Application of the Regularization Strategy in Solving Numerical Differentiation for Function with Error Xian-Zhou GUO 1,a and Xiang-Mei ZHANG 2,b,*

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1 d Aual Iteratioal Coferece o Advaced Material Egieerig (AME 06) Alicatio of the egularizatio Strategy i Solvig Numerical Differetiatio for Fuctio with Error Xia-Zhou GUO,a ad Xiag-Mei ZHANG,b,* Hebei Uiversity of Techology, Tiaji City, 30040, P Chia Hebei Uiversity of Techology, Tiaji City, 30040, P Chia a xiazhou_guo@hebuteduc, bxmei_zhag@hebuteduc *Corresodig author Keywords: Noisy fuctio, Exact fuctio, Differetiatio, egularizatio strategy, egularizatio arameter Abstract I may ractical roblems, it is sometimes ecessary to evaluate the derivative of fuctio whose values are give aroximately Firstly, the roblem of estimatig the derivative of a fuctio observed with error is studied It resets a roer regularizatio strategy ad exlai how to choose regularizatio arameter Secodly, the regularizatio strategy above to the umerical differetiatio is alied ad discussed i the imlemetatio of the umerical method ad the tests which it has erformed i order to ivestigate the accuracy ad stability of the umerical differetiatio rocedure Fially, some umerical examles will further illustrate that this method is reasoable, effective ad reasoable Itroductio I recet years, iverse roblems i mathematical hysics have bee oe of the fastest growig areas However, iverse roblems are closely liked to the ill-osed, ad due to a great deal of difficulty to umerical solutio At the begiig of 960s, the regularizatio strategy is roosed by Tikhoov creatively[] From the o, Dih Nho Hao studied the mollificatio method for ill-osed roblems[] Y F Wag, W Q Liag, J T Zhao coducted a lot of research[3,4,5] As it is kow, the choice of the regularizatio arameter is a key matter for esurig roer regularizatio There are may roblems that ca be described by umerical differetiatio i the atural sciece ad egieerig techology field It is easy to imagie may differet situatios--- mostly ivolvig ordiary ad artial differetial equatios related with the questio of umerical differetiatio of oisy (measured) data So it is sometimes ecessary to evaluate the derivative of fuctio whose values are give aroximately How to get the umerical solutio of these fuctios with oisy data has become a secial course I this aer, the oerator with Gaussia kerel is studied ad its reasoable regularizatio arameters i a efficiet maer is itroduced Therefore, a lot of umerical examles show that the rocess has good stability ad high accuracy The geeral method to solve the ill-osed roblem is to aroximate the solutio of the origial roblem with a set of well-osed roblems How to establish a effective regularizatio method is a imortat art of the research o the roblem of ill-osed roblems i the field of iverse roblem J J Cao, Y F Wag ad B F Wag[6,7] coducted a lot of research egularizatio Strategy Theorem : Let gα L ( ) ad gα ( x)dx =, () if f L ( ), where < +, the lim gα f f =; 0 () if f L ( ), the lim( gα f )( x) = f ( x), where x is the cotiuous oit of f Proof: Set gα * f = fα, fα ( x) f (= x) [ f ( x y) f ( x)]gα ( y)dy 06 The authors - Published by Atlatis Press 87

2 By the geeralized Mikowski iequality, fα f = [ ( ) ( )] ( ) [ ( ) ( ) ] ( ) f x y f x g y dy dx f x αt f x dx g t dt α To every t, lim f( x αt) f( x) dx = 0 ad f( x αt) f( x) dx f By domiated covergece theorem, [ f ( x α t) f ( x) dx] g() t dt 0 Whe α 0 () has already roved Because f L ( ) ad x is a cotiuous oit of f, the f ( x) f( x) ( f( x y) f( x)) g ( y) dy = f( x t) f( x) g( t) dt α α α To every t, lim f( x αt) f( x) = 0 ad f( x αt) f( x) f By domiated covergece theorem, f( x α t) f( x) g() t dt 0, whe α 0 () has already roved By Theorem, g α is eeded to choose roerly, the g α f f wheα 0 The Gaussia kerel gα () t is defied by gα ( t) = ex( t / α ), t, α π where α > 0 deotes a arameter The gα () t dt = ad the covolutio + 88

3 + Tα f() t = ( gα f)( t) = gα ( t s) f () s ds = gα () s f ( t s) ds, t exists ad is a L -fuctio for every f t + () L ( ) Furthermore, by Youg s iequality, Tα f g, ( ) L α f g L α f f f L L L L = = is obtaied Therefore, the oerator f gα f is uiformly bouded i L ( ) with resect toα So regard Tα f() t as regularizatio oerator However, i ractical alicatio, f() t caot be accurately give The oisy (measured) fuctio f () t which satisfied the error boud is obtaied: f( t) f ( t), I i the data iterval I = [0,] Theorem : ( Error estimatio): T f () t f() t = T f () t T f() t + T f() t f() t α α α α Tα f() t Tα f() t + Tα f() t f() t + Cα where α is a costat So Tα f () t is used to comute the aroximate derivative of the exact f() t The origial ill-osed roblem of fidig f is relaced by ew roblem of fidig Tα f The ew roblem is well-osed, ad deeds o a arameter > 0 For give Tα f, regularizatio arameter α is choosed by Morozov deviatio ricile or by GCV[8,9] I the umerical comutatio, Newto method is used as follows: Ste: Set iitial value α 0, the we comute f0 = f( α0), f0' = f '( α0), where f( α) = T f f α f0 Ste: Comute α = α0, f = f( α), f ' = f '( α) f Ste3: If α satisfied η < ε or η < ε, the α = α ; else go to ste 4 α α0 ; α < C Where η = α α0, C is a cotrol costat ; α C α Ste4: if f ' 0 =, the over, 0 ' otherwise alterate ( α 0, f0, f0') by ( α, f, f'), the go to Ste Numerical esult I this sectio, the regularizatio strategy is alied above to the umerical differetiatio, ad discussed the imlemetatio of the umerical method ad the tests which it erformed i order to ivestigate the accuracy ad stability of the umerical differetiatio rocedure[0,] I the examles, the exact data fuctio is deoted by f() t ad the oisy fuctio f () t f () t is obtaied by addig a radom error or a high frequecy disturbace error to f() t 89

