Asymptotic analysis of numerical steepest descent with path approximations

Transcription

1 Asymptotic analysis of numeical steepest descent with path appoximations Andeas Asheim and Daan Huybechs Repot TW536, Mach 29 Ò Katholieke Univesiteit Leuven Depatment of Compute Science Celestijnenlaan 2A B-31 Hevelee (Belgium)

2 Asymptotic analysis of numeical steepest descent with path appoximations Andeas Asheim and Daan Huybechs Repot TW536, Mach 29 Depatment of Compute Science, K.U.Leuven Abstact We popose a vaiant of the numeical method of steepest descent fo oscillatoy integals by using a low-cost explicit polynomial appoximation of the paths of steepest descent. A loss of asymptotic ode is obseved, but in the most elevant cases the oveall asymptotic ode emains highe than a tuncated asymptotic expansion at simila computational effot. Theoetical esults based on numbe theoy undepinning the mechanisms behind this effect ae pesented. Keywods : oscillatoy integal, steepest descent, numeical integation AMS(MOS) Classification : Pimay : 65D3, Seconday : 3E2, 41A6.

3 Asymptotic analysis of numeical steepest descent with path appoximations Andeas Asheim, Daan Huybechs Mach 16, 29 Abstact We popose a vaiant of the numeical method of steepest descent fo oscillatoy integals by using a low-cost explicit polynomial appoximation of the paths of steepest descent. A loss of asymptotic ode is obseved, but in the most elevant cases the oveall asymptotic ode emains highe than a tuncated asymptotic expansion at simila computational effot. Theoetical esults based on numbe theoy undepinning the mechanisms behind this effect ae pesented. AMS(MOS) Classification: Pimay : 65D3, Seconday : 3E2, 41A6. Keywods: oscillatoy quadatue, steepest descent, numeical integation. 1 Intoduction Conside a highly oscillatoy integal of the fom I[f] = 1 1 f(x)e iωg(x) dx, (1.1) whee ω is a lage paamete and f and g ae smooth functions called the amplitude function and oscillato of the integal espectively. Such integals, often efeed to as Fouie-type integals, appea in a wide aea of applications, e.g., highly oscillatoy scatteing poblems in acoustics, electomagnetics o optics [5, 3, 13, 2]. Numeical evaluation of Fouie-type integals with classical techniques becomes expensive as ω becomes lage, which coesponds to a highly oscillatoy integal. Typically, a fixed numbe of evaluation points pe wavelength is equied to obtain a fixed accuacy, which makes the computational effot at least linea in ω [6]. Depatment of Mathematical Sciences, NTNU, 7491 Tondheim, Noway. Andeas.Asheim@math.ntnu.no Depatment of Compute Science, KU Leuven, 31 Leuven, Belgium. Daan.Huybechs@cs.kuleuven.be. This autho is a PostDoctoal Fellow of the Reseach Foundation - Flandes (FWO). 1

4 Asymptotic techniques on the othe hand yield appoximations that become moe accuate as ω inceases, making them supeio fo ω sufficiently lage. One of these techniques, the pinciple of stationay phase [2, 25], states that I[f] asymptotically depends only on f and g in a set of special points as ω. These points ae the endpoints, hee x = 1 and x = 1, and stationay points - points whee the deivative of g vanishes. At stationay points the integal is locally non-oscillatoy. The integal has an asymptotic expansion in invese powes of ω, with coefficients that depend on the deivatives of f and g at these citical points [15]. x= 1 x=1 Figue 1: The contous of the imaginay pat of the oscillato g(x) = x 2 in the complex plane and the coesponding paths of steepest descent. Two paths emege fom the endpoints x = 1 and x = 1. They ae connected by a path passing though the stationay point at x =. A set of paticulaly effective ways of obtaining the contibution fom a special point ae the saddle point methods[25, 19, 8]. Based on Cauchy s integal theoem, the path of integation can be defomed into the complex plane without changing the value of the integal, povided that f and g ae analytic [9]. The method of steepest descent is obtained by following a path whee g has a constant eal pat and inceasing imaginay pat, which endes the integal (1.1) non-oscillatoy and exponentially deceasing. This pocedue yields sepaate paths oiginating fom each special point that typically connect at infinity (see Figue 1 fo an illustation). The esult is sepaate contibutions coesponding to each special point. Evey one of these contibutions is a non-oscillatoy integal that can be witten as ψ(q)e ωq dq, (1.2) 2

