Efarmog Algorjmou gia ton Prosdiorism twn Mikrofusikn

Idiottwn twn Swmatidwn me Qrsh Majhmatikn Teqnikn

Antstrofhc Skdashc

Stfanoc S. Samarc

Diatmhmatik Prgramma Metaptuqakn Spoudn{Fusik kai Teqnologikc Efarmogc}

Sqol Efarmosmnwn Majhmatikn kai Fusikn EpisthmnEjnik Metsbio Poluteqneo

Epiblpwn KajhghtcAnap. Kaj. Alxandroc Papaginnhc

Tomac Fusikc

2012

..

iii

Perlhyh

T .

, - , . .

, lidar, laser . - , , lidar . Fredholm .

-, , , . , -. .

, , Galerkin. - SVD, . , TSVD, SSVD Tikhonov, , -. , , L .

, Muller, Veselofski, Griaznov Kol-gotin. , , .

v

Euqaristec

J ,

., . , -

.

. , ,

.

vii

ix

Perieqmena

v

vii

xiii

xv

xvii

1 11.1 : ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 ,

, . . . . . . . . . . . . . . . . . . 41.4 Lidar: . . . . . . . . . . . . . . . . . . . . 41.5 Lidar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Fredholm . 112.1 Fredholm : . . . . . . . . . . . . . . . . . . . . 112.2 (SVE) Picard . . . . . . . . . . . . . . . . . . . 132.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 . . . . . . . . . . . . . . . . . . . . . . . . 19

3 233.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5 SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.6 Picard . . . . . . . . . . . . . . . . . . . . . 35

4 : 414.1 . . . . . . . . . . . 414.2 SVD (TSVD) . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 SVD (SSVD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 Tikhonov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.6 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

xi

5 595.1 . . . . . . . . . . . . . . . . . . . . 595.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.4 . . . . . . . . . . . . . . . . . . . . . 70

6 Lidar - 756.1 . . . . . . 756.2 . . . . 776.3 RAP . . . . 80

6.3.1 Discrepancy Pri-nciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.3.2 GCV . . . . . . 886.3.3 . . . . . . . . . . . . . . . . . 90

- 103

Katlogoc Sqhmtwn

1.1 CO2 (ppmv) - Mauna Loa, 1958-2009 (ppmv) . . . . . . 21.2 (Wm2) , 1750-2000 . . . . . 31.3 - -

TOMS NASA . . . . . . . . . . . . . . . . . . . . . . . . 51.4 lidar . . . . . . . . . . . . . . . . . . . . . 5

2.1 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Riemann - Lebesque f(t) = sin(t) . . . . . . 132.3 i, i . . . . . . . 152.4 Picard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Ursell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 shaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 baart 283.3 i ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Picard ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6 Picard Ursell. . . . . . . . . . . . 343.7 Picard ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.8 Picard , -

Picard . . . . . . . . . . . . . . . . . . . . . 373.9 Picard L. Fox

E. T Goodwin . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 . . . . . . . . . . . . . . . 42

4.2 TSVD . . . . . . . . . . . . 454.3 SSVD

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 TSVD

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5 Picard . 50

4.6 []i Tikhonov i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.7 Tikhonov x , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.8 TSVD Tikhonov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

xiii

4.9 L Tikhonov deriv2 shaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1 - k shaw. . . . . . . . . . . . . . . . . . . . . . . 60

5.2 Picard shaw, TSVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 TSVD shaw . . . . . . . . . . . . 635.4 (discrepancy principle)

shaw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.5

shaw, Ax b2 = n1/2 Axk b2 = e2. . . . . . . . . 645.6

shaw, Ax b2 = n1/2 dp = 2. . 655.7 5.5 5.6), e2. 665.8 L . . 695.9 generalized cross validation -

Tikhonov shaw gravity . . . . . . . . . . . . . . . . . 725.10 generalized cross validation -

Tikhonov . . . . . . . . . . . . . . . . . . 725.11 generalized cross validation -

TSVD . . . . . . . . . . . . . . . . . . . . 73

6.1 (volume distribution) rmin = 0.04 m.d. 40, 30, 20 10%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2 m.d.=20% inversion windows. . . . . . . . 836.3 l.rmin = 0.04 m.d. 40, 30, 20

10%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.4 l.rmin = 0.07, m.d 10% 20%

m.l.r.e. 10% 20%. . . . . . . . . . . . . . . . . . . . . . . 866.5 l.rmin = 0.13, m.d.=20%, m.l.r.e.=20%. 876.6 m.d.=10% m.l.r.e.=20% inversion

windows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.7 left rmin run 1. . 926.8 m.d.=30% discrepancy principle (run 1)

m.d.=40% GCV (run 2) m.l.r.e.=10%. . . . . . . . . . . . . 936.9 rmin = 0.07 m.d.=20% run 1, 2 rmin = 0.17

rmin = 0.17, m.d.=10% m.l.r.e.=10% . . . . . . . . . . . . . . . . . . . . . . . 946.10 left rmin run 2. . 956.11 Raman lidar . . . . . . . . . . 966.12 Range corrected Raman lidar -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.13 l.rmin = 0.04 m.d. 30, 20, 12 5%. . . . . . 986.14 r l.rmin = 0.04 m.d. 30,

20, 12 5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.15 l.rmin = 0.04, m.d.=5% m.l.r.e.=10%. . . 996.16 l.rmin = 0.07, m.d.=12% m.l.r.e.=10%. . . 1006.17 m.d.=12% m.l.r.e.=10% l.rmin

0.04, 0.07, 0.13, 0.17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.18 l.rmin

0.04 0.07 m.d.=5% m.l.r.e.=10%. . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.19 left rmin run 3. . 102

Katlogoc Pinkwn

1.1 (- - ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

6.1 run 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 run 1: (. . 6.1), discrepancy principle. 856.3 run 2: (. . 6.1), GCV . . . . . . . 896.4 run 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.5 run 3: (. . 6.4), discrep. principle. 97

xv

Eisagwg kai Knhtra

T (aerosols particles) ( , , ) ( -

, , , . (aerosol) (.. ). , - . () , - , - - . (radiative forcing) .

. ,, , . laser (advanced laser remote sensing). , (single scattering albedo), . lidar, , . lidar , .. lidar , ., lidar .

EARLINET (European Aerosol Research Lidar Network) - lidar , 20000 , lidar, lidar . - EARLINET ,

Oi ekfrseic {aiwromena swmatdia} {swmatdia twn aerolumtwn} enai tautshmec kai enallssontai sto kemeno.'Otan de dhmiourgetai sgqush mpore na anafrontai apl kai wc {aerolmata}. O roc {metastajc} rqetai na perigryeiton dunamik akmh qaraktra (makru ap jermodunamik isorropa) twn poiotikn qarakthristikn tou. Bl. [HW99].O roc anafretai sth bibliografa kai wc Uetc kai afor kje ptsh enapjesh sto dafoc prontwn tou datoc

(se ugr stere morf, epimerismnh) ta opoa prorqontai ap sumpknwsh twn udratmn thc atmsfairac, pwc Broq,Qionnero Qionbroqo Qionluto, Yekdec, Qalzi, Qini, Qionkokkoi, Pagobelnec, Pagkokkoi kai o Ualopgocpou dhmiourgetai mwc sto dafoc.Anafretai se mroc tou fsmatoc radiokumtwn me sqetik megla mkh kmatoc.Metabol thc olikc isqoc thc aktinobolac metax strwmtwn thc atmsfairac. Metritai se Watt/m2 kai posoti-

kopoietai kurwc sthn troppaush.Leukthta memonwmnhc skdashc orzetai wc to phlko, = Cscatt

Cscatt+Cabs, pou Cscatt kai Cabs, h energc diatom

skdashc kai aporrfhshc ap ta aerolmata antstoiqa. Bl. [PA10].

xvii

( )

., ,

Fredholm ,

() =

rmaxrmin

r2Q/ext(r, ,m)n(r)dr

Q Mie (. ), (ext) (extinctioncoefficients) () (backscatter coefficients), (), . , rmax, rmin . , n(r) ( ), , m . , n(r). , , , , . , Fredholm , , , .

() , - . , , ( Raman) , - . , . - , , .

, . , . . , , ( ) () .

, - . . ( ) - , , . , , , . , - : , , . ( () ).

, , , , . , , , , . (5) . SVD ,

Gia mh sfairik swmatdia (pwc h sknh ap th Sahara) h jewrhtik pleur thc skdashc tou fwtc den qei anaptuqjeidiatera akma, opte den enai dunat h antistrof thc oloklhrwtikc exswshc Fredholm a' edouc.

xviii

( )

.

Fredholm . . DetlefMuller, ( ) .

, , - . Regularization Tools Matlab, Octave. Christian Hansen, . [HC94] () Netlib http://www.netlib.org/numeralgo.

- Fredholm , , , , -, lidar .

xix

http://www.netlib.org/numeralgo

Keflaio 1

Pagksmia Klimatik Allag

1.1 Pagksmia Klimatik Allag: Rpoi, Jrmansh thcGhc kai Episkphsh Msw enc Aplo Montlou

H , , 4.6 -, (CO2), (N2), (H2O)

(H2). , , , , 400 . - . , .

() , , . , , . ( ) - , , , (trace gases: CO2, CH4, O3, N2O,NOx, .). - CO2 /, , ., 6 .

(- CO2) ( CO2 ). CO2 . CO2, . CO2 CO2. 40-50% CO2 , . [SG02]. CO2 0.35% , , - Mauna Loa .., 1958-2009 (RobertSimmon, NOAA Climate Monitoring & Diagnostics Laboratory) 1.1.

. , , , F ,

F = (1R)Fs/4,

R (albedo), . , Fs = 1370Wm2 (

1

1.2. ( )

1.1: CO2 ( ppmv) Mauna Loa .., 1958-2009 (Robert Simmon, NOAA ClimateMonitoring & Diagnostics Laboratory).

). , FT , ( ) Stefan Boltzmann, FT = T 4s , < 1 Ts . , CO2 0.3K(Wm2)

1,

1.4K. :

, (1 5 km),

(, , -) ( ) -. , ( ) . [PA10]

1.2 Katakrthsh Aktinobolac

H (radiative forcing), , ,

(WMO, 2003). - , (positive radiative forcing) (negative radiative forcing), - .

CO2 , CH4, O3, N2O,NOx, H2O, - . , 1) (. [AC92, KT02]), ( direct aerosol effect, 2) (cloud condensation nuclei: CCN) , - ,

2

1.2. ( )

1.2: ( Wm2) (CO2, CH4, O3, N2O,NOx, , .), 1750 ( ) 2000 (IPCC, 2007).

(. [LF01]), ( - indirect aerosol effect). , , . , .

, , , , . , (organic carbon: OC) (black carbon: BC). OC -, BC .

OC - ( direct cooling effect). , BC (mineral dust) , ( direct warming effect).