4 That is f ( ti) = f( ti)( + σ i), where ti = ih; i = 0,, ; h =, ad σ i is a uiform radom variable with values i [-,], such that max f ( t ) f( t ) or f ( t ) = f( t )( + si( ωt )), 0 i i i i i i where ti = ih; i = 0,, ; h =, ad ω is a erturbatio frequecy Examle : The first examle is rather oscillatory o [0,] We choose f( t) = si(0 πt) ad the exact derivative f / ( t) = 0π cos(0 πt) I table, we give the error betwee the exact derivative of f( t) = si(0 πt) ad the solutio obtaied with the regularizatio strategy I order to illustrate the umerical aroximatio to the derivative is alied f / ( x ) deoted f / ( x), error f t f t / /, h = ( ( i) ( i) ) i= Table This is the the error betwee the exact derivative ad regularizatio derivative error = 00, = 00, = 00, = 00, ω = 00 ω = 00 ω = 00 ω = Examle : The exact fuctio f ( t) = ormcdf ( t,,3), solve ρ ( x), such that ρ() t dt = f () t The comlete algorithm for the regularizatio strategy is as follows: Ste: Provide the exact discrete data fi () t of f() t by MATLAB Ste : Obtai the oisy fuctio f ( t i ) by addig a high frequecy disturbace error to f( t i ) Ste 3: Comute the aroximate solutio ρ ( x) of ρ ( x) Ste 4: Comare ρ ( x) with ρ ( x) I exeriece the exact ρ( x) ad error fuctio error are give as follow: ρ π ( x) = ex( ( t 3) ), error = t t ( ρ( i) ρ ( i)) i= Table This is the the error betwee the exact derivative ad regularizatio derivative(ω =7) error (ω =7) =0 =00 =000 regularizatio stratery bisectio method iterolatio slie extraolatio discrete regularizatio(α ukow) discrete regularizatio(α kow) e e

5 Table3 This is the the error betwee the exact derivative ad regularizatio derivative(ω =00) error (ω =00) =0 =00 =000 regularizatio stratery bisectio method iterolatio slie extraolatio discrete regularizatio(α ukow) discrete regularizatio(α kow) e e-004 Coclusios I may ractical roblems, it is sometimes ecessary to evaluate the derivative of fuctio whose values are give aroximately Firstly, the roblem of estimatig the derivative of a fuctio observed with error is studied It resets a roer regularizatio strategy ad exlai how to choose regularizatio arameter Secodly, the regularizatio strategy above to the umerical differetiatio is alied ad discussed i the imlemetatio of the umerical method ad the tests which it has erformed i order to ivestigate the accuracy ad stability of the umerical differetiatio rocedure Fially, some umerical examles will further illustrate that this method is reasoable, effective ad reasoable[] Ackowledgemets The aer is suorted by the Natioal Natural Sciece Foudatio of Chia(Grat No 303, No 07003, No7087, No0007) efereces [] A NTikhoov, O solvig icorrectly osed roblem ad method of regularizatio, Dokl, Acad, Nauk USSP, 963, 5(3), [] Dih Nho Hao, A mollificatio method for ill-osed roblems, Numer Math 994, 68, [3] W Q Liag, Y F Wag ad C C Yag, Determiig the fiite differece weights for the acoustic wave equatio by a ew disersio-relatioshi-reservig method, Geohysical Prosectig, Vol 63, -, 05 [4] J T Zhao, Y F Wag ad C X Yu, Diffractio imagig by uiform asymtotic theory ad the double exoetial fittig, Geohysical Prosectig, Vol 63, , 05 [5] J J Cao ad Y F Wag, Seismic data restoratio with a fast L orm trust regio method, J Geohys Eg, Vol, No 4, 04500, doi:0088/74-3//4/04500, 04 [6] W Q Liag, Y F Wag ad C C Yag, Comariso of umerical disersio i acoustic fiite-differece algorithms, Exloratio Geohysics, htt://dxdoiorg/007/eg307, 04 [7] C X Yu, Y F Wag, J T Zhao ad Z L Wag, Double arameterized regularizatio iversio method for migratio velocity aalysis i trasversely isotroic media with a vertical symmetry axis, Geohysical Prosectig, Vol 6, , DOI: 0/ , 04 [8] Pullia A,iboldi S Time-domai simulatio of electroic oises IEEE Trasactios o Nuclear Sciece,004,5(4),

6 [9] Viagre B M, Che Y Q Fractioal calculus alicatio i automatic cotrol ad robotics 4st IEEE CDC,Tutorial Worksho, Las Vegas,00 [0] Pocco A,West B J Fractioal calculus ad the evolutio of fractal heomea Physica A,999,65(3~4), [] Ahmad W M,Harb A M O oliear cotral desig for autoomous chaotic systems of iteger ad fractioal orders Chaos,Solitios ad Fractals, 03,6,-9 [] Magi L Fractioal calculus i bioegieerig,critical reviews i biomedical egieerig Begell House Publicatios, 004,3,-378 9