5 whee ψ is a smooth function, = 1 fo endpoint contibutions, and > 1 fo stationay points. These integals ae usually teated with standad asymptotic techniques like Watson s Lemma. The lage class of saddle point methods also contains methods that follow othe paths with simila chaacteistics as the steepest descent paths, e.g., Peon s method[25]. The asymptotic expansion of I[f] in geneal diveges, but it can yield vey accuate appoximations if ω is vey lage. Still, divegence implies that the eo is uncontollable, which is poblematic in the context of numeical computations. Recent eseach has howeve poduced seveal numeical methods that exhibit convegence. The Filon-type methods [15, 14, 16] ae based on polynomial intepolation of the amplitude f and can delive eos that ae O(ω p ) fo any p, much like tuncated asymptotic expansions, but with contollable eo fo fixed ω. Filon-type methods equie that moments w k = I[x k ] ae available, a seious dawback in some cases. Combining asymptotic expansions and Filon-type methods[1] can economise on, but not eliminate the need fo moments. Methods that do not ely on moments ae the Levin-type methods, due to Levin[18] and extended by Olve[22, 23]. Levin-type methods do not wok in the pesence of stationay points, but a wok-aound is povided in [21]. We efe the eade to [11] fo a detailed oveview of these and othe numeical methods. One of the altenatives is the numeical method of steepest descent [12], which is a numeical adaptation of the above descibed method of steepest descent. Relying on classical numeical integation methods applied to an exact decomposition of the integal, the numeical method of steepest descent has contollable eo wheeve the exact decomposition is available, and asymptotic eo decay O(ω p ) fo any p. The paths of steepest descent can howeve be difficult to compute, as thei computation coesponds to solving a non-linea poblem that can in pactice only be solved iteatively. The method of this pape is simila in spiit but based on the pactical obsevation that the exact choice of path is not essential. This obsevation esonates with the theoy behind saddle point methods. A Taylo expansion of the path of steepest descent, which can explicitly be deived fom a Taylo expansion of the oscillato function g, is in many cases sufficient. Iteative methods to solve a non-linea poblem can theefoe be entiely avoided. We obtain a numeical scheme which is elatively simple to implement and cheap to evaluate. The method exhibits high asymptotic ode, and the ode is in fact highe than one would get fom a tuncated asymptotic expansion using exactly the same numbe of deivatives of g. It is the pupose of this pape to analyse the asymptotic ode of the poposed explicit numeical saddle-point method. Unlike the numeical adaptation of the steepest descent method and the othe methods fo highly oscillatoy integals mentioned above, the asymptotic ode does not follow fom standad esults in asymptotic analysis. A seemingly iegula elation between the numbe of deivatives of g that ae used and the numbe of 3

6 quadatue points along the appoximate paths of steepest descent can only be explained in tems of elementay numbe theoy. The main esult of this pape is fomulated and poved in 4 in Theoem The numeical method of steepest descent In this section we give a bief oveview of the numeical method of steepest descent. Fo a moe thoough teatment, see [25] fo the classical method of steepest descent, and [12] fo paticulaities on the numeical vesion. In the following, we will fo simplicity assume that all paths may extend to infinity, which implies among othe things that f and g should be analytic in a sufficiently lage potion of the complex plane. We note that this equiement can be significantly elaxed if so desied[1]. 2.1 Paths of steepest descent Fo the oscillatoy integal (1.1) the path of steepest descent h x (p) oiginating fom the point x can be found by solving the equation g(h x (p)) = g(x) + ip. (2.1) Subject to the bounday condition h x () = x, equation (2.1) is uniquely solvable fo small p if g (x). Along the path of steepest descent we have e iωg(hx(p)) = e iωg(x) e ωp, which means that the line integal P I[f; h x, P] = e iωg(x) f(h x (p))h x(p)e ωp dp, is non-oscillatoy and exponentially deceasing. Along paths of steepest descent oiginating fom diffeent points, g x (h(p)) may have diffeent, constant eal pats, hence the paths may neve meet. A connecting path must theefoe be intoduced. If thee ae no stationay points, the paths may connect at infinity by letting P. In that case, the connecting path has no contibution to the value of the integal. In the pesence of stationay points in [ 1, 1] howeve, the total path must pass though all of these points and thei contibutions ae not negligible. Any value ξ [ 1, 1] such that g (ξ) = is called a stationay point. We call ξ a stationay point of ode 1 if g (i) (ξ) = fo i = 1, 2,..., 1, and g () (ξ). 1 The canonical example is g(x) = x. At a stationay point, equation (2.1) may have seveal solutions. In paticula, if ξ is a stationay point of ode 1 >, then thee ae diffeent paths, h ξ,j, j = 1,...,, emeging fom ξ. Since the total path passes tough ξ only once, exactly 1 Note that with this definition an endpoint is a stationay point of ode. 4