1.2 ( Wm2) (CO2, CH4, O3, N2O,NOx, ,.), 1750 ( ) 2000, . (IPCC: Intergovermental Panel on Climate Change) . 1850 . (error bars). CO2 (+1.4 Wm2), CH4 (+0.5 Wm2), 3 (+0.4 Wm2) BC, (+0.35Wm2). . , .

3

1.3. , , ( )

(-0.2 -2), OC (-0.1Wm2 -0.4Wm2, ). , indirect aerosol effect ( 2Wm2)(Houghton, et al., 2001)

1.5-4.5C 21 ( 0.1 C 0.4 C /). ( C) 2070 Max Planck Institut fur Meteorologie,Hamburg, ), 4-5C . , .

1.3 Katakrufh kai Orizntia Katanom Aerwn, Entopi-smc Aiwromenwn Swmatidwn sthn Atmsfaira, E-pdrash sthn Pagksmia Klimatik Allag

O . ( ) (

/ .), ( .) N2, CFC (CO2, CH4 .). O3 CO ( ) ( / ) ( ) . .

. - . - lidar (, ) , AERONET ( ) ( ) . EARLINET(. [Bo03]) lidar, lidar 2000-2003 60 - 22 lidar (Ansmann et al., 2003, Papayannis et al., 2004a, Papayannis etal., 2005).

1.3 () - aer, (km1sr1) 400 m 8000 m , 30/08/2003, 11:52 16:00 UT, lidar (Papayannis et al., 2004b). TOMS NASA, (aerosol index), 1.3 () 12:00 UT.

1.4 Teqnik Lidar: Arq Leitourgac kai Edh Metrsewn

H lidar, (LIght Detection And Ranging).

laser , , , , , -, (remote sensing) .

H bisfaira kaletai to exwterik perblhma tou planth. Perilambnei ton ara, to dafoc, to oikologik ssthmapou enswmatnei louc touc zwntanoc organismoc kai tic metax touc sqseic, perilambanmenhc thc allhlepdrashc toucme ta stoiqea thc lijsfairac (petrmata), thc udrsfairac (ner), kai thc atmsfairac (arac), [CE04].Suntelestc opisjoskdashc (skdashc se gwna = ) gkou R volume backscattering coefficient dnetai proseggi-

stik ap th sqsh, R = NR( = ), pou R h energc diatom skdashc (Rayleigh) kai N = 2.55 1019mol/cm3. O dekthc aiwromenwn swmatidwn enai nac dekthc pou aniqnuei thn parousa UV aporrofhtikn aerolumtwn,

pwc sknh kai aijlh, [NASA, GES DISC (Goddard Earth Sciences Data and Information Services Center)].

4

1.4. Lidar: ( )

1.3: : - aer, (km1sr1) 30/08/2003, 11:52 16:00 UT. : TOMS NASA, (aerosolindex) 12:00 UT.

1.4: lidar: laser (ns) laser, lidar.

laser , - laser. 1.4 lidar. , , , . laser. lidar. lidar, - laser, ( 3 7 pm) ( 10 30 s min.). lidar,

5

1.4. Lidar: ( )

, :

. ( ) (.. ) .

(5-1000 m) (1-10 s).

( m 100-120 km).

1-2-3 .

.

lidar :

N(, r) = Ne() ct

2 (, r) A () () O(r) r2 e2(,0,r), (1.4.1)

, N(, r) r,Ne() , A , () - lidar, c , t laser, () , O(r) lidar r,(, r) (, 0, r) (aer), (mol) (c) :

(, r) = aer(, r) + mol(, r),

(, 0, r) =

r0

[mol(, r) + aer(, r

) + c(, r)] dr,

(r, ) (extinction coefficient) . , r = ct/2 lidar.

laser , . Rayleigh Mie, (- ) (- ) (aer) , . Raman , (Raman - Stokes) (Raman -anti-Stokes) -. Raman DIAL (DifferentialAbsorption Lidar) . lidar , , . 1.1 ( - ) (, -, , .).

O suntelestc autc ekfrzei thn pijanthta to pedo thc ekpempmenhc dsmhc laser na brsketai msa sto optikpedo tou thleskopou lyhc gia sma lidar pou prorqetai ap apstash r. Tupik ta smata lidar pou qrhsimopoiontaienai gia O(r) = 1 (full overlap).To optik pqoc orzetai ap th sqsh (0, r) =

r0 (, r

)dr.H jewra skdashc pou anaptqjhke ap ton Gustav Mie dnei thn analutik lsh gia skedazmenh aktinobola gia

tuqaec timc tou mkouc kmatoc, thc aktnac thc sfarac kai tou migadiko suntelest dijlashc. Entotoic, sto parn,ja diathrsoume aut ton {paraplanhtik} orism pou qrhsimopoietai sth bibliografa. Bl. [WE04, Mi08].Anafretai se mh parathrsimec kbantikc katastseic mikro qrnou zwc se pragmatikc all mh eustajec katast-

seic, bl. [GL53], sel. 61. Uprqei mhdenik pijanthta na breje to ssthm mac, katpin mtrhshc, se aut thn {eikonik}katstash, apl msw autc parqetai h aparathth szeuxh metax arqikn kai telikn katastsewn, bl. [WD99], sel.22.

6

1.5. Lidar ( )

1.1: (- - ). - -

-

1 = 2 , , , , -

( Raman)1 = 2 + R

, 3, , .

DIAL 1, 2 SO2, O3, NO2, NO, CO2, Hg, HCl, NH3, HCs,CO, H2O

K, Na, Li, Ca, Fe (LIF) OH

1.5 Eplush thc Basikc Exswshc Lidar

G lidar (multiple scattering), single scattering.

, lidar (1.4.1), :

Pss(r) =K(r)

r2(r)e2(r), (1.5.1)

Pss(r) , ss - , K(r) 1.4.1.

S(r) = ln(Pss(r)

r2

K(r)

) r,

dS(r)

dr=

1

(r)

d(r)

dr 2(r), (1.5.2)

(r), (r). , . d(r)dr = 0, (r)

(r) = 12

dS(r)

dr (r) = 1

2S(r) + a0.

, J. D. Klett (1981 1985, . [Kl81]) :

(r) = Cu(r), (1.5.3)

C, u . 1.5.3, . u = 1, C = (r)(r) lidar (lidar ratio). . 1.5.3 . 1.5.2,

dS(r)

dr=

u

(r)

d(r)

dr 2(r). (1.5.4)

Riccati ( Bernoulli)

(r) =exp(S(r)pS(rm)u )

1(rm)

+ 2u rmr

exp(S(r)S(rm)u ) dr

, (1.5.5)

Afor fainmena memonwmnhc skdashc.Sthn pragmatikthta eisgetai h nnoia tou diorjwmnou smatoc lidar P (, r), ap to opoo qei afaireje o jruboc

upobjrou (background noise).

7

1.5. Lidar ( )

rm (.. 8-10 km), ().

(0) 1.5.3, . , , lidar , .

. Wiskonsin (. [WE04] . 5, [ST83]) lidar Rayleigh Mie. Rayleigh Doppler, Mie . , (high rejection power notch filter) Mie, , . 1.5.1, , . (. [WE04, AW92]). , , . Ansmann,Wandinger, Riebesell, . (. [AW92]) , , Raman , Raman laser ( ).

Pss(r, R) =K(r)

r2NN2(r)

dRm(, , r)

de[(r,0)+(r,R)], (1.5.6)

0 R laser 2 - Raman -, NN2(r) d

Rm(,,r)/d 2 - Raman

( = ). (r, 0) (r, R) ( mol(r, 0) mol(r, R) - ) (.[AW92]) 0 R , . . 1.5.6 aer(r, 0), . 1.4.1 1.5.6 (r, 0), (r, R) (r, 0).

. - lidar (. [MA72, AB73]) . lidar . - lidar (.. Monte Carlo Methods, Stochastic and Phenomenological Methods, .), , , (effective mediumtheorem) Katsev (Quasi Small Angle) (. [KZ97]).

. - . - lidar - . .[MA72, AB73, B73, C75, PC73, Ch50, ZI91, I78].

Ed enai epitaktik angkh metrsewn uyhlc akrbeiac, diti to sma Raman enai txeic megjouc mikrtero ap tosma twn aerolumtwn. Gia sgkrish, h N2 Raman diatom enai mikrterh ap thn atmosfairik Rayleigh diatom kat napargonta 1000.H skdash se gwna akribc 180 thc aktinobolac ap sfairik swmatdia diathre th grammik plwsh thc phgaac

dsmhc laser.Fusik fainmeno metaforc engeiac se morf hlektromaghtikc aktinobolac.

8

1.6. ( )

1.6 Antstrofo Prblhma

'O , ( -) lidar .

. , - ( ) - Raman lidar . : Raman lidar .

() =

rmaxrmin

r2Qext(r, ;m)n(r) dr, () =

rmaxrmin

r2Q(r, ;m)n(r) dr, (1.6.1)

n(r), m . , rmin rmax , Qext/ Mie . - , () (. [BH83])

K(m) = r2Q(r, ,m) =

2

n=1(2n+ 1)(1)n(an bn)2

,K

(m)ext = r

2Qext(r, ,m) =2< [

n=1(2n+ 1)(an + bn)] ,

(1.6.2)

= 2/ an, bn (n 1) :

an = an(x;m) =mn(mx)

n(x)n(x)

n(mx)

mn(mx)n(x)n(x)n(mx),

bn = bn(x;m) =n(mx)

n(x)mn(x)

n(mx)

n(mx)n(x)mn(x)n(mx),

(1.6.3)

, n(t) =

t2 Jn+ 12 (t), n(t) =

t2 Yn+ 12 (t), n(t) = n(t) n(t), Jn, Yn -

Bessel = r. , . lidar, -

Raman lidar 3+2 ,. (355, 532 1064 nm) 2 (355, 532 nm). - 6+2 . 3 . 3+2 6+2, . 3+2 EARLINET .

, , lidar. , 1.6.1. , . , , .

Optikc parmetroi jewrontai epshc kai to optik pqoc () kai h sunrthsh fshc (p).Kurwc jewrome sfairik swmatdia, dhl. optik dedomna lidar me mikroc lgouc ekplwshc (depolarization ratios).

9

Keflaio 2

Oloklhrwtik Exswsh Fredholma' Edouc.

2.1 Oloklhrwtik Exswsh Fredholm a' Edouc: Idithtec

O 1.6.1 Fredholm

Kf =

10

K(s, t)f(t)dt = g(s), 0 s 1. (2.1.1)

, K g , f . f g K K. K . [a, b] [0, 1], , , .. l : [0, 1] 7 [a, b], l(t) = (ba)t+a.

f K , g -. f , .

F X,Y . F (x) = y x Hadamard :

1. x X, y Y ,

2. ,

3. ,

. , (3), (1) (2) .

(), g,. gg , g g 0, . , f, K(f) = g, f , f f . (3) .

, / , - , . , , .

