18.3 Integral Equations with Singular Kernels

Transcription

1 18.3 Integrl Equtions with Singulr Kernels 797 This procedure cn be repeted s with Romberg integrtion. The generl consensus is tht the best of the higher order methods is the block-by-block method (see [1). Another importnt topic is the use of vrible stepsize methods, which re much more efficient if there re shrp fetures in K or f. Vrible stepsize methods re quite bit more complicted thn their counterprts for differentil equtions; we refer you to the literture [1,2 for discussion. You should lso be on the lookout for singulrities in the integrnd. If you find them, then look to 18.3 for dditionl ides. CITED REFERENCES AND FURTHER READING: Linz, P. 1985, Anlyticl nd Numericl Methods for Volterr Equtions (Phildelphi: S.I.A.M.). [1 Delves, L.M., nd Mohmed, J.L. 1985, Computtionl Methods for Integrl Equtions (Cmbridge, U.K.: Cmbridge University Press). [ Integrl Equtions with Singulr Kernels Mny integrl equtions hve singulrities in either the kernel or the solution or both. A simple qudrture method will show poor convergence with N if such singulrities re ignored. There is sometimes rt in how singulrities re best hndled. We strt with few strightforwrd suggestions: 1. Integrble singulrities cn often be removed by chnge of vrible. For exmple, the singulr behvior K(t, s) s 1/2 or s 1/2 ner s =cn be removed by the trnsformtion z = s 1/2. Note tht we re ssuming tht the singulr behvior is confined to K, wheres the qudrture ctully involves the product K(t, s)f(s), nd it is this product tht must be fixed. Idelly, you must deduce the singulr nture of the product before you try numericl solution, nd tke the pproprite ction. Commonly, however, singulr kernel does not produce singulr solution f(t). (The highly singulr kernel K(t, s) =δ(t s) is simply the identity opertor, for exmple.) 2. If K(t, s) cn be fctored s w(s)k(t, s), wherew(s)is singulr nd K(t, s) is smooth, then Gussin qudrture bsed on w(s) s weight function will work well. Even if the fctoriztion is only pproximte, the convergence is often improved drmticlly. All you hve to do is replce guleg in the routine fred2 by nother qudrture routine. Section 4.5 explined how to construct such qudrtures; or you cn find tbulted bscisss nd weights in the stndrd references [1,2. You must of course supply K insted of K. This method is specil cse of the product Nystrom method [3,4, where one fctors out singulr term p(t, s) depending on both t nd s from K nd constructs suitble weights for its Gussin qudrture. The clcultions in the generl cse re quite cumbersome, becuse the weights depend on the chosen t i s well s the form of p(t, s). We prefer to implement the productnystrom method on uniform grid, with qudrture scheme tht generlizes the extended Simpson s 3/8 rule (eqution 4.1.5) to rbitrry weight functions. We discuss this in the subsections below. 3. Specil qudrture formuls re lso useful when the kernel is not strictly singulr, but is lmost so. One exmple is when the kernel is concentrted ner t = s onsclemuch smller thn the scle on which the solution f(t) vries. In tht cse, qudrture formul cn be bsed on loclly pproximting f(s) by polynomil or spline, while clculting the first few moments of the kernel K(t, s) t the tbultion points t i. In such scheme the nrrow width of the kernel becomes n sset, rther thn libility: The qudrture becomes exct s the width of the kernel goes to zero. 4. An infinite rnge of integrtion is lso form of singulrity. Truncting the rnge t lrge finite vlue should be used only s lst resort. If the kernel goes rpidly to zero, then