7 two of these paths ae elevant. We denote these two paths by h ξ,j1 and h ξ,j2. Each of these paths coesponds to an integal of the fom P I[f; h ξ,j, P] = e iωg(ξ) f(h ξ,j (p))h ξ,j (p)e ωp dp. Again, letting P eliminates contibutions fom connecting paths, with cetain assumptions. Witing I[f; h x ] = lim P I[f; h x, P], the integal (1.1) is epesented as a sum of contibutions I[f] = I[f; h 1 ] I[f; h ξ1,j 1 ]+I[f; h ξ1,j 2 ]+... I[f; h ξn,j 1 ]+I[f; h ξn,j 2 ] I[f; h 1 ], whee ξ 1,...,ξ n ae stationay points. We will concentate on integals of the type I[f; h], heeafte efeed to as steepest descent integals. 2.2 Numeical evaluation of steepest descent integals Steepest descent integals can be appoximated efficiently with Gaussian quadatue. This is the obsevation behind the numeical method of steepest descent, which we shall biefly explain hee. Fo convenience, we intoduce the notation f x (p) = f(h x (p))h x(p). The contibution fom an endpoint becomes I[f; h x ] = e iωg(x) f x (p)e ωp dp = eiωg(x) ω ( t ) f x e t dt. (2.2) ω Since f x (t/ω) is smooth, this integal can be computed efficiently with classical Gauss-Laguee quadatue fo the weight function e t [6]. Applying an n-point quadatue yields an appoximation with eo O(ω 2n 1 ) [12]. Tuncating the asymptotic expansion afte n tems yields only O(ω n 1 ) asymptotic eo, but equies the same numbe of evaluations of f. Fo the contibution fom a stationay point things ae a little diffeent. When ξ is a stationay point of ode 1 >, h ξ (p) behaves as p 1/ nea p = and h ξ (p) has a p ( 1)/ singulaity [9]. This singulaity can be canceled by the substitution p = q. The contibution is now witten I[f; h ξ ] = e iωg(ξ) f ξ (q )q 1 e ωq dq (2.3) = eiωg(ξ) ω f ξ ( t ω )t 1 e t dt. This is an integal of the fom (1.2). Since f ξ ( t ω )t 1 is a smooth function, the integal can be efficiently appoximated by Gaussian quadatue with 5

8 weight function e t. We note that it may be beneficial to mege the two contibutions fom a stationay point into a single integal ove the whole eal line. Fo example, in the case of a fist ode stationay point ( = 2), classical Gauss-Hemite quadatue can be applied[7]. In this exposition, howeve, we will only wok with integals on the half-space. The esult of applying an n-point Gaussian quadatue leads to an appoximation with an eo which is O(ω (2n+1)/ ) as ω [7]. In contast, tuncating the asymptotic expansion afte n tems yields only O(ω (n+1)/ ) asymptotic eo, but equies the same numbe of evaluations of f. 3 A numeical saddle point method Finding the path of steepest descent means solving equation (2.1). This is a non-linea equation and solving it amounts to computing the invese function g 1, which in pactical applications may be difficult to achieve. The ationale in this section is that in many cases it is sufficient to have only a ough appoximation of the exact steepest descent path. If not, then the ough appoximation is still useful as a stating value fo, e.g., Newton iteations to solve the non-linea equation numeically. Hee, we obtain a local appoximation of the path by means of its Taylo seies aound x. Only deivatives of g at x ae used to constuct this appoximation. This appoximate path may divege away fom the actual steepest descent path deep into the complex plane. Howeve, this is not a poblem in pactice povided ω is lage: because the quadatue points cluste towads x as ω gows, as can be seen fom equations (2.2) and (2.3), a good appoximation close to the eal axis is geneally sufficient. 3.1 Local paths at endpoints In the case of the steepest descent path emeging fom an endpoint, we assume that the path is of the fom h x (p) = x + a j p j. (3.1) j=1 Note that we aleady incopoated the bounday condition h x () = x. Substitution into equation (2.1) gives ( g x + a j p j) = g(x) + ip. j=1 Taking the Taylo expansion of g aound x yields the equation ( j=1 a jp j ) k g (k) (x) = ip. (3.2) k! k=1 6

