A REGULARIZED GMRES METHOD FOR INVERSE BLACKBODY RADIATION PROBLEM

Transcription

1 Wu, J., Ma. Z.: A Reguarized GMRES Method for Inverse Backbody Radiation Probe THERMAL SCIENCE: Year 3, Vo. 7, No. 3, pp A REGULARIZED GMRES METHOD FOR INVERSE BLACKBODY RADIATION PROBLEM by Jieer WU a and Zheshu MA b* a Schoo of Matheatics and Physics, Jiangsu University of Science and Technoogy, Zhenjiang, China b Departent of Power Engineering, Jiangsu University of Science and Technoogy, Zhenjiang, China Origina scientific paper DOI:.98/TSCI3678W The inverse backbody radiation probe is focused on deterining teperature distribution of a backbody fro easured tota radiated power spectru. This probe consists of soving a first kind of Fredho integra equation and any nuerica ethods have been proposed. In this paper, a reguarized generaized inia residua ethod is presented to sove the inear i-posed probe caused by the discretization of such an integra equation. This ethod projects the origina probe onto a ower diensiona subspaces by the Arnodi process. Tikhonov reguarization cobined with the generaized cros vaidation criterion is appied to stabiize the nuerica iteration process. Three nuerica exapes indicate the effectiveness of the reguarized generaized inia residua ethod. Key words: backbody radiation, inverse probe, generaized inia residua ethod, reguarization Introduction During theoretica study of backbody radiation probes, we often use a set of inaccurate experienta data to cacuate other physica data. The inverse backbody radiation (BRI) probe is one of the exapes. According to Panck's aw, the atheatica ode of the backbody radiation can be expressed as []: hv 3 wv ( ) T c hv/kt e d () where frequency v [V, V ], w(v) is the tota radiated power spectru, T the absoute teperature and the range of T usuay goes fro to K, a(t) the area teperature distribution, c the speed of ight, k the Botzann's constant, and h the Panck's constant. The direct probe of backbody radiation is to cacuate w(v) bya(t) whie the BRI probe is to obtain a(t) by soving in integra eq. (). The BRI probe is iportant in reote sensing appications. For convenience etting G(n) =c w(v)/hv 3 and then expression () is equivaent to: G( n) e n/ h kt dt K(, v T) a( T) dt where integra kerne K(n T) =(e hv/kt ). Equation () is a first kind of Fredho integra equation and is an inherenty i-posed probe []. Since 98, this probe has attracted any * Corresponding author; e-ai: azheshu@6co ()