2 798 Chpter 18. Integrl Equtions nd Inverse Theory Guss-Lguerre [w exp( αs) or Guss-Hermite [w exp( s 2 ) qudrture should work well. Long-tiled functions often succumb to the trnsformtion s = 2α α (18.3.1) z +1 which mps <s< to 1 >z> 1so tht Guss-Legendre integrtion cn be used. Here α>is constnt tht you djust to improve the convergence. 5. A common sitution in prctice is tht K(t, s) is singulr long the digonl line t = s. Here the Nystrom method fils completely becuse the kernel gets evluted t (t i,s i). Subtrction of the singulrity is one possible cure: K(t, s)f(s) ds = = K(t, s)[f(s) f(t) ds + K(t, s)[f(s) f(t) ds + r(t)f(t) K(t, s)f(t) ds where r(t) = K(t, s) ds is computed nlyticlly or numericlly. the right-hnd side is now regulr, we cn use the Nystrom method. (18.1.4), we get f i = λ (18.3.2) If the first term on Insted of eqution N w jk ij[f j f i+λr if i + g i (18.3.3) j=1 j i Sometimes the subtrction process must be repeted before the kernel is completely regulrized. See [3 for detils. (And red on for different, we think better, wy to hndle digonl singulrities.) Qudrture on Uniform Mesh with Arbitrry Weight It is possible in generl to find n-point liner qudrture rules tht pproximte the integrl of function f(x), times n rbitrry weight function w(x), over n rbitrry rnge of integrtion (, b), sthe sumof weights times n evenlyspcedvluesof the function f(x), sy t x = kh, (k +1)h,...,(k+n 1)h. The generl scheme for deriving such qudrture rules is to write down the n liner equtions tht must be stisfied if the qudrture rule is to be exct for the n functions f(x) =const,x,x 2,...,x n 1, nd then solve these for the coefficients. This cn be done nlyticlly, once nd for ll, if the moments of the weight function over the sme rnge of integrtion, W n 1 x n w(x)dx (18.3.4) h n re ssumed to be known. Here the prefctor h n is chosen to mke W n scle s h if (s in the usul cse) b is proportionl to h. Crrying out this prescription for the four-point cse gives the result w(x)f (x)dx = 1 [(k 6 f(kh) +1)(k+2)(k+3)W (3k 2 +12k+ 11)W 1 +3(k+2)W 2 W [ 2 f([k +1h) k(k+2)(k+3)w +(3k 2 +1k+6)W 1 (3k +5)W 2 +W f([k +2h) [ k(k+1)(k+3)w (3k 2 +8k+3)W 1 +(3k+4)W 2 W f([k +3h) [ k(k+1)(k+2)w +(3k 2 +6k+2)W 1 3(k +1)W 2 +W 3 (18.3.5)

3 18.3 Integrl Equtions with Singulr Kernels 799 While the terms in brckets superficilly pper to scle s k 2, there is typiclly cncelltion t both O(k 2 ) nd O(k). Eqution (18.3.5) cn be specilized to vrious choices of (, b). The obvious choice is = kh, b =(k+3)h, in which cse we get four-point qudrture rule tht generlizes Simpson s 3/8 rule (eqution 4.1.5). In fct, we cn recover this specil cse by setting w(x) =1, in which cse (18.3.4) becomes W n = h n +1 [(k +3)n+1 k n+1 (18.3.6) The four terms in squre brckets eqution (18.3.5) ech become independent of k, nd (18.3.5) in fct reduces to (k+3)h f(x)dx = 3h kh 8 f(kh)+9h 8 f([k+1h)+9h 8 f([k+2h)+3h f([k+3h) (18.3.7) 8 Bck to the cse of generl w(x), some other choices for nd b re lso useful. For exmple, we my wnt to choose (, b) to be ([k +1h, [k +3h)or ([k +2h, [k +3h), llowing us to finish off n extended rule whose number of intervls is not multiple of three, without loss of ccurcy: The integrl will be estimted using the four vlues f(kh),...,f([k +3h). Even more useful is to choose (, b) to be ([k +1h, [k +2h), thus using four points to integrte centered single intervl. These weights, when sewed together into n extended formul, give qudrture schemes tht hve smooth coefficients, i.e., without the Simpson-like 2, 4, 2, 4, 2 lterntion. (In fct, this ws the technique tht we used to derive eqution , which you my now wish to reexmine.) All these rules re of the sme order s the extended Simpson s rule, tht is, exct for f(x) cubic polynomil. Rules of lower order, if desired, re similrly obtined. The three point formul is w(x)f(x)dx = 1 [ 2 f(kh) (k +1)(k+2)W (2k +3)W 1 +W 2 +f([k +1h) [ k(k+2)w +2(k+1)W 1 W [ 2 f([k +2h) k(k+1)w (2k +1)W 1 +W 2 Here the simple specil cse is to tke, w(x) =1,sotht (18.3.8) W h n= n+1 [(k +2)n+1 k n+1 (18.3.9) Then eqution (18.3.8) becomes Simpson s rule, (k+2)h f(x)dx = h kh 3 f(kh)+ 4h 3 f([k +1h)+h f([k +2h) (18.3.1) 3 For nonconstnt weight functions w(x), however, eqution (18.3.8) gives rules of one order less thn Simpson, since they do not benefit from the extr symmetry of the constnt cse. The two point formul is simply (k+1)h kh w(x)f(x)dx = f(kh)[(k +1)W W 1+f([k +1h)[ kw + W 1 ( ) Here is routine wwghts tht uses the bove formuls to return n extended N-point qudrture rule for the intervl (, b) =(,[N 1h). Input to wwghts is user-supplied routine, kermom, tht is clled to get the first four indefinite-integrl moments of w(x), nmely F m(y) y s m w(s)ds m =,1,2,3 ( ) (The lower limit is rbitrry nd cn be chosen for convenience.) Cutionry note: When clled with N<4,wwghts returns rule of lower order thn Simpson; you should structure your problem to void this.