9 The coefficients can now be obtained by seies invesion. The fist few coefficients ae given explicitly by, with evaluation in x implied, g a 1 = i g, a 2 = 1 2 (g ) 3, a 3 = i 1 ( 12 (g ) 5 2g g (3) (g ) 2), (3.3) a 4 = 1 1 ( 24 (g ) 7 g (4) (g ) 2 + 1g g g (3) 15(g ) 3). In geneal, a k is given in tems of deivatives of g up to ode k. We define the local path h x by tuncating the seies of h x afte m tems, m 1 h x (p) = x + a j p j. (3.4) j=1 This means that the left and ight hand side of (2.1) match up to ode m, g( h x (p)) = g(x) + ip + O(p m ), p. (3.5) Fom this path we can define the steepest descent integal with an appoximated path, using the notation f x (p) = f( h x (p)) h x(p) and g x (p) = g( h x (p)), I[f; h x, P] = P f x (p)e iω gx(p) dp. (3.6) We shall late evaluate this integal numeically. The numeical appoximation will seve as an appoximation to the infinite integal I[f; h x ], we shall see that this is indeed justified in Local paths at stationay points We now tun ou attention to paths passing though stationay points. Let x be a stationay point of ode 1, meaning that g (x) =... = g ( 1) (x) =, but g () (x). Expanding the path stating at x in intege powes of p is not possible, since h x (p) is singula at p =. This can also be seen fom equation (3.2): the fist 1 tems in the expansion of g in the left hand side would be zeo, which makes it impossible to match the ight hand side of the equation. Howeve, poceeding as in 2.2, the substitution p = q eliminates this poblem. Thus, we assume a path of the fom h x (p) = x + a j p j/. (3.7) j=1 Note that the function h x (q ) is analytic in q. Plugging this ansatz into equation (2.1) fo the path of steepest descent yields ( j=1 a jp j/ ) k g (k) (x) = ip. k! k= 7

10 The fist coefficient is easily obtained, i! a 1 = g () (x). (3.8) The squae oot in this expession has banches in the complex plane, coesponding to the diffeent paths nea the stationay point. Moe coefficients can be computed ecusively. In the case of an ode one stationay point, the fist fou coefficients ae, with evaluation in x implied, 2i a 1 = ± g, a 2 = i g (3) 3 (g ) 2, 2i i ( a 3 = ± g 36(g ) 3 3g g (4) 5(g (3) ) 2), (3.9) a 4 = 1 1 ( 27 (g ) 5 4(g (3) ) 3 45g (3) g (4) g + 9g (5) (g ) 2). Explicit expessions fo the coefficients can be found fo geneal. We efe the eade to [24] fo a geneal desciption of such explicit expessions. As in the endpoint case, we fom an appoximated path by tuncating (3.7) afte m tems, m 1 h x (p) = x + a j p j/. (3.1) j=1 This means that the both sides of (2.1) match up to ode +m 1, g( h x (p)) = g(x) + ip + O(p +m 1 ), p. (3.11) This expession agees with (3.11) fo = 1. Next, we fom the integal I[f; h x, P] = = P Q f x (p)e iω gx(p) dp. q 1 fx (q )e iω gx(q) dq. (3.12) with Q = P 1/. 3.3 Numeical evaluation As noted in section 2.2, it is advantageous to evaluate the half-space integal I[f, h x ] with Gaussian quadatue. Though the integal I[f, h x, P] is finite, we intend to apply Gaussian half-space quadatue hee as well. 8

11 Fo the numeical evaluation of steepest descent integals with appoximated paths, we ewite (3.12) as I[f; h x, P] = Q q 1 fx (q )e iω gx(q )+ωq e ωq dq. (3.13) Note that (3.6) is a special case of (3.13) with = 1, so that we can teat the cases of endpoints and stationay points simultaneously. A change of vaiables q = ω 1/ t gives the fom I[f; h x, P] = ω Qω 1/ t 1 fx (t /ω)e iω gx(t /ω)+t e t dt. This integal can be evaluated with the same Gaussian half-space quadatue ules with weight function e t that wee used on the exact steepest descent integals. To be pecise, if that quadatue ule is given by points x i and weights w i, then we popose the appoximation I[f; h x, P] Q[f; h x ] := ω n i=1 w i x 1 i ( ) x f i x e iω gx x i ω «+x i. (3.14) ω We expect that this quadatue ule povides a good appoximation to I[f; h x, P]. This is what we examine next in 4. 4 Asymptotic eo analysis Thus fa, we have pesented a way of obtaining a numeical appoximation of I[f; h x, P]. We will show fist in 4.1 that this finite saddle-point integal is a good (asymptotic) appoximation to the infinite steepest descent integal I[f; h x ]. Next, we shall investigate in 4.2 the numeical appoximation of I[f; h x, P] by Gaussian quadatue. Theoem 4.3 gives the asymptotic ode of this appoximation. Its poof follows in 4.3 and The eo of using tuncated appoximate paths In the method outlined in section 3, we eplaced the exact path of steepest descent h x oiginating at x with an appoximation h x that is valid only nea x. By ou assumptions of analyticity, the path taken does not change the value of the integal. Howeve, since the appoximate path may divege away fom the exact path fo lage P, the limit P may esult in both paths leading into diffeent sectos of the complex plane. It is clea that the integal along the appoximate path should be tuncated at finite P to avoid this. In the following theoem and coollay, we pove that the diffeence between the exact steepest descent integal I[f; h x ] and the tuncated integal I[f; h x, P] is exponentially small as ω. This implies that using a numeical appoximation of I[f; h x, P] is justified. 9