2 Wu, J., Ma. Z.: A Reguarized GMRES Method for Inverse Backbody Radiation Probe 848 THERMAL SCIENCE: Year 3, Vo. 7, No. 3, pp schoars' attention. The first foruation for this probe was proposed by Bojarski [] in 98. The Lapace transfor together with an iterative process was presented. Chen and Li [3], Dai and Dai [4] proved the existence and uniqueness of the soution of BRI. Sun and Jaggard [5], Dou and Hodgson [6], and Li and Xiao [7] discussed Tikhonov reguarization ethods. Li [8] proposed conjugate gradient ethod. Dou and Hodgson [9] epoyed axiu entropy ethod. Ye et a. [] deveoped universa function set ethod. Wu and Dai [] presented a reguarizing Lanczos ethod. Generaized inia residua (GMRES) agorith ethod [] is a coon too for soving inear systes. Unti recenty GMRES ethod has been appied to discrete i-posed probes. For instance Jensen and Hansen [3] systeaticay studied the characteristics of the reguarization GMRES ethod. Cavettiet et a. [4, 5] discussed reguarization GMRES ethod and appied the L curve condition nuber to sove inear i-posed probes and iage restoration processing. In this paper, we discrete the integra eq. () and introduce the GMRES ethod to sove the obtained inear discrete i-posed syste. This ethod is based on the Arnodi process, which yieds a sequence of sa east squares probes by approxiating the origina discrete i-posed probe. Tikhonov reguarization [] cobined with generaized cross vaidation (GCV) criterion [6] are used to stabiize the iteration process. Nuerica resuts iustrate the potentia of the proposed ethod. Discretization and reguarization In practice, the range of T usuay goes fro K to K, and v goes fro Hz to 4 Hz. Assuing the range of T is [T, T ], then eq. () can be expressed approxiatey as: Gv ( ) T dt hv/kt T e Let n [V, V ], we choose n coocation points: v = V + (V V )/(n ), =,,... n on the [V, V ], then eq. (3) becoes: n Gv ( ) dt,,,..., n (4) hv / kt j e b The nuerica quadrature ruer f ( x) dx ( b a) f ( a) with n intervas of equa ength on [T a, T ] is discreted as: n t j n T T a( tj ) G( n ) dt hv kt hv j t / j e (5) n / kt j j e If the vectors x and b are defined by x =[a(t ),..., a(t n )] T, b =[G(v ),..., G(v n )] T, and if the n n square atrix [A] is defined by A =(d/e hv /kt j ) n n where d =(T T )/n then eq. (5) can be written as: Ax = b (6) It is we known that the discretization of integra in eq. (3) gives rise to an discrete inear i-posed syste, which eans that: () the syste (6) ight not have a soution, () the soution ight not be unique, and (3) the soution if it exists and is unique does not depend continuousy on the right-hand side. In addition the atrix A is i-conditioned. Straightforward soution of the syste (6) is typicay not eaningfu. In order to avoid this difficuty, the inear syste (6) can be repaced by a nearby syste which is we-conditioned and the coputed soution is a good approxiation. This repaceent is known as reguarization. (3)

3 Wu, J., Ma. Z.: A Reguarized GMRES Method for Inverse Backbody Radiation Probe THERMAL SCIENCE: Year 3, Vo. 7, No. 3, pp One of the ost coon ethods of reguarization is Tikhonov reguarization [], which repaces the syste (6) by the iniization probe in Ax b x or equivaenty: x (A T A + I) x = A T b (7) where is a reguarization paraeter and denotes the -atrix nor. Cobining the GMRES ethod with Tikhonov reguarization [], syste (6) is projected onto a Kryov subspace. The projected probe is aso i-posed. Since the diension of the projected probe is usuay sa reative to n, reguarization of the projected probe is uch ess expensive. Reguarized GMRES ethod The GMRES ethod which based on the Arnodi process is a popuar iterative ethod for soving arge inear syste with a non-syetrica non-singuar atrix. In exact arithetic, for a given starting vector r, the GMRES ethod projects syste (6) to Kryov subspace: K (A, r ) = span{r, Ar,... A r } and deterines iterates x x + K (A, r ), =,,..., which satisfy: b Ax in b Ax. x K ( A, b) We use notation (x, y) =x T y, x, y R N in the foowing Arnodi process. Agorith. (The Arnodi process) () et x be a starting vector. Copute r = b Ax and v = r / r ; () for j =,..., ; (.) copute h = Av j ; (.) for i =,..., j copute h ij =(h,v i ) and h = h h ij v i, (.3) copute h j+,j = h, and (.4) copute v j+ = h/h j+,j. The Arnodi process generates an orthonora atrix V + =[v, v,..., v + ] whose couns are orthonora bases of K (A, r ), and an upper Hessenberg atrix H = (h ij ) R,. In atrix for we have: AV VH h v e T, (8) where e is the first canonica vector. ~ H Let H h, v e T, the GMRES ethod coputes the approxiation x = x + V y, where y soves the east squares probe: b Ax in b A ( x V y in ~ r e H y yr yr (9) Generay with the increasing iteration, the argest and the saest singuar vaue of atrix H ~ wi approxiate those of atrix A, respectivey. This eans that the probe (9) inherits properties of the syste (6) and is aso a sa i-posed probe. Therefore we use Tikhonov reguarization ethod to reguarize (9) and sove: ( ~ ~ ~ H T H I) y r H T e () Suppose that the singuar vaue decoposition (SVD) of H ~ is given by: ~ H P QT () where P =[p,..., p + ] and Q =[q,..., q ] are atrices with orthonora couns, and the diagona atrix W = diag(w,..., w ). In ters of the decoposition of () the soution of () can be expressed as:

4 Wu, J., Ma. Z.: A Reguarized GMRES Method for Inverse Backbody Radiation Probe 85 THERMAL SCIENCE: Year 3, Vo. 7, No. 3, pp w P y i i r q i () i w w i i There exists different ways of choosing the reguarization paraeter. Here the GCV criterion is epoyed. This ethod is to find the paraeter that iniizes the GCV function: ( I HH ) r e G( ) [ tr( I HH )] where H ( HH I) H T. Here the sybo H is H ~ in eq. (8) and tr(a) denotes the trace of A. Proposition. Denote K i = /(w i + ), i =,,...,, then: G( ) r ( K P ) P i K i i Proof. Foowing eq. () it is iediate to see that: I HH P I I ( ) P T Since tr(ab) = tr(ba), we have: I ( ) tr( I HH I ) tr Ki i Moreover, the nuerator is: I ( I) ( I HH r e r P T e r ( K i P i ) P i thus eq. (3) hods generay. The proposition suggests that one can estiate paraeter by finding the iniu vaue of eq. (3). If is known, then the approxiate soution x = x + V y can be found by eq. (). The agorith suarizes how the coputations for the reguarized GMRES can be organized. Agorith. (Reguarized GMRES ethod) () copute the inear syste (6) by discretizing eq. (3), () choose a starting vector x and carry out steps of the Arnodi process, (3) copute the SVD of H ~, (4) sove eq. () using GCV criterion, (5) set x = x + V y k, and (6) check convergence. If not converged et = + and continue iterating. In the agorith, the iteration nuber shoud not be too arge. Usuay we can set an upper iit for the nuber of iteration. If the Kryov subspace diension increases up to the axiu iteration nuber and the residua nor b Ax is not sa enough, we can appy the restarted GMRES ethod to get the iterate x. Nuerica resuts The reguarized GMRES ethod is appied to three exapes. These exapes are seected fro [9]. The nuerica resuts obtained by the reguarized GMRES ethod and the exact soution are given in different figures. A coputations were done in MathCAD. i (3)

5 Wu, J., Ma. Z.: A Reguarized GMRES Method for Inverse Backbody Radiation Probe THERMAL SCIENCE: Year 3, Vo. 7, No. 3, pp For nuerica error estiation, we define the reative error as: g = a(t) x / a(t), where a(t) is the exact teperature distribution and x the approxiate soution cacuated by the reguarized GMRES ethod. Exape is a Gaussian teperature distribution given by: a(t) = exp[ (T d) /5], T [, 8], where d is a paraeter. For a given n = 5, we discrete (3) and obtain the inear syste (6). It is easy to know that the coefficient atrix A with d = 45 is i-conditioned because the argest singuar vaue of atrix A is and the saest singuar vaue is. Let d =, 45, and 6. Appication of the reguarized GMRES ethod to these distributions resuts the teperature distribution a(t) as shown in fig. -3. These figures dispay the coparisons of the approxiate soution deterined by Figure. Coparison of exact and coputed teperature distributions for d = and g =.36 the reguarized GMRES ethod indicated by the dotted curve and the exact soution (soid curve). Obviousy the overa agreeent between cacuated and exact vaues dispayed in fig. and fig. is exceent. For the case of d = 6 the resuted distribution shown in fig. 3 has soe disagreeent and the corresponding reative error g = =.48. This resut is good and can be acceptabe. Figure. Coparison of exact and coputed teperature distributions for d= 45 and g =.7 Figure 3. Coparison of exact and coputed teperature distributions for d = 6, g =,48 Figure 4. Coparison of exact and coputed teperature distributions for d = 45, g =.9 Exape is the doube Gaussian teperature distribution given by: a(t) = exp(t 3) /9 + exp(t 6) /9, T [, 8]. The coputed resuts in fig. 4 shows good, but the cacuated distributions a(t) indicated by the dotted curve have soe disagreeent at the right part. Exape 3 is that of a rectanguar teperature case: 5. T 3 T 45, 3 T T In this exape the distributions a(t) is continuous, but not differentia at soe coocation points. Figure 5 shows the Figure 5. Coparison of exact and coputed teperature distributions, g =.86