4 8 Chpter 18. Integrl Equtions nd Inverse Theory void wwghts(flot wghts[, int n, flot h, void (*kermom)(double [, double,int)) Constructs in wghts[1..n weights for the n-point equl-intervl qudrture from to (n 1)h of function f(x) times n rbitrry (possibly singulr) weight function w(x) whose indefiniteintegrl moments F n(y) re provided by the user-supplied routine kermom. int j,k; double wold[5,wnew[5,w[5,hh,hi,c,fc,,b; Double precision on internl clcultions even though the interfce is in single precision. hh=h; hi=1./hh; for (j=1;j<=n;j++) wghts[j=.; Zero ll the weights so we cn sum into them. (*kermom)(wold,.,4); Evlute indefinite integrls t lower end. if (n >= 4) Use highest vilble order. b=.; For nother problem, you might chnge for (j=1;j<=n-3;j++) this lower limit. c=j-1; This is clled k in eqution (18.3.5). =b; Set upper nd lower limits for this step. b=+hh; if (j == n-3) b=(n-1)*hh; (*kermom)(wnew,b,4); for (fc=1.,k=1;k<=4;k++,fc*=hi) Eqution (18.3.4). w[k=(wnew[k-wold[k)*fc; wghts[j += ( Eqution (18.3.5). ((c+1.)*(c+2.)*(c+3.)*w[1 -(11.+c*(12.+c*3.))*w[2 +3.*(c+2.)*w[3-w[4)/6.); wghts[j+1 += ( (-c*(c+2.)*(c+3.)*w[1 +(6.+c*(1.+c*3.))*w[2 -(3.*c+5.)*w[3+w[4)*.5); wghts[j+2 += ( (c*(c+1.)*(c+3.)*w[1 -(3.+c*(8.+c*3.))*w[2 +(3.*c+4.)*w[3-w[4)*.5); wghts[j+3 += ( (-c*(c+1.)*(c+2.)*w[1 +(2.+c*(6.+c*3.))*w[2-3.*(c+1.)*w[3+w[4)/6.); for (k=1;k<=4;k++) wold[k=wnew[k; Lst intervl: go ll the wy to end. Reset lower limits for moments. else if (n == 3) Lower-order cses; not recommended. (*kermom)(wnew,hh+hh,3); w[1=wnew[1-wold[1; w[2=hi*(wnew[2-wold[2); w[3=hi*hi*(wnew[3-wold[3); wghts[1=w[1-1.5*w[2+.5*w[3; wghts[2=2.*w[2-w[3; wghts[3=.5*(w[3-w[2); else if (n == 2) (*kermom)(wnew,hh,2); wghts[1=wnew[1-wold[1-(wghts[2=hi*(wnew[2-wold[2)); We will now give n exmple of how to pply wwghts to singulr integrl eqution.