'Ena prblhma kaletai uperkajorismno an oi exisseic tou enai perissterec ap touc agnstouc, en kaletai upo-kajorismno an oi gnwstoi enai perissteroi ap tic exisseic.

11

2.1. Fredholm : ( Fredholm . )

. , minx Ax b2,

A =

0.16 0.100.17 0.112.02 1.29

, b = A( 11

)+

0.010.030.02

= 0.270.25

3.33

(0.01,0.03, 0.02)T b

(11

),

. Ax = b.

xLS =

(7.018.40

) .

A. , , , .

2.1.1, - () f(t) g(s) - . . f g A, f . , K(s, t) f , g f .

Riemann - Lebesque . f(t) = sin(2t), = 1, 2, . . .,

g(s) =

10

K(s, t)f(t)dt 0, . (2.1.2)

(2.1.2) : f ( ) g .

, (gravity surveying model). 2.1.1, K s t, K(s, t) = h(st), h . (deconvolution problem) 1

0

h(s t)f(t) dt = g(s), 0 s 1, (2.1.3)

. - K(s, t) = d

[d2+(st)2]3/2 . d

f(t), 0 1 t . - , . , s 0 1, , g(s)., ,

g(s) =

10

d

[d2 + (s t)2]3/2f(t) dt, 0 s 1. (2.1.4)

, , f(t) = H(t)+H(0.3 t), H(t) (Heaviside). , 2.1 , 3 d.

f(t) - K, (Riemann - Lebesque. , f(t) = sin 2t = 1, 2, . . ., () g. 2.2, . , . f(t) g(s) ,

12

2.2. (SVE) Picard ( Fredholm . )

2.1: -. f(t) ( ) , g(s) ( ) 3 d.

2.2: Riemann - Lebesque f(t) = sin(t). g .

. g - f , . , g, f , .

2.2 Anptugma Idizouswn Timn (SVE) kai Sunjkh Pi-card

T ., . -

, , , . , -

13

2.2. (SVE) Picard ( Fredholm . )

lidar, , Fred-holm . 1.6.2. , , , . , .

, , . a (a), K K (adjoint operator) :x X y Y , X,Y Hilbert

Kx, y = x,Ky.

, Fredholm K (s)

[K](s) =

10

K(s, t)(t) dt.

( ) , A A.

, , , [0, 1] ,

, 1

0

(t)(t)dt,

2-

2 , 1/2 = 1

0

(t)2dt

K, 2.1.1, ,

10

10

[K(s, t)]2dsdt . K

(SVE, Singular Value Expansion)

K(s, t) =

i=1

ii(s)i(t). (2.2.1)

i i

i, j = i, j = ij , i = 1, 2., . . .

i i () L2[0, 1], f(t), g(s)

f(t) =

i=1

i, fi(t) g(s) =i=1

i, gi(s) (2.2.2)

, K

Ki =ii, (2.2.3)

Ki =ii, (2.2.4)

Kx =

i=1

ix, ii, (x X) (2.2.5)

Kx =

i=1

iy, ii, (y Y ). (2.2.6)

To disthma [0, 1] eisgetai gia eukola pwc kai prohgomena, en ja mporosame na qrhsimopoisoume to geniktero[a, b], bl. kai 2.1 1h par.

14

2.2. (SVE) Picard ( Fredholm . )

2.3: i, i, i ( ). C. Hansen(. [HC10]) .

i ,

1 2 3 0

0 . q K , - , i O(iq1/2), K .

(degree of ill-posedness) a i = O(ia) :

(i) (mildly ill posed) , 0 pa 1

(ii) (moderately ill-posed), a > 1,

(iii) (severely ill-posed), i = O(eai).

i, () i, i. 2.3.

. ,

Ki = ii 1

0

K(s, t)i(t) dt = ii(s), i = 1, 2, . . . . (2.2.7)

SVE , .

f g 2.2.2 . .2.1.1 . 2.2.2 . 2.2.7,

g(s) =

10

K(s, t)

i=1

i, fi(t) dt

=

i=1

i, f 1

0

K(s, t)i(t) dt

=

i=1

ii, fi(s), (2.2.8)

- Lebesque.

{Exsou} jemelidhc enai kai h exswsh

Ki = ii 10K(s, t)i(t) dt = ii.

.Aut enai anamenmeno, afo oi i, i apotelon bseic tou diou qrou (L2[0, 1]).

15

2.2. (SVE) Picard ( Fredholm . )

2.4: Picard. , - . , i SVE, i, g i, g/i g f . , , g, . [HC10].

. 2.2.2 2.2.8, i ii . , Fourier.

. 2.2.2 . 2.2.8 i 6= 0, i, f = i, g/i. , , , f(t) , , Picard

f22 = 1

0

[f(t)]2dt =

10

( i=1

i, gi

i(t)

)2dt =

i=1

(i, gi

)2 10

2i (t) dt

i=1

(i, gi

)2 N i,gi 0. , . 2.4 . Picard . Fredholm , SVE.

, , SVE, . Fredholm

10K(s, t)f(t) dt = g(s), 0 s 1 ,

K(s, t) =

{s(t 1), s < t,t(s 1), s t.

(2.2.10)

: g, f g, . f(t) = g(t), 0 t 1.

K(s, t) = 22

i=1

sin(is) sin(it)

i2(2.2.11)

16

2.3. ( Fredholm . )

i =1

(i)2, i(s) =

2 sin(is), i(t) =

2 sin(it), i = 1, 2, . . . . (2.2.12)

Fredholm , . , 2- .

F. Ursell[UF74] 10

1

s+ t+ 1f(t) dt = 1, 0 s 1. (2.2.13)

K(s, t) = (s+ t+ 1)1 g(t) = 1. SVE,

gk(s) =

ki=i

i, gi(s). (2.2.14)

2.5 gk k ,

g gk2 0 k .

, . 1

0(s+ t+ 1)1fk(t) dt = gk(s),

fk(t) =

ki=1

i, gi

i(t), (2.2.15)

, 2.5, fk2 k, k fk2 k . .

, , SVE . Laplace, . , Laplace

L[f(t)] = g(s) =

0

estf(t) dt, s 0. (2.2.16)

K(s, t) = est. , a0

(est)2 ds =

a0

e2st ds =1 e2ta

2t 1

2t, a, (2.2.17)

01t dt .

0

0

(est)2 ds dt . SVE .

2.3 Amfishmea sth Lsh

E Fredholm .

g, . ,

Topik oloklhrsimh sunrthsh (locally integrable function) kaletai h sunrthsh pou enai oloklhrsimh se kjesumpagc uposnolo tou pedou orismo thc.

17

2.3. ( Fredholm . )

2.5: Urshell. : 2- ggk (2.2.14) . : 2- fk (2.2.15). : fk, k = 1, . . . , 8. . [UF74].

2 Hadamard, .

(nullspace ambiguity) , . fnull 6= 0

Kfnull = 0 1

0

K(s, t)fnull(t) dt = 0.

fnull , afnull a . , K(s, t). , . fnull , . . 2.2.7, null(K) = span{i|i =0}. , , . (i > 0), . K(s, t) = s+ 2t fnull = 3t2 1,

11

(s+ 2t)(3t2 1) dt = 0, 1 s 1

Legendre 2.

- . , , .

, , , (formulation ambiguity). , . .

18

2.4. ( Fredholm . )

2.4 Fasmatikc Idithtec Idizouswn Sunartsewn

A Fourier eiks/

2, i ,

.

K

[Kf ](s) =

K(s, t)f(t) dt.

:

1. K ( K = K) C1([, ] [, ]). K (2.) .

2. K(, t)K(, t)2 = 0. - , O(1).

K , j(x) j(x) j > 0

[Kj ] (s) = jj(s), [Kj ] (s) = jj(s), j = 1, 2, . . . .

B, k = , . . . ,, j = 1, 2, . . . ,

Bkj =

j ,

eiks2

. , eiks/2 L2([, ]), j L2([, ]). . B, .. j = a,

. . . ,a,

elis2

, . . . ,

a,

12

, . . .

a,

elis2

, . . . a

eiks/2, k = , . . . ,. j k.

B < , . - V, V k = 0, j = 1. , . , . ([HK06]), . .. ([AG80, KS09]) . B j .

Cauchy-Schwarz ,

Bkj = j2

eiks2

2

Bkj 1

B, , , . Parseval , H Hilbert , en H, H ( ), x H

n

|x, en|2 = x2 (2.4.1)

Grammik jkh perblhma (linear span) kaletai to snolo lwn twn grammikn sunduasmn twn en.

19

2.4. ( Fredholm . )

H = L2([, ]), x = uj , ek = eiks/

2,

k=

B2kj = 1 (2.4.2)

Bessel Parseval en, = . (2.4.1) . x = eiks/

2 ej = uj ,

j=1

B2kj 1 (2.4.3)

, , (2.4.2) (2.4.3) . B. k 6= 0,

j ,eiks

2

=

1

jKj ,

eiks2

=1

j

j ,K

eiks

2

=

j(t)

K(s, t)eiks

2ds dt. (2.4.4)

,

K(s, t)eiks

2ds =

1

ik

{[K(, t)K(, t)

](1)k

2

K

s

eiks2

ds

}, (2.4.5)

eki = eki = cos(k) = (1)k. (2.) 1 , . (2.4.4) Cauchy-Schwarz,

j ,

eiks2

= 1|k|jj ,

K

s

eiks2

Bkj 1kjKs eiks2

2

(2.4.6)

,

1 = maxf2=1

Kf2.

f = eis

2(k = 1),

1

K eis2

2

=

Ks eis2

2

,

. (2.4.5).

Riemann-Lebesque k , Ks eis22 0.

|k| > 1

1>Ks eis2

2

,

2.4.6

Bkj

2.4. ( Fredholm . )

< , 1 k, . , j, j

Fourier eiks

2 |k| j. -

, , : Fourier ( j) , ( j) .

21

Keflaio 3

Diakritopohsh GrammiknAntstrofwn Problhmtwn

T SVE , , . -

. , , , . , - . SVE, , SVD (singularvalue decomposition), , .

3.1 Mjodoc Tetragwnismo

A fj f t1, t2, . . . , tn, .,

fj = f(tj), j = 1, 2, . . . , n.

fj f , f , . ,(quadrature methods, Nystrom) : 1

0

(t) dt =

nj=1

j(tj) + En,

, En -, t1, t2, . . . , tn 1, 2, . . . , n . , (midpoint rule) [0, 1]

tj =j 12n

, j =1

n, j = 1, 2, . . . , n. (3.1.1)

- , () f(tj). s:

(s) =

10

K(s, t)f(t) dt =

nj=1

jK(s, tj)f(tj) + En(s),

23

3.2. ( )

s. , (collocationrequirement), g g m :

(si) = g(si), s = 1, 2, . . . ,m.

