ill-poed equation, thu in order to olve it one need to ue the Tikhonov (it i known that thi operator i injective under ome aumption on D ). Having found the regularized olution with regularization parameter and given an approximation of the cattered wave u obtained b inerting the Tikhonov regularization olution of (0) into (9), the unknown boundar D i then determined b requiring that the ound-oft boundar condition i u u 0 on D be atified in a leat quare ince, i.e b minimizing i u u L ( D) over a uitable et of admiible curve. 3. Far field equation and the linear ampling method We now define the far field operator F:L [0,] L [0,] b (Fg)( ): 0 u (, )g( )d From the repreentation (8) for () u and the fact i that u depend continuoul on u in C ( D) we ee that u(, ) i continuou on [ 0,] x [ 0, ]. Thi fact combined with Theorem.5 prove the following reult. Theorem 3. The far field operator F:L [0,] L [0,] i compact The far field operator i an important object in the tud of invere obtacle cattering problem conidered here. In particular it contain information about he obtacle D and i related to i ik the cattering operator S b S I e 4 F. k B uperpoition Fg i the far field pattern of the cattered (0) field due to the Herglotz fuction ikrco( ) vg : 0 e g( ) d A incident wave Theorem 3. The far field operator correponding to the cattering problem (3) (5) i injective with dene range, provided that k i not a Direchlet eigenvalue of in D(i.e. v k v 0 in D, v=0 on D ha onl the trivial olution v=0). The linear ampling method look for olution to the far field equation [], [], [4], [8], [0]. (fg)( ) (,z), for zr () To hoh wh the olution of () can be ued to recontruct D, we aume that g z olve () and z D. Then it follow from rellich lemma [3], [6] that u (x, )g z( )d (x,z) 0 for z R \ D From the boundar condition u=0 on D we ee that v gz (x) (x,z) 0 (3) For xd where vgz i Herglotz wave fuction with kernel g z. We can now conclude from (3) that v gz become unbounded a z xd and hence lim gz L [0,] zd i.e D i characterized b point z where the olution of () become unbounded. The far field equation i everil ill-poed due to the compactne of the far field operator F. Thu one olve the regularized equation ( IFF)g F (,z), zr In fact, onl the noi far field pattern u(, ) i known in practice which mean that the noi far field operator F i available which i given b (F g)( ): 0 u (, )g( ) d AKTET Vol. V, Nr 3, 0 33

6 Where i the noie level. Thu, one olve the following regularized equation ( ( )I F F )g F (,z), zr where the Tikhonov regularization parameter () i choen b the Morozov dicrepanc principle a explained in Section. REFERENCES [] T. Aren, Wh linear ampling work, Invere Problem, 0 (004), [] F. Cakoni, D. Colton and P. Monk, The direct and invere cattering problem for partiall coated obtacle, Invere Problem 7 (00), 997{05. [3] F. Cakoni and D. Colton, Qualitative Method in Invere Scattering Theor, Springer, Berlin, 006. [4] F. Cakoni, D. Colton, The determination of the urface impedance of a partiall coated obtacle from far _eld data, SIAM J. Appl. Math. 64 (004), [5] D. Colton, Partial Di_erential Equation, Dover Publication, NY, 004. [6] D. Colton and R. Kre Invere Acoutic and Electromagnetic Scattering Theor, nd edition, Springer, Berlin, 998. [7] D. Colton and R. Kre Integral Equation Method in Scattering Theor, Wile, New York, 983. [8] D. Colton and A. Kirch, A imple method for olving invere cattering problem in the reonance region, Invere Problem, (996), [9] R. Kre, Linear Integral Equation Springer, New York, 999. [0] A. Kirch and N. Grinberg, The Factorization Method for Invere Problem, Oxford Lecture Serie in Mathematic and it Application, 36 Oxford Univerit Pre, Ox-ford, 008. [] A. Kirch and R. Kre, Uniquene in invere obtacle cattering, Invere Problem 9 (993), [] HH. Qin and F. Cakoni, Nonlinear integral equation for hape recontruction in the invere interior cattering problem, Invere Problem, 7 (0), [3] W. McLeanWStrongl Elliptic Stem and Boundar Integral Equation. Cambridge Univerit Pre, Cambridge (000). [4] O. Scherzer, Handbook of Mathematical Method in Imaging Springer, Berlin, Heidel-berg, New York, AKTET Vol. V, Nr 3, 0