5 18.3 Integrl Equtions with Singulr Kernels 81 Worked Exmple: A Digonlly Singulr Kernel As prticulr exmple, consider the integrl eqution f(x)+ π with the (rbitrrily chosen) nsty kernel K(x, y) = cos x cos y K(x, y)f(y)dy =sinx ( ) ln(x y) y<x y x y x ( ) which hs logrithmic singulrity on the left of the digonl, combined with squre-root discontinuity on the right. The first step is to do (nlyticlly, in this cse) the required moment integrls over the singulr prt of the kernel, eqution ( ). Since these integrls re done t fixed vlue of x, we cn use x s the lower limit. For ny specified vlue of y, the required indefinite integrl is then either or F m(y; x) = F m(y; x) = y x y x s m (s x) 1/2 ds = s m ln(x s)ds = y x x y (x + t) m t 1/2 dt if y>x ( ) (x t) m ln tdt if y<x ( ) (where chnge of vrible hs been mde in the second equlity in ech cse). Doing these integrls nlyticlly (ctully, we used symbolic integrtion pckge!), we pckge the resulting formuls in the following routine. Note tht w(j +1)returns F j(y; x). #include <mth.h> extern double x; Defined in qudmx. void kermom(double w[, double y, int m) Returns in w[1..m the first m indefinite-integrl moments of one row of the singulr prt of the kernel. (For this exmple, m is hrd-wired to be 4.) The input vrible y lbels the column, while the globl vrible x is the row.we cn tke x s the lower limit of integrtion. Thus, we return the moment integrls either purely to the left or purely to the right of the digonl. double d,df,clog,x2,x3,x4,y2; if (y >= x) d=y-x; df=2.*sqrt(d)*d; w[1=df/3.; w[2=df*(x/3.+d/5.); w[3=df*((x/3. +.4*d)*x + d*d/7.); w[4=df*(((x/3. +.6*d)*x + 3.*d*d/7.)*x+d*d*d/9.); else x3=(x2=x*x)*x; x4=x2*x2; y2=y*y; d=x-y; w[1=d*((clog=log(d))-1.); w[2 = -.25*(3.*x+y-2.*clog*(x+y))*d; w[3=(-11.*x3+y*(6.*x2+y*(3.*x+2.*y)) +6.*clog*(x3-y*y2))/18.; w[4=(-25.*x4+y*(12.*x3+y*(6.*x2+y* (4.*x+3.*y)))+12.*clog*(x4-(y2*y2)))/48.;

6 82 Chpter 18. Integrl Equtions nd Inverse Theory Next, we write routine tht constructs the qudrture mtrix. #include <mth.h> #include "nrutil.h" #define PI double x; Communictes with kermom. void qudmx(flot **, int n) Constructs in [1..n[1..n the qudrture mtrix for n exmple Fredholm eqution of the second kind. The nonsingulr prt of the kernel is computed within this routine, while the qudrture weights which integrte the singulr prt of the kernel re obtined vi clls to wwghts. An externl routine kermom, which supplies indefinite-integrl moments of the singulr prt of the kernel, is pssed to wwghts. void kermom(double w[, double y, int m); void wwghts(flot wghts[, int n, flot h, void (*kermom)(double [, double,int)); int j,k; flot h,*wt,xx,cx; wt=vector(1,n); h=pi/(n-1); for (j=1;j<=n;j++) x=xx=(j-1)*h; Put x in globl vrible for use by kermom. wwghts(wt,n,h,kermom); cx=cos(xx); Prt of nonsingulr kernel. for (k=1;k<=n;k++) [j[k=wt[k*cx*cos((k-1)*h); Put together ll the pieces of the kernel. ++[j[j; free_vector(wt,1,n); Since eqution of the second kind, there is digonl piece independent of h. Finlly, we solve the liner system for ny prticulr right-hnd side, here sin x. #include <stdio.h> #include <mth.h> #include "nrutil.h" #define PI #define N 4 Here the size of the grid is specified. int min(void) /* Progrm fredex */ This smple progrm shows how to solve Fredholm eqution of the second kind using the product Nystrom method nd qudrture rule especilly constructed for prticulr, singulr, kernel. void lubksb(flot **, int n, int *indx, flot b[); void ludcmp(flot **, int n, int *indx, flot *d); void qudmx(flot **, int n); flot **,d,*g,x; int *indx,j; indx=ivector(1,n); =mtrix(1,n,1,n); g=vector(1,n); qudmx(,n); Mke the qudrture mtrix; ll the ction is here. ludcmp(,n,indx,&d); Decompose the mtrix. for (j=1;j<=n;j++) g[j=sin((j-1)*pi/(n-1)); Construct the right hnd side, here sin x. lubksb(,n,indx,g); Bcksubstitute. for (j=1;j<=n;j++) Write out the solution. x=(j-1)*pi/(n-1);