:

nj=1

jK(si, tj)f(tj) = g(si) En(si), i = 1, 2, . . . ,m.

m n . , m = n. , En(si) , .

nj=1

jK(si, tj)fj = g(si), i = 1, 2, . . . , n (3.1.2)

, n n fj = f(tj)1K(s1, t1) 2K(s1, t2) nK(s1, tn)1K(s2, t1) 2K(s2, t2) nK(s2, tn)

......

...1K(sn, t1) 2K(sm, t2) nK(sn, tn)

f1f2...fn

=

g(s1)g(s2)

...g(sm)

, (3.1.3) Ax = b, A n n , aij = jK(si, tj), b = g(si) x = f(tj) - , i, j = 1, . . . , n.

3.2 Mjodoc Anaptugmtwn

A

f (n)(t) =

nj=1

jj(t), (3.2.1)

1(t), . . . , n(t) . , . , , , f (n) .

- . Galerkin Petrov-Galerkin, - . , , i j ,

f(t) = f (n)(t) + Ef (t), f(n)(t) span{1, . . . , n}, (3.2.2)

g(s) = g(n)(s) + Eg(t), g(n)(s) span{1, . . . , n}. (3.2.3)

Ef , Eg . j (3.2.1), f (n) . ,

(s) =

10

K(s, t)f (n)(t) dt =

nj=1

j

10

K(s, t), j(t) dt,

24

3.2. ( )

f, g

(s) = (n)(s) + E(s), (n) span{1, . . . , n}.

i j . , , , , . , .

Galerkin g span{1, . . . , n}, g(n), Eg. , (n) g(n) g span{1, . . . , n}, E, Eg span{1, . . . , n}, . E, i = 0 Eg, i = 0, i = 1, . . . , n.

Galerkin j , , (n)

g(n) :

(n) = g(n) (s) g(s) = E(s) Eg(s).

E, Eg span{1, . . . , n}, Galerkin

i, g = 0 , = i, g i = 1, . . . , n. (3.2.4)

. 3.2.4 - . Hn = span{1, . . . , n} H .

f (n) . 3.2.1 Galerkin 3.2.4, ,

i, g =i,

10

K(s, t)f (n)(t) dt

=

nj=1

i

i,

10

K(s, t)j(t) dt

, i = 1, . . . , n. (3.2.5)

3.2.5 nn Ax = b, - x 3.2.1,xi = i, A b

aij =

10

10

i(s)K(s, t)j(t) ds dt (3.2.6)

bi =

10

i(s)g(s) ds. (3.2.7)

, . , K (K(s, t) = K(t, s)) i = i, A Rayleigh-Ritz.

Galerkin, - (top hat functions / scaled indicator funtions) h = 1/n

i(t) =

{h1/2, t [(i 1)h, ih]0 ,

i = 1, . . . , n. (3.2.8)

, -, (i2 = 1).

i(t) = i(t) i(t) = i(t) i = 1, . . . , n.

25

3.3. ( )

Galerkin. , A b

aij =h1 ih

(i1)h

jh(j1)h

K(s, t) ds dt

bi =h1/2

ih(i1)h

g(s) ds.

.

3.3 Katllhlh Epilog Mejdou

M , .

, . -

, aij , K . , - K f , .. K (singular) . , - . , .([WT92, AS64]). , , fj tj . , 3.1 . shaw Regularization Tools(RT) n = 24, . , n . shaw .

K(s, t) = [cos(s) + cos(t)]

[sin(u)

u

]2, u = [sin(s) + sin(t)] (3.3.1)

f(t) =1ec1(tt1)2 + a2e

c2(tt2)2 , 2 s, t

2. (3.3.2)

a1 = 2, c1 = 6, t1 = 0.8, a2 = 1, c2 = 2, t2 = 0.5. A x. . b = Ax.

, .

Dnoume endeiktik do suqnc ekfnseic idizontoc purna. (i) Anmala shmea (singularities) tou purna pou enaisuqn antimetwpsima me allag metablhtc. P.q. an uprqei idiomorfa tou K(s, t) t1/2sto t = 0, aretai jtontacz = t1/2. ii) Idiomorfa kata mkoc thc diagwnou s = t (bl. p.q prblhma deterhc paraggou kai ex. 2.2.10). 'Enac trpocna xeperaste (o aplosteroc, swc qi o kalteroc) enai me afaresh thc idiomorfac wc exc, 1

0K(s, t)f(t) dt =

10K(s, t)[f(t) f(s)] dt+ f(s)

10K(s, t) dt

Kai stic do periptseic diakritopoiome katpin me mjodouc tetragwnismo, pijann me thn epiplon qrsh diorjwtikcparamtrou. Parathrome ti meletme thn anmalh sumperifor tou K, en ja prepe na dome th sunolik oloklhrwtaposthta K(s, t)f(t). Kat genik kanna, mwc, nac idizon purnac den pargei idizousa lsh f(t). Bpl. ent. 2.1

jewreste gia pardeigma K(s, t) = (s t) pou enai apl o tautotikc telestc, K = I, dhl. Kf = f .Apokptontac to peiro eroc enc disthmatoc oloklrwshc se ma peperasmnh tim erouc ja prpei na enai h

stath lsh antimetpishc. Ant auto metasqhmatzoume to dasthma oloklrwshc, sunjwc sto dasthma [1, 1]. P.q.oi metasqhmatismo t = z

1z2 t = a +z

1z t = a 1zz

apeikonzoun ta diastmata [,], [a,] kai [, a] sto[1, 1] antstoiqa. Endeiktik, p.q. gia thn prth perptwsh, ja qoume

K(s, t)f(t) dt =

11

K(s,z

1 z2)f(

z

1 z2)1 + z2

1 z2dz

Katpin qrhsimopoiome kapoia tetragwnik mjodo, pijann eisgontac kpoia parmetro beltwshc.

26

3.3. ( )

3.1: shaw

RT[HC10] . , .

, , , SVE SVD , . , (.. spline, thin plate smoothing, ).

t, f (n) j(t), . 3.2 ( baart ), i (3.2.8). , (scaling) , . , n, . , baart RT n = 40, , ()

K(s, t) = es cos(t), g(s) = 2sinh(s)

s f(t) = sin(t),

t [0, ] s [0, 2 ]. Galerkin (box functions) . A , . b. Simpson.

, , bi = g(si) . , si. Galerkin i -. , i(s) = i(s) (. .(3.2.8) ), , . . , , i(s) = (s si) si g, 1

0

(s s1)g(s) ds = g(si).

Enallaktik onomasa twn sunartsewn pou orsthkan sthn ex. 3.2.8. Sunartseic box orzontai kai msw thc sunr-thshc bmatoc heaviside

uab(t) = H(t a)H(t b) =

0, an t < a

1, an a t b0, an a > t

27

3.4. SVD ( )

3.2: baart

RT[HC10] i (3.2.8). , . , , h1/2 .

3.4 H paragontopohsh idizouswn timn SVD

S SVE Fredholm . ,

, . , (Singular Value Decomposition).

, - , . , . , A Cmn m n SVD

A = UV =

ni=1

iii . (3.4.1)

, Rnn i

= diag{1, . . . , n}, 1 2 n 0.

:

AF

mi=i

nj=1

a2ij

1/2 = (trace(AA))1/2 = (trace(V 2V ))1/2 ( Frobenius)=(trace(2))1/2 =

(ni=1

2i

)1/2A2 max

x2=1Ax2 = 1. (2- )

U Cmn V Cnn

U = [1, . . . , n], V = [1, . . . , n],

, .

i i = i i = ij , i, j = 1, . . . , n,

UU = V V = I.

, A ( ),

A1 = V 1U (3.4.2)

28

3.4. SVD ( )

, A12 = 1n , 2-

cond(A) = A2A1 =1n.

m n, A1 Moore-Penrose (Moore-Penrose pseudoinverse), A = (AA)1A.

AA = U2U, AA = V 2V , (3.4.3)

1. A ( ) AA AA.

2. A AA.

3. A AA

SVD AA AA, . G. H Golub W. Kahan Q-R. , . [GK65].

, , SVE, - ,

Ai = ii, Ai2 = i, i = 1, . . . , n. (3.4.4)

A

A1i = 1i i, A

1i2 = 1i i = 1, . . . n. (3.4.5)

x = A1b. U, V x, b

x = V V x =V

i x...

nx

= ni=1

(i x)i, (3.4.6)

b =

ni=1

(i b)i. (3.4.7)

. (3.4.6) . (3.4.4), Ax

Ax =

ni=1

Ai(i b) =

ni=1

i(i x)i. (3.4.8)

Ax b . (3.4.7) (3.4.8)

i(i x) =

i b. (3.4.9)

j , .

x =

nj=1

cjj , (3.4.10)

. 3.4.9 i 6= 0 (. A1) , cj

ii

nj=1

cjj = i b i

nj=1

cjij = i b

ci =i b

i. (3.4.11)

29

3.5. SVD ( )

. (3.4.10) (3.4.11)

x =

ni=1

b

ii. (3.4.12)

(3.4.5) , - SVE. i b - .

m > n, (3.4.1) SVD(thin/reduced SVD) U . SVD m > n, SVD (full SVD). U , (. UU = I), U Cmm, U , U. , :

A =U V

=[U U

] [ 0

]V T ,

U mm m n . A (. m < n), SVD

A = U S V =

mi=1

iii ,

U Cmm, V Cnm = diag{1, . . . , m} Rmm. , cond(A) = 1m . , SVD

csvd RT, SVD ( ) (CSVD, compact SVD) [U, diag(S), V ]. , .. m n m n n (U).

SVD SVD O(mn2) ( m n) (flops). SVD , , , SVE.

3.5 Anlush thc paragontopohshc SVD

H SVE SVD - Galerkin SVE. -

, A . 3.2.6, i A i K. , Galerkin , n

2n = K22 A2F , K22 = 1

0

10

|K(s, t)|2 ds dt,

'Ola ta ektc diagwnou stoiqea enai 0, all o pnakac enai mh tetragwnikc.Geniktera, wc sumpagc paragontopohsh SVD noetai h graf

M = UrrVr ,

pou r = rank(A) (r min(m,n)) kai oi pnakec Ur,r kai Vr enai antstoiqa oi r prtec stlec twn pinkwn U, kai V .H paragontopohsh SVD upologzetai se 2 bmata. Arqik metatrpetai o pnakac se didiagnio (mjodoc Golub-Kahan-

Lanczos), to opoo kostzei O(mn2) flops kai katpin upologzetai hp SVD tou didiagniou pnaka me qrsh epanalhptiknmejdwn kai kstoc O(n) flops. 'Ara sunolik to kstoc anrqetai se O(mn2) flops.

30

3.5. SVD ( )

(n)i n n ,

0 i (n)i n, i = 1, . . . , n (3.5.1)

(n)i

(n+1)i i, i = 1, . . . , n (3.5.2)

, A K n. , SVD.

(n)j (s) =

ni=1

iji(s), (n)j (t) =

ni=1

iji(t). (3.5.3)

, , .