6 Wu, J., Ma. Z.: A Reguarized GMRES Method for Inverse Backbody Radiation Probe 85 THERMAL SCIENCE: Year 3, Vo. 7, No. 3, pp cacuated resut (dotted curve) for this exape. The agreeent in the part of ower teperature is satisfactory but the osciations appear in the right region. This phenoenon indicates that the discontinuity and the intrinsic instabiity of the physica probe effect the reconstructed resut. Concusions In this paper the reguarized GMRES ethod is introduced to recover the a(t) fro the tota power spectra easureents of its radiation. Fro the iited nuerica resuts we find that the proposed agorith is nuericay stabe and can recover the a(t) which is continuay differentia. References [] Tikhonov, A. N., Arsenin, U. Y., Soutions of I-Posed Probes, John Wiey and Sons, New York, USA, 977 [] Bojarski, N.N., Inverse Back Body Radiation, IEEE Trans. Antennas and Propagation, 3 (98), 4, pp [3] Chen, N., Li, G., Theoretica Investigation on the Inverse Back Body Radiation Probe, IEEE Trans. Antennas and Propagation, 38 (99), 8, pp [4] Dai, Xi., Dai, J., On Unique Existence Theore and Exact Soution Forua af the Inverse Back-Body Radiation Probe, IEEE Trans. Antennas and Propagation, 4 (99), 3, pp [5] Sun, X., Jaggard, D. L., The Inverse Backbody Radiation Probe: A Reguarization Soution, J. App. Phys., 6 (987),, pp [6] Dou, L., Hodgson, R. J. W., Appication of the Reguarization Methods to the Inverse Back Body Radiation Probe, IEEE Trans. Antennas and Propagation, 4 (99),, pp [7] Li, C., Xiao., T., The Fast and Stabe Agoriths for the Nuerica Inversion of Back Body Radiation, Chinese Journa of Coputationa Physics, 9 (),, pp. -6 [8] Li, H. Y., Soution of Inverse Backbody Radiation Probe with Conjugate Gradient Method, IEEE Trans. Antennas and Propagation, 53 (5), 5, pp [9] Dou, L., Hodgson, R. J. W., Maxiu Entropy Method in Inverse Back Body Radiation Probe, J. App. Phys, 7 (99), 7, pp [] Ye, J., The Back-Body Radiation Inversion Probe, Its Instabiity and a New Universa Function Set Method, Physics Letters A, 348 (6), 3-6, pp [] Wu, J., Dai, H., Reguarizing Lanczos Method for Inverse BIack Body Radiation Probe, Journa of Nanjing University of Aeronautics & Astronautics, 39 (7),, pp [] Saad, Y., Schutz, M., GMRES: A Generaized Minia Residua Agorith for Soving Nonsyetric Linear Systes, SIAM J. Sci. Statist. Coput., 7 (986), 3, pp [3] Jensen, T. K., Hansen, P. C., Iterative Reguarization with Miniu-Residua Methods, BIT Nuerica Matheatics, 47 (7),, pp. 3- [4] Cavetti, D., et a., On the Reguarizing Properties of the GMRES Method, Nuerische Matheatik, 9 (), 4, pp [5] Cavetti, D., et a., LMRES,L-Curves and Discrete I-Posed Probes, BIT, 4 (),, pp [6] Goub, G. H., et a., Generaized Cross-Vaidation as a Method for Choosing a Good Ridge Paraeter, Technoetrics, (979),, pp. 5-3 Paper subitted: March 6, Paper revised: Apri 9, Paper accepted: May 4,