7 18.3 Integrl Equtions with Singulr Kernels 83 1 f (x) x 2 n = 1 n = 2 n = 4 Figure Solution of the exmple integrl eqution ( ) with grid sizes N =1,2,nd4. The tbulted solution vlues hve been connected by stright lines; in prctice one would interpolte smllnsolution more smoothly. printf("%6.2d %12.6f %12.6f\n",j,x,g[j); free_vector(g,1,n); free_mtrix(,1,n,1,n); free_ivector(indx,1,n); return ; With N =4, this progrm gives ccurcy t bout the 1 5 level. The ccurcy increses s N 4 (s it should for our Simpson-order qudrture scheme) despite the highly singulr kernel. Figure shows the solution obtined, lso plotting the solution for smller vlues of N, which re themselves seen to be remrkbly fithful. Notice tht the solution is smooth, even though the kernel is singulr, common occurrence. CITED REFERENCES AND FURTHER READING: Abrmowitz, M., nd Stegun, I.A. 1964, Hndbook of Mthemticl Functions, Applied Mthemtics Series, Volume 55 (Wshington: Ntionl Bureu of Stndrds; reprinted 1968 by Dover Publictions, New York). [1 Stroud, A.H., nd Secrest, D. 1966, Gussin Qudrture Formuls (Englewood Cliffs, NJ: Prentice-Hll). [2 Delves, L.M., nd Mohmed, J.L. 1985, Computtionl Methods for Integrl Equtions (Cmbridge, U.K.: Cmbridge University Press). [3 Atkinson, K.E. 1976, A Survey of Numericl Methods for the Solution of Fredholm Integrl Equtions of the Second Kind (Phildelphi: S.I.A.M.). [

8 84 Chpter 18. Integrl Equtions nd Inverse Theory 18.4 Inverse Problems nd the Use of A Priori Informtion Lter discussion will be fcilitted by some preliminry mention of couple of mthemticl points. Suppose tht u is n unknown vector tht we pln to determine by some minimiztion principle. Let A[u > nd B[u > be two positive functionls of u, so tht we cn try to determine u by either minimize: A[u or minimize: B[u (18.4.1) (Of course these will generlly give different nswers for u.) As nother possibility, now suppose tht we wnt to minimize A[u subject to the constrint tht B[u hve some prticulr vlue, sy b. The method of Lgrnge multipliers gives the vrition δ δu A[u+λ 1(B[u b)= δ δu (A[u+λ 1B[u) = (18.4.2) where λ 1 is Lgrnge multiplier. Notice tht b is bsent in the second equlity, since it doesn t depend on u. Next, suppose tht we chnge our minds nd decide to minimize B[u subject to the constrint tht A[u hve prticulr vlue,. Insted of eqution (18.4.2) we hve δ δu B[u+λ 2(A[u )= δ δu (B[u+λ 2A[u) = (18.4.3) with, this time, λ 2 the Lgrnge multiplier. Multiplying eqution (18.4.3) by the constnt 1/λ 2, nd identifying 1/λ 2 with λ 1, we see tht the ctul vritions re exctly the sme in the two cses. Both cses will yield the sme one-prmeter fmily of solutions, sy, u(λ 1 ). As λ 1 vries from to, the solution u(λ 1 ) vries long so-clled trde-off curve between the problem of minimizing A nd the problem of minimizing B. Any solution long this curve cn eqully well be thought of s either (i) minimiztion of A for some constrined vlue of B, or (ii) minimiztion of B for some constrined vlue of A, or (iii) weighted minimiztion of the sum A + λ 1 B. The second preliminry poinths to do with degenerte minimiztion principles. In the exmple bove, now suppose tht A[u hs the prticulr form A[u = A u c 2 (18.4.4) for some mtrix A nd vector c. IfAhs fewer rows thn columns, or if A is squre but degenerte (hs nontrivil nullspce, see 2.6, especilly Figure 2.6.1), then minimizing A[u will not give unique solution for u. (Toseewhy, review 15.4, nd note tht for design mtrix A with fewer rows thn columns, the mtrix A T A in the norml equtions is degenerte.) However, ifweddny multiple λ times nondegenerte qudrtic form B[u, for exmple u H u with H positive definite mtrix, then minimiztion of A[u +λb[uwill led to unique solution for u. (The sum of two qudrtic forms is itself qudrtic form, with the second piece gurnteeing nondegenercy.)