(n)j (s) j(s)

(n)j (t) j(t), n. (3.5.4)

,

max{1 (n)1 2, 1 (n)1 2}

(2n

1 2

)1/2.

. , .

, R C . , - 3.5.4, SVD Ti b i, g SVE. g(n) - g n, . g(n) g n, ci g =

i=1 cii

3.2.7

ci = i, g = 1

0

i(s)g(s) ds = bi, (3.5.5)

g(n) i (. 3.2.3)

g(n)(s) =

nk=1

bkj(s). (3.5.6)

, (n)j , g(n) j , g, .

(n)j , g(n) j , g, n. (3.5.7)

. (3.5.3), . (3.5.6)

(n)j , g(n) =

10

ni=1

iji(s)

nk=1

k(s) ds

=

ni=1

ij

nk=1

bki, k =ni=1

ijbi = Tj b.

. SVE SVD A Picard .

, - (). n n

31

3.5. SVD ( )

3.3: i 6464 () ().

si = ti , A (. . 2.1.4, 3.1.1, 3.1.2, 3.1.3)

aij =d

n

[d2 +

(i jn

)2]3/2, i, j = 1, . . . , n,

A . gravity - RT, n = 64, d = 0.25,s [0, 1] f(t) = sint + 0.5 sin(2t). gravity 64 64 , SVD. 3.3 . i 55 , ( - ). 2.3288 1018, .

3.3 , . . (2.2.10), (2.2.11) (2.2.12). , (. .2.2.10) . 3.3 . , . deriv2 n = 64 RT, f(t) = t () g(s) = s

3s6 . deriv2

64 64 , SVD. 3.4 SVD

. , , . i, .

'Eqoume diadoqik

A =AT U V T = V UT

= (UTV )(UTV ) UTV = I U = V.

32

3.5. SVD ( )

3.4: 3.3

i. . (64) , 10, .

, SVD Ti b ( ) Ti bi

(-). Picard (Picard plot), . picard RT, Picard - . 2d+ 1 i, d . (3.3) 3.4, picard U,, V SVD 64 64 (b) , (. d = 0). 3.5. |Ti b| i i 35 .

Ti bi

, , i < 35, i 35 , , Ti b. ,

. - SVD . , . x = A1b Gauss, , , .

Picard . , Ursell (. . 2.2.13). ursell RL Galerkin (3.2.8) n = 14. 3.6

33

3.5. SVD ( )

3.5: Picard , .

3.6: Picard Ursell, . ( picard) 3.5.

3.5. Ti b . Ursell , , , .

34

3.6. Picard ( )

3.6 Jruboc sta Dedomna kai Diakrit Sunjkh Picard

A .

. , . , - . 2.1.1 g(s) g(s) = g(s) + g(s), g(s) f(t).

Ax = b , , . . .

b. . A . , . , , .

, , xexact, bexact = Axexact. , A, bexact xexact K, g f Fredholm ( ), .,

b = bexact + e, bexact = Axexact. (3.6.1)

e . () .

(Gaussianwhite noise), . e Rm ( ei E[ei] = 0) . -. Picard.

randn Octave . randn - ( ), 0 1. - . m, . randn(m,1) ( m=size(b exact)). eta -, - , .

Wc sma jewretai o forac pou kwdikopoietai h zhtomenh plhrofora kata th diadikasa miac mtrhshc.O leukc jruboc, pre to nom tou kat analoga me to leuk fwc. Enai o sunduasmc twn qwn lwn twn diafo-

retikn suqnottwn, allic enai na tuqao sma me eppedh puknthta fasmatikc isqoc (flat power spectral density).Sth statistik, leukc jruboc enai ma akolouja mh susqetizmenwn tuqawn metablhtn me mhdenik msh tim kaipeperasmnh diaspor. Gkaoussian kanonik katanom (Gaussian or normal distribution) kaletai h suneqc katanom pijanthtac pou qei

sunrthsh puknthtac pijanthtac (Gkaoussian sunrthsh) se sqma {kampnac} kai dnetai ap

f(x; , 2) =1

2e 1

2( x

)2,

pou enai h msh tim kai h tupik apklish.Ed qrhsimopoietai o sumbolismc gia thn tupik apklish kai gia thn msh tim ant gia ton sunjh sumbolism

kai antstoiqa, gia apofgoume th sgqush me tic idizousec timc.An den knoume thn epilog m=size(b exact) ja prpei na pollaplasisoume ap arister me nan mh tetragwnik

monadiao pnaka Imn, pou n=size(b exact) ste na gnei diac distashc me to b exact. Kai stic do periptseic (m b exact) h entol sto Octave enai, ones(size(b exact),m)*randn(m,1). To m enai sthn euqreia tou qrsth. An epilxoumem < size(b exact) eisgoume jrubo mno stic m prtec sunistsec tou b exact.

35

3.6. Picard ( )

3.7: Picard , , . :e2 106 . : e2 104. : e2 102.

eta*randn(m). -

(relative noise level), rnl = bbexact2

bexact2 =e2

bexact2 . Octave Matlab

e = randn(m, 1);

e = e/norm(e);

e = rnl norm(b exact) e;b = b exact+ e;

3.5 picard b. gravity deriv2 n = 64, .

3.7, Picard eta ( ). () n = 64., , 3.7 3.5. , e2 106 e2/bexact2 107 Fourier |Ti b| , i 20, . i < 20, . 3.7 Picard |Ti b| i.

3.8 , deriv2 n = 64 picard . 3.8 Picard , . rnl. rnl = 105, , Picard ,

36

3.6. Picard ( )

3.8: : Picard . : Picard rnl = 105 rnl = 103 .

(i 50) . . , . rnl = 103, . , , |Ti b| . , i 15, .

SVD. SVD SVE, Picard. , . Picard :

i - . Picard , , |Ti b| i.

, 3.7, |Ti b| > i i = 1, 2, 3, 4, Pi-card i = 1, . . . 20. |Ti b| .

Picard Urshell, 3.6, . L. Fox E. T Goodwin . ,

K(s, t) =(s2 + t2

)1/2, g(s) =

1

3

((1 + s2)3/2 s3

)f(t) = t. (3.6.2)

foxgood RT - . 3.9 Picard n = 40 Fredholm . 3.6.2. .

37

3.6. Picard ( )

3.9: Picard L. Fox E. T Goodwin . : Picard . : Picard rnl = 105. , Picard.

Picard i = 18 |Ti b| < i i = 18, . . . , 40. i = 8 . e2/bexact2 = 105. , i = 6.

. . (covariance matrix) e .

Cov(e) E[(e E[e]) (e E[e])T

]= E(eeT ) = 2I,

E[] 2 .

E[ei] = 0 E[e2i ] =

2 E[|ei|] =

2 0.8 , (3.6.3) e Rm

E[e] = 0, E[e22] = 2m, E[e2] =

2(m+12 )

(m2 ), (3.6.4)

.

2(m+12 )

(m2 )

m, . m = 100

9, 975. bexact e , E[b] = E[bexact].

b

Cov(b) E[((b E[b])(b E[b])T

)] = E[eeT ] = 2I. (3.6.5)

Gia thn Gkaoussian katanom isqei kti pio isquro:

E[|X |p] = p(p 1)!! {

2/, an o p enai perittc

1, an o p enai rtioc

E[|X |p] = p2p( p+1

2)

,

pou X h tuqaa metablht, () h sunrthsh Gmma kai to dipl paragontik n!! sumbolzei to ginmeno lwn twn perittnap to 1 wc to n. H deterh sqsh isqei gia kje (mh akraio) p > 1.

38

3.6. Picard ( )

(multivariate statistics) M r

Cov(Mr) = MCov(r)MT . (3.6.6)

. ,

x = A1b = A1(bexact + e) = xexact +A1e, (3.6.7)

3.6.6, x

Cov(x) = A1Cov(b)AT = 2(ATA)1,

Cov(x)2 =2

2n. (3.6.8)

n A. A (.cond(A) 1) , x = A1b .

SVD, . UT b = UT bexact + UT e

Ti b = Ti b

exact + Ti e, i = 1, . . . , n.

3.6.6

Cov(UT e) = UTCov(e)U = 2UTU = 2I, (3.6.9)

UT e -, 3.6.3 3.6.4 UT e.

bexact Picard, . Ti b

exact . e bexact, bexact b. Ti b

exact Ti e . Ti b

Ti b = Ti b

exact + Ti e

{Ti b

exact, |Ti bexact| > |Ti e|Ti e, |Ti bexact| < |Ti e|.

(3.6.10)

. |Ti b| , bexact Picard i. SVD , |Ti bexact| < . |Ti b| , Picard. Picard.

- . , e [

(3),

3].

. , . ( ) , (signal correlated noise), Poisson (Poisson noise), (broad-bandcoloured noise) ..

, Picard Ti bexact =

a, i = 1, . . . , n a > 0 , -- . a > 1 Ti b .

39

Keflaio 4

Upologistik distash : Mjodoikanonikopohshc

M ,

. .

(regularization methods) , . . -. , SVD

xreg =

ni=1

iTi b

i,

i . spectral filtering methods, 2.4, SVD . : SVD, SVD Tikhonov. , , . A Rmn m n, > =,

, x =ni=1

Ti bi

, , cond(A) = 1/n.

4.1 Angkh gia Kanonikopohsh twn Diakritn Antstro-fwn Problhmtwn

E , . ,

. , xexact

x Axexact = bexact, Ax = b = bexact + e,

,

xexact x2xexact2

= cond(A)e2bexact2

, e2 b2, - . .

41

4.2. SVD (TSVD) ( : )

4.1: ( ) ( ) () n = 64 () n = 32. ( ), .

4.1 . 4.1 ()

g(s) =

{(4s3 3s)/24, s < 12(4s3 + 12s2 9s+ 1)/24, s 12

(4.1.1)

f(t) =

{t, t < 121 t, t 12

(4.1.2)

n = 64 ( deriv2(64,3)) e2 = 5 105. , xexact . , 4.1 () . n = 32 ( gravity(32)) e2 = 107, ( 109) . n e2 = 0, .. n = 53 . . xexact, . , .

4.2 H mjodoc thc Apokommnhc SVD (TSVD)

A , .

H sunrthsh deriv2 ap to programmatistik pakto Regularization Tools mpore na ulopoihje gia tra paradegmata.Bzoume ton axonta arijm touc wc to risma pou akolouje aut thc distashc tou pnaka diakritopohshc pou epilgoume.'Hdh qoume qrhsimopoisei to pardeigma (1) prohgomena, pou f(t) = t, kai ed qrhsimopoietai to pardeigma (3).

42

4.2. SVD (TSVD) ( : )

. 3.6.10 SVD Ti b/i Ti e/i ( e b) ,. . 3.7. Ti b/i Ti bexact/i ( bexact = Axexact ), , . . , Picard, . SVD , i ( )

Ti b

i

Ti b

exact

i= Ti x

exact.

- SVD. , SVD . , - xk SVD (TSVD, truncated SVD) k

xk =

ki=1

Ti b

ii. (4.2.1)

, . 4.2.1 TSVD. k , -. k Picard. , 3.7 k = 20, k = 12 k = 8 .

TSVD . xk . 4.2.1, . , TSVD Ak r (rank(A) =r)

Ak =[1 . . . k

] 1 . . .k

[ 1 . . . k ]T = ki=1

iiTi . (4.2.2)

cond(Ak) = 1/k cond(A) = 1/n . , , Ax = b minAx b2 minAkx b2. , , Akx = b. , Ak r < n . . 2-, .

minx2 Akx b2 = min. (4.2.3)

4.2.3 TSVD . 4.2.1, SVD, x .

xk, TSVD

xk =[1 . . . k

] 11

. . .1k

[ 1 . . . k ]T b = ( ki=1

iiTi

)b (4.2.4)

Apodeiknetai ti h genik lsh qei th morf

x =ki=1

Ti b

ii +

ni=k+1

ii,

gia aujareto i.

43

4.2. SVD (TSVD) ( : )

Ak Ak

x = Akb,

Ak =ki=1 ii

Ti .

Vk =[1 . . . k

], Uk =

[1 . . . k

], k =

1 . . .k

Ak = UkkVTk A

k = Vk

1k U

Tk

UTk U = VTk V = I.

e Cov(e) = 2I, TSVD

Cov(xk) = AkCov(b)(A

k)T = 2

ki=1

2i iTi , (4.2.5)

3.6.6, . 3.6.5 Ak(A

k)T = Vk(

1)2V Tk .

Cov(xk)2 =2

2k,

k ( ) n, Cov(xk) ( ) Cov(x) .

xk, x = A1b (bias). A1b . 3.6.7, . 3.6.4

E[x] = E[xexact] + E[A1e] = xexact,

E[MX] = ME[X] () m n n k ., TSVD :

E[xk] = E

[x

ni=k+1

Ti b

ii

]= xexact

ni=k+1

Ti xexacti. (4.2.6)

x = Ab =V 1UT b

V Tx =1i UT b

Ti xi =Ti b

i,

E

[n

i=k+1

Ti b

ii

]= E

[n

i=k+1

(Ti x)i

]=

ni=k+1

(Ti xexact)i.

Sth statistik, merolhya bias enc ektimht (kannac upologismo mac dedomnhc posthtac basismnoc se dedomnaparatrhshc) orzetai wc h diafor thc mshc timc kai thc pragmatikc timc thc metablhtc proc ektmhsh. H merolhyaenai na edoc {sflmatoc} pou de diorjnetai epanalambnontac to perama kai parnontac to mso ro twn apotelesmtwn.

44

4.3. SVD (SSVD) ( : )

4.2: TSVD k, ( ).

, Picard, |Ti xexact| ( - . 4.2.6)

ni=k+1

(Ti xexact)i

2

=

[n

i=k+1

(Ti xexact)2

]1/2

, xexact2 =[n

i=1(Ti x

exact)2]1/2

. 4.2 TSVD

(n = 64) k, ( ). k = 10, ( e2 = 0.01) k = 14, 16 18. TSVD tsvd RT.

TSVD k i. , . [HC98]. , |Ti b| i, (scaling) |Ti b| i, . , , , . SVD . Ti b.

4.3 H mjodoc thc Epilektikc SVD (SSVD)

H TSVD SVD . -

H txh enc pnaka kajorzetai ap twn arijm twn mh mhdenikn idizouswn timn. Wstso tan autc enai polkont sthn akrbeia thc mhqanc ja prpei na tjetai na rio gia to poiec na jewre mhdn o upologistc, orzontactsi thn arijmhtik txh enc pnaka. To rio aut exarttai ap th mhqan kai enai max(m,n) eps(norm(A)) 2.2 1016eps(norm(A)), gia nan mn pnaka A. H arijmhtik txh diafrei ap thn {pragmatik} txh tou A tan autcqei pol meglo dekth katstashc kai rank(A) < max(m,n).

45

4.3. SVD (SSVD) ( : )

Picard. , , . , . . , , SVD (T2 x

exact = T4 xexact = T6 x

exact = = 0)., .

TSVD B. Rust (. [BR98]), - SVD (SSVD, selective SVD). , . SSVD - . , SVD , SSVD x

x =|Ti b|>

Ti b

ii. (4.3.1)

SVD Ti b/i |Ti b| .

[ ]i =

{1, |Ti b| > ,0, .

SSVD Ti b . 3.6 , Ti b (. . 3.6.9) (. . 3.6.3 3.6.4). ,

= s,

s , 3 5, .

SSVD Octave ( Matlab) ssvd. ssvd csvd ( A), b (b = bexact+e) (tau) SSVD x (x tau) (residual norms) eta = x2 rho = Ax b2. , b x Ax bexact2 xexact x2.

function [x tau,rho,eta,g] = ssvd(U,s,V,b,tau)% SSVD Selective SVD

% [x tau,rho,eta,g] = ssvd(U,s,V,b,tau)% SVD% tau x tau % x tau = [ x tau(1), x tau(2), ... ] .

% [n,p] = size(V); ltau = length(tau);if (min(tau)norm(b))error( )endifx tau = zeros(n,ltau);eta = zeros(ltau,1); rho = zeros(ltau,1);r=zeros(ltau,1);beta = U(:,1:p)*b;

Epiplon (proairetik) mpore na upologiste kai na dinusma g pou qei san stoiqea touc dektec twn rwn pousuneisfroun sth lsh. An doje dinusma tau gia tautqrono legqo polln kanonikopoihmnwn lsewn, tte to g enaipnakac me stlec ta diansmata pwc prohgoumnwc.

46

4.3. SVD (SSVD) ( : )

xi = beta./s;

% tau for j=1:ltaufor i=1:pif (abs(beta(i))>=tau(j) & tau(j)>0) % SSVDx tau(:,j) = x tau(:,j)+V(:,i)*xi(i);m(i,j)=xi(i);if nargout==4g(i,j)=i; % i SSVDendifelser(i,j)=beta(i);endifendforrho(j)=norm(r); % Ax b2eta(j) = norm(m); % x2endfor

if (nargout > 1 & size(U,1) > p)rho = sqrt(rho. 2 + norm(b - U(:,1:p)*beta)2);endif

endfunction

SSVD , (Ti b

exact) , Ti e . , , 2 4 SVD . (3) deriv2 (sawtooth functions) . 4.1.1 4.1.2. n = 60 e2 = 108. UT bexact 2 , 1016 . .

4.3 ( ssvd) SSVD , 7 - . = 105 0.00498046 4 (1, 3, 5, 7). = 5 109 = 0.5 108 0.000345319 23 ( 1 45 2).

, TSVD 4.4. 10 TSVD 0.00353939. k = 56 0, 000396772, TSVD( 56 60 ). SSVD TSVD .

Picard SSVD, 4.5. , . , , TSVD SSVD Ti b

exact .

47

4.4. Tikhonov ( : )

4.3: SSVD ( (3) ) 7 , ( ). = 0.5 = 0.5 108. x = xexact.

4.4 Mjodoc Kanonikopohshc Tikhonov

H TSVD TSVD xk k, -

SVD. , SVD , k . .

Tikhonov. , Andrey Tikhonov . Tikhonov (-) . , Tikhonov x

minx

{Ax b22 + 2x22

}(4.4.1)

Ax b22 () ( b). , , x , . , , b, , (data fitting problems), .

x22 .

48

4.4. Tikhonov ( : )

4.4: TSVD ( (3) ) 7 , ( ). k = 10, k = 54. SSVD, TSVD ( 54/60 !) SSVD. x = xexact.

. , x, .

2. , x2 ( x 0 ). , , . . (oversmoothing), (undersmoothing) . = 0 .

, x , . 4.4.1

minx

{Ax b22 + 2x22

}= min

x

{(Ax b)T (Ax b) + (x)Tx

}= min

x

{[Ax bx

]T [Ax bx

]}= min

x

{[ AI]x

[b0

]2

}, (4.4.2)

49

4.4. Tikhonov ( : )

4.5: Picard ( (3) ). .

() . - [

AI

]T [AI

]x =

[AI

]T [b0

]

(ATA+ 2I)x = AT b x = (ATA+ 2I)1AT b. (4.4.3) Tikhonov. Tikhonov 4.4.2.

SVD Tikhonov x. - SVD A, A = UV T I = V V T , 4.4.3

x =(V 2V T + 2V V T )1V UT b

=V (2 + 2I)1V TV UT b

= V (2 + 2I)1UT b. (4.4.4)

,

x =

ni=1

[]i

Ti b

ii, (4.4.5)

[]i i = 1, . . . , n

[]i =

2i2i +

2

1, i 2i2 , i .

(4.4.6)

50

4.4. Tikhonov ( : )

4.6: []i Tikhonov i.

4.6 . = 0.001

[]i i. i i = i = 10. , (1), SVD . , , i SVD .

, []i 2i ,

Ti b/i Ti e/i., , Tikhonov ,

TSVD. , k. , Tikhonov TSVD . Tikhonov tikhonov RT tsvd. 4.7 Tikhonov (gravity, n = 64 e2 = 0.01 ), 4.2 TSVD. Tikhonov 9 103 10 ( Octave Matlab: logspace(1,-3,9)). SVD. , 4.4.2.

( ) , Octave Matlab surf. 4.8 (9) TSVD() Tikhonov () . 4.8 TSVD Tikhonov : k xk x. . , . .. [HC86].

Tikhonov. Lagrange. ,

minxAx b22 x22 2. (4.4.7)

A1b2 , . , -

51

4.4. Tikhonov ( : )

4.7: Tikhonov x , , ( ). = 0.00316228 = 0.001 SVD.

4.8: TSVD () Tikhonov () .

( )

52

4.5. ( : )

minxx22 Ax b22 2. (4.4.8)

Lagrange Tikhonov .

TSVD Tikhonov. x = (ATA + 2I)1AT b Cov(e) = 2I, ( . 4.2.5) Tikhonov

Cov(x) = 2

ni=1

([]i )

2i i

Ti ,

Cov(x)2 2

(2)2.

n, Tikhonov, (. 3.6.8). ,

E[x] =

ni=1

[]i (

Ti x

exact)i = xexact

ni=1

(1 []i )(Ti x

exact)i.

1 i[] = 2/(2i + 2) Picard, xexact2.

4.5 Jewra Diataraqn

O , ,

. - . . , .

Ax = b Ax = b,

A b A b,

A = A+ A b = b+ b,

. A A, b. , Axexact = bexact. x x x x2/x2. . . x = A1b x = A1b ( A

) . A2 < n,

x x2x2

cond(A)1

(b2b2

+A2A2

),

= A2A12 =A2n

.

53

4.6. L ( : )

A2 < n < 1, . A, cond(A) = 1/n. . -

min Ax b2 min Ax b2 ( m > n rank(A) = n) . A2 < n, n,

x x2x2

cond(A)1

(b2bn2

+A2A2

+ b bn2bn2

),

bn = Ax = A2/n. , cond(A) = 1/n . TSVD. (. [HC90].) xk xk TSVD k,

A2 < kk+1. ,

xk xk2x2

k1 k

(b2bk2

+A2A2

+k

1 k kb bk2bk2

),

bk = Axk, k = cond(Ak) =1k

=A2k

, k =A2k

k =k+1k

.

TSVD k = 1/k A. Tikhonov. (. [HC89]) x x Tikhonov -

. , A2 < . ,

x x2x2

1

(b2b2

+ 2A2A2

+ b b2b2

),

b = Ax, =1

=A2

=A2

.

Tikhonov = 1/ TSVD, A.

TSVD Tikhonov . , , . , b bk = b Axk b b = b Ax. A SVD . k = k+1/k TSVD, . k (1) 1 k . (cluster) . Tikhonov,

, SVD []i = 2i /(

2i +

2) . Tikhonov , . .

4.6 H kamplh L

S SVD . SVD

Sustda (cluster) idizouswn timn kaletai na uposnolo twn idizouswn timn pou enai pol kont h ma sth llhkai enai apomonwmnec ap tic uploipec.

54

4.6. L ( : )

. , SVD . , , SVD .

TSVD ,

xk22 =ki=1

(Ti b

i

)2 xk+122 (4.6.1)

Axk b22 =n

i=k+1

(Ti b)2 + 2 Axk+1 b22, (4.6.2)

= (I UUT )b2 b A., Tikhonov

x22 =ni=1

(

[]i

Ti b

i

)2(4.6.3)

Ax b22 =ni=1

((1 []i )

Ti b)2

+ 2 (4.6.4)

,

= x22 = Ax b22.

[]i ,

d

d(

[]i ) =

22i(2i +

2)2= 2

2i2i +

2

2

2i + 2

= 2

[]i (1

[]i ),

([]i )2

d

d[(

[]i )

2] = 2[]i

d

d(

[]i ) =

4

([])2(1 []i ).

dd

= 4

ni=1

[([]i )

2](1 []i )(Ti b

i

)2< 0 , (4.6.5)

dd

=4

ni=1

(1 []i )2

[]i (

Ti b)

2 > 0, . (4.6.6)

,

=4

ni=1

(1 []i )2 1

2i + 2

[]i

2i

(Ti b

i

)2

=24

ni=1

(1 []i )([]i )

2

(Ti b

i

)2 = 2. (4.6.7)

55

4.6. L ( : )

(4.6.5) (4.6.6) . = 2 dd =

2, x22 Ax b22. , . ,

d2

d2=

d

d(2) = 2 2, (4.6.8)

d2d2 .

(, ). . c , (. ),

c =

[()2 + ()2]3/2

=2()2

[()2 + ()2]3/2

,

. , c > 0 (, ) .

,

0 x2 A1b2 0 Ax b2 b2.

, . 4.4.4

Ax =UVT V (2 + 2I)1UT b

=U2(2 + 2I)1UT b (4.6.9)

=

ni=1

[]i (

Ti b)i, (4.6.10)

Ax22

Ax22 =ni=1

(

[]i

)2(Ti b)

2 ni=1

(Ti b)2 = Ax22 = b22

Ax22 b22

bAx2 b2. 4.6.9 , Ax b2:

Axb = U2(2 +2I)1UT bUUT b = U(2(2 +2I)1I)UT b =ni=1

([]i 1)(

Ti b)i, (4.6.11)

Ax b2 =

[ni=1

(1 []i )2(Ti b)

2

]1/2. (4.6.12)

, , U , UUT = In. UUT 6= I, UUT b .4.6.4.

, , ,

min Ax b22 x22 ,

(. 4.4.2), . x.

56

4.6. L ( : )

4.9: L Tikhonov. - x2 Ax b2 . deriv2 () shaw (). .

x (Axb2, x2 . .

( Octave Matlab: loglog). ,

(1

2log ,

1

2log ) = (log Ax b2, log x2),

L Tikhonov. L .

4.9 L . tikhonov Ax b2 x2 loglog. , ( (1) ) n = 64 e2/bexact2 = 0.01 30 105 1. ( ) shaw n = 20, = 0.001 8 = [1, 3e1, 1e2, 3e3, 3e4, 1e4, 3e5, 1e 5]. L , ( ), . , , .

Tikhonov . L.

57

4.6. L ( : )

x SVD bexact . ( . 5, . [HC01])

x2 xexact2 () Ax b2

, Tikhonov , .

x2 1 Ax b2 e2 ().

L , . .

(log e2, log xexact2)

, L 0. , L 0 x2 A1b2 ( ).

:

1. L - Tikhonov, .

2. L ( ) .

L papers, Lawson Hanson, [LH95].

58

Keflaio 5

Epilog ParamtrouKanonikopohshc

S , SVD.

TSVD Picard . : . , , SVD ( ) . , . - , . , , .

. , , . , , L (discrepancyprinciple). , (generalized crossvalidation method), . ( 0).

5.1 Sflmata Kanonikopohshc kai Sflmata Diataraqn

- . Tikhonov, TSVD. [] Tikhonov,

[]1

. . .

[]n

, []i =

2i /(

2i +

2) ( TSVD 0 1).

x = V []1UT b. (5.1.1)

TSVD, [] [k] = diag[1, . . . , 1, 0, . . . , 0].

59

5.1. ( )

5.1: - k shaw n = 20 = 106. k = 10.

, :b = Axexact + e. Tikhonov

xexact x = xexact V []1UT b= xexact V []1UTAxexact V []1UT e= (I V 1UTUV T )xexact V []1UT e= V (I [])V Txexact V []1UT e. (5.1.2)

:

xbias = V (I [])V Txexact =ni=1

(1 []i )(Ti x

exact)i, (5.1.3)

() , . - () x .

:

xpert = V []1UT e. (5.1.4)

, [], .

TSVD,

xbias =

ni=k+1

(Ti xexact)i, pert =

ki=1

Ti e

ii.

k - , , . k. k

60

5.1. ( )

n, SVD , . , k , SVD , , . , , xbias xpert, 5.1. shaw (. 3.3.1) n = 20 = 106

k. k = 10.

Tikhonov .

[]i 1 ( [] I)

, . , .

. TSVD, k, . , , .

, Picard, Ti b , , , (. . 3.6.10). k

|Ti b|. Picard, |Ti b|i

, , i < kn.

, , 4.6.1 () . k < k k > k

k < k : xk22 ki=1

(Ti b

exact

i

)2,

k > k : xk22 ki=1

(Ti b

i

)2+

ki=k+1

(

i

)2 xexact22 + 2

kk+1

2i .

k = k . 4.6.2 ,

k

k < k Axk b22 kk+1

(Ti b)2 + (n k)2

kk+1

(Ti bexact)2

k > k : Axk b22 (n k)2.

k < k (), , k > k() . k = k . 5.2 shaw (n = 64, = 106). Picard ( ). k < k k > k ., 13 TSVD 5.3.

k = k, SVD , .|Ti b| > . k, . , , , Ti b. , .

61

5.2. ( )

5.2: : Picard shaw (n = 20), ( = 106). : TSVD xk2 k < k k > k. : Axk b2 .

5.2 H Arq thc Asumfwnac

A k = k SVD - , (n k)1/2,

k = k. , : k (n k)1/2, k . , . E[e2] = n1/2 (. 2 . 3.6.4). 5.4 . dp .. 2 dbe2 .

. k , dpe2. , k Axk b2 dpe2:

k = kdp Axk b2 dpe2 Axk+1 b2. (5.2.1)

62

5.2. ( )

5.3: 13 TSVD shaw. 13 ( xpert .)

5.4: . Axk b2 (n k)1/2 k, Axk b2 e2 k.

, , Tikhonov, xk x :

= dp Ax b2 = dpe2. (5.2.2)

k .

, , e2 0. 5.2.2 (.. Newton-Raphson). Tikhonov, Ax b2 = 2 (. . 4.6.6) .

63

5.2. ( )

5.5: : shaw, Axk b2 = n1/2. , n1/2 5% n1/2 10%. : , Axk b2 = e2. , , ( 5%) e2, .

e2 ( ) (). , , kdp dp e2 ( k ). , dp, , , L Ax b2 = dpe2. , Ax b2 e2. L (. ). , .

5.5 , . dp = 1 shaw n = 100 = 102. . 5.2.2 e2 n1/2, 5% 10% n1/2 . 5%(), , 10%, . ( . 5.7), e2 ( ) ( 50% ). . ( 2 ) e2 . 0.089.

64

5.3. L ( )

5.6: shaw, Ax b2 = n1/2 dp = 2. e2 : 5% 10% .

, 5.5 5% 10% e2 , . 5%. e2 , 5.5, 5.2.2 n1/2, 2 5% e2. , > 5% , .

. - dp = 2 5.6 5% 10%. 5.7 5.5, e2( 5.5): n1/2 e2 1 e2 dp = 2.

discrep RT . SVD (U,, V ), () b () ( x0) x . bAx2 2 = 0 Newton-Raphson.

5.3 To Kritrio thc Kamplhc L

H L 4.6 , xbias xpert . ,

L. , TSVD k |Ti b|.

k k, xk

65

5.3. L ( )

5.7: ( 5.5 5.6), e2: n1/2 e2 1 e2 dp = 2. , - 5.2.2 n1/2, / e2. , 5% 10% 1 2.

SVD . , xk2 Axk b2. (. . 5.2), k :

xk2 xexact2, k , xk2 k 0.

Axk b2 k 0, b2 k = 0.

k k, xk k. , k:

xk2 ( ) k, - .

Axk b2, , e2, k n.

, , L 2 , , . L . , , L .

66

5.3. L ( )

, k, , .

L , L, . , . .

L. Tikhonov, L . , . c (log(Ax b2), log x2).

L Tikhonov. - 4.6,

= log x22 = log Ax b22, (5.3.1)

L ( 12 ,12 ). ()

=

=

, (5.3.2)

= ()2

2 =

()2

2. (5.3.3)

,

c = 2

[()2 + ()2]3/2,

2 L . - 5.3.1, 5.3.3, (4.6.6), (4.6.8) , L , :

c =2

()2

2 ()2

2

[(

)2 + (

)2]3/2

= 22[ ()2] 2[ ()2]

[()2 + ()2]3/2

=22[ ()2] 2[ 2 4()2]

[()2 + (2)2]3/2

=22()3 + 222()242()3

3[2 + (2)2]= 2

2+ 2+ 4

(2 + 42)3/2.

= x22 = Ax b22 , . , SVD

=4

xT z, (5.3.4)

z

z = (ATA+ 2I)1AT (Ax b).

K = ATA = V 2V T M = ATA+ 2I.H kampulthta mac parametrikc kamplhc sthn analutik gewmetra ekfrzei to rujm metabol thc katejunshc tou

(monadiaou) efaptmenou diansmatoc thc kamplhc. Tuqntec mhdenismo thc kampulthtac diaqwrzoun ta kurt ap takola mrh thc kamplhc, en ta akrtata antiproswpeoun tic gwnec thc kamplhc.

67

5.3. L ( )

xz

xT z = xTM

1Kx x22, (5.3.5)

. 4.4.3. , 4.6.5

=4

[ni=1

([]i )

3

(Ti b

i

)2

ni=1

([]i )

2

(Ti b

i

)2]

=4

[ni=1

([]i )

3

(Ti b

i

)2 x22

], (5.3.6)

x .4.6.3. (5.3.5) (5.3.6),

xTM1Kx =

ni=1

([]i )

3

(Ti b

i

)2. (5.3.7)

M1K =[I + 2(ATA)]1

=[V 22V T + 2V 2V T ]1

=V T2(2 + 2I)1V = V T[]V,

[] (ATA)1 = (V 2V T )1 =V 2V T .

xTM1Kx = (V

Tx)T[]V Tx, (5.3.8)

([]). , yTDy D ( dk, k = 1, . . . , n)

yTDy =

nk=1

diy2i .

, y = V Tx

(V Tx)T[]V Tx =

nk=1

[]i [(V

Tx)(i)]2 ((i): i V Tx)

=

nk=1

[]i (

Ti x)2

=

nk=1

[]i

Ti nj=1

[]j

Tj b

jTj

2

=

ni=1

([])3(Ti b

i

)2,

5.3.4. 4.4.2 Tikhonov 4.4,

z Tikhonov

minz

[ AI]z

[Ax b

0

]2

.

68

5.3. L ( )

5.8: L . , , ( =0.0013895).

, z Tikhonov Ax b.

c, , L c:

= L c . (5.3.9)

5.8 L () () . n = 60, = 103, Tikhonov 15 105 10. L , ( = 0.0013895). lcfun RT.

TSVD, () k = 1, 2, . . . L - () . TSVD L:

k = kL L . (5.3.10)

L /2. , L L. L L, . . [CG02, HC01] .

L . ()

Sthn pragmatikthta aut h sunrthsh qrhsimopoie wc orsmata na dinusma me tic paramtrouc kanonikopohshc, thnafel lsh, touc rouc Ti b kai tic idizousec timc gia na upologsei thn arnhtik kampulthta (c).

69

5.4. ( )

, , . , L , SVD Ti x

exact 0. , xexact , ( ) i. L , SVD . , , k , L .

k. , , L. .

5.4 H Genikeumnh Mjodoc Diastaurwmnhc Epikrwshc

H -. k Ax Axk

bexact. TSVD,

[k] = diag[1, . . . , 1 k

, 0, . . . , 0] =

[Ik 00 0

].

, bexact Axk:

Axk bexact = AV []1UT (bexact + e) bexact

= U

[Ik 00 0

]UT bexact + U

[Ik 00 0

]UT e UUT bexact

= U

[Ik 00 0

]UT e U

[0 00 Ink

]UT bexact.

, SVD,

Axk bexact22 =ki=1

(Ti e)2 +

ni=k+1

(Ti bexact)2 k2 +

ni=k+1

(Ti bexact)2.

k, |Ti bexact| < i > k:

k < k : Axk bexact22 k2 +k

i=k+1

(Ti bexact)2

k > k : Axk bexact22 k2.

k < k k, , k k + 1, Tk b

exact ( ), k > k k. , .k k. .

, bexact . (cross validation) , . cross validation, ( ) , . ,

70

5.4. ( )

i b Tikhonov

x(i) =

((A(i))TA(i) + 2In1

)1(A(i))T b(i), (5.4.1)

A(i) b(i) A b, i i . 5.4.1, x bi (

A), A(i, :)x(i) . , :

min

1

m

mi=1

(A(i, :)x

(i) bi

)2.

, ( ),

min

1

m

mi=1

(A(i, :)x bi

1 hii

)2, (5.4.2)

x Tikhonov, hii A(ATA+ 2I)1AT . 5.4.2 , -

Tikhonov. : hii A, -, .

- (GCV, generalized cross validation) hii . , 5.4.2

min

1

m

mi=1

(A(i, :)x bi

1 trace (A(ATA+ 2I)1AT ) /m

)2. (5.4.3)

Ax b22 (1 trace(A(ATA+ 2I)1AT )/m)2. SVD A,

trace(A(ATA+ 2I)1AT

)= trace

(UV T (V 2V T + V V T2)1V UT

)= trace

(U2(2 + 2I)1UT

)= trace(U[]UT )

= trace([]) =

mi=1

[]i .

m, GCV :

= GCV G() =Ax b22(

mni=1

[]i

)2 . (5.4.4) TSVD, , trace([k]) = k, GCV

k = kGCV G() =Axk b22(m k)2

. (5.4.5)

GCV , -, GCV. 5.10 5.11 GCV Tikhonov TSVD,. . ,

71

5.4. ( )

5.9: generalized cross validation Tikhonov . : . shaw n = 40, = 0.01. : . gravity n = 64 = 0.001.

5.10: generalized cross validation Tikhonov . GCV GCV. : GCV, G(), .: , ( x ).

72

5.4. ( )

5.11: generalized cross validation TSVD . kGCV GCV. : GCV, k G(k), .: , .

5.10, GCV, G(k) ( ). , GCV , G(k) . gravity n = 60 = 0.01. GCV gcv RT. U, SVD (tikhonov ) tsvd TSVD) - TSVD , . . 5.11: () GCV GCV ( - , . 5.9), . GCV .

GCV -

Uprqei kai trth epilog mejdou me to anagnwristik dsvd, pwc kai se prohgomenec sunartseic pou qoume deiwc tra. Afor th mjodo thc genikeumnhc SVD enc zegouc pinkwn A,L, h opoa xefugei ap ta ria tou parntoc.

73

5.4. ( )

Tikhonov, G() 5.4.4

Gw() =Ax b22(

m wni=1

[]i

)2 , w . , GCV.

GCV Grace Wabba, [WG75] .

74

Keflaio 6

Ankthsh MikrofusiknParamtrwn ap Dedomna LidarPollapln Mhkn Kmatoc

H , Fredholm, -

. . , , . , . , .

, - lidar . , , Retrieval of Aerosol Parameters (RAP) DetlefMuller, Igor Veseloskii, Vadum Griaznov Alexei Kolgotin (. [KM08, MW99, LH02, BK06]). , , . . .

6.1 To Mh Kalc Topojethmno Prblhma AnkthshcMikrofusikn Paramtrwn

T Fredholm

,

() =

rmaxrmin

Kn/ext(, r;m, s)n(r) dr, (6.1.1)

r , m , , s , rmax rmin , , , , n , K Kext ( n ). , , , , Mie. ,

75

6.1. ( Lidar )

. , 1.6.2. / , .

K

Kx = y,

K : X Y HilbertX,Y .

y(t) =

dc

K(t, s)x(s) ds, t [a, b],

lidar. t, s, c, d x, y,K , r, rmin, rmax n,,K/ext, .

X = L2[a, b], Y = L2[c, d] K [a, b] [c, d]. K L2([a, b] [c, d]), L2[a, b] [a, b]. K , x ., Hadamard. ,

x = Ky

( x), ,

KKx = Ky,

K K K MoorePenrose.

x = (KK)1Ky =

j=1

y, ii

i(t),

{j , j , j} K (. . 2.2, . 2.2.3 - 2.2.6). j

. , , , y , y y2 . K Hilbert X Y , K X , . x . , , , Raman lidar .

Raman lidar , .

Sumpagc telestc enai nac grammikc telestc ap na qro Banach X se nan llo Y , ttoioc ste h apeiknishopoioudpote fragmnou uposunlou tou X enai h sumpagc kleistthta enc uposunlou tou Y . Eidiktera: 'Estw Knac fragmnoc, grammikc telestc se na qro Hilbert H. 'Estw akma fn ma peirh omoimorfa fragmnh akoloujadhl. M > 0, t.w fn M,n. O A kaletai sumpagc telestc an uprqei upakolouja {Afnk} thc {Afn}, h opoaenai akolouja Cauchy, bl. kai [HH73].Diaqwrsimoc kaletai nac topologikc qroc an periqei na arijmsimo pukn uposnolo dhl. na uposnolo me

arijmsimo pljoc stoiqewn tou opoou h kleistthta enai loc o qroc. H Diaqwrisimthta enai shmantik nnoia giathn Arijmhtik Anlush diti poll jewrmata qrhsimopoion kataskeuastikc apodexeic mno gia diaqwrsimouc qrouc,oi opoec mporon na metatrapon se algrijmouc.Genikc, kje sunrthsh thc morfc

x =

j=1

y, ii

i(t) + q,

pou q null(K), enai pijan lsh.

76

6.2. ( Lidar )

, , , . - .

, - . 3+ 2 () 3 2 () 2 , . 6 + 2, . , 3 + 2 (6) 6 + 2. -, , . (effective radius) , .

6.2 Leptomreiec sthn Efarmog sto Prblhma Ankth-shc Mikrofusikn Paramtrwn

T . 6.1.1 ((r)) n(r) -

() =

rmaxrmin

K/ext(, r;m)(r) dr

=

rmaxrmin

3

4rQ/ext(, r;m)(r) dr, (6.2.1)

(r) = 4r3

3 n(r). , , . 300 1100nm 0.1 2m K/ext K

n/ext, 6.2.1

6.1.1. , K/ext K (r) x(s).

lidar Raman, ( 3+2), . Aer (355nm), Aer (532nm), Aer (1064nm) Aer (355nm), Aer (532nm). (mono-/multimodallogarithmic-normal distribution) (M = 1 M 2, ) - n(r)

n(r) =

Mj=1

nt,j

r

2 ln jexp

((ln r ln rmed,j)2

2 ln2 j

), (6.2.2)

nt,j , j rmed,j j . , .

Merikc plhroforec gia th logarijmik kanonik katanom katanom log-normal katanom Galton/McAlister,Gibrat/Cobb-Douglas.Sth jewra pijanottwn, h katanom log-normal enai ma suneqc katanom pijanthtac miac tuqaac metablhtc (t.m.), thcopoiac o logrijmoc qei kanonik katanom. An h t.m. X qei kanonik katanm, tte h t.m. Y = eX qei katanomlog-normal. 'Omoia, an h t.m. Y qei katanom log-normal tte h t.m. X = log Y qei kanonik katanom. Ma t.m. mpore namontelopoihje ste na qei katanom log-normal an enai ginmeno polln anexrthtwn t.m., h kajema ap tic opoec enaijetik. Se ma katanom log-normal oi parmetroi kai enai antstoiqa o msoc kai h tupik apklish tou logarjmouthc t.m.. Se autc tic katano