12 Γ h x (P) h x (P) Figue 2: Illustation of the exact (continuous) and appoximate (dashed) steepest descent paths. The cuve Γ connects a tuncation of these two paths. Theoem 4.1. Let x [ 1, 1] be a point of ode 1. Assume f and g ae analytic, and let h x (p) be an m-tem appoximation to the exact path h x (p) as in (3.1) with m > 1. Then a constant P > exists, such that I[f; h x, P] I[f; h x, P] = O(ω n ), n >, P < P. Poof. By Cauchy s integal theoem we have I[f; h x, P] I[f; h x, P] = f(s)e iωg(s) ds, Γ whee Γ is any simple path connecting h x (P) and h x (P). In the following, we choose Γ to be the staight line. We intend to show that the integand is exponentially small along all of Γ. Let us expand g in a Taylo seies aound h x (P). We have g(x + δ) = O(δ ) and g (j) (x + δ) = O(δ j ). Since h x (p) = O(p 1/ ), we find that g (j) (h x (p)) = O(p ( j)/ ). We have by constuction that h x (p) h x (p) = O(p m/ ) and theefoe, γ h x (P) = O(P m/ ), γ Γ, P. To conclude, we note that g(γ) = g(h x (P) + γ h x (P)) = j= g(j) (h x (P))(γ h x (P)) j = g(x) + ip + O(P ( 1+m)/ ) + O(P ( 2+2m)/ ) +... If m > 1 and if P is sufficiently small, then the tem ip dominates the othe tems and g has positive imaginay pat along Γ. It follows that the integand is exponentially small along all of Γ. 1

13 Coollay 4.2. Unde the assumptions of Theoem 4.1 I[f; h x ] I[f; h x, P] = O(ω n ), n >, ω. Poof. We have I[f; h x ] = I[f; h x, P] + P ψ(q)e ωq dq, whee ψ(q) is analytic in q. It follows fom epeated integation by pats that I[f; h x ] I[f; h x, P] = O(ω n ), n > ω. The esult follows fom this and Theoem Asymptotic eo of the numeical appoximation Since eplacing the paths does not lead to a loss in asymptotic ode, the ode of the oveall method elies on the ode of the numeical appoximation of I[f; h x, P]. We evaluate the latte by a quadatue ule. A quadatue ule with n points and with ode d with espect to the weight function e x satisfies the conditions x j e x dx = n k=1 w k x j k, j =,...,d 1. (4.1) Using such a ule fo the steepest descent integal leads to an asymptotic eo of size O(ω (d+1)/ ) [7, Th.2]. When using an appoximate path, we have the following esult. Note that by intege division d\β, we mean that the eal quantity d/β is ounded towads the neaest smalle intege. Theoem 4.3. Assume x is eithe a egula point, = 1, o a stationay point of ode 1 >. An appoximation I[f; h x, P] to the steepest descent integal I[f; h x ] is constucted by eplacing the path h x with its m-tem Taylo expansion h x, with m > 1. Let Q[f; h x ], given by (3.14), denote the appoximation to I[f; h x, P], obtained though an n-point quadatue ule of ode d that satisfies the conditions (4.1). Define β = + m 1, k = d\β and l = d mod β. Then { I[f; h x ] Q[f; h O(ω d+1 +k ), if l m 1, x ] = O(ω d+1 l (m 1) (4.2) +k+ ), if l > m 1, fo ω. In paticula, fo = 1 we have I[f; h x ] Q[f; h x ] = O(ω d 1+d\m ). 11

14 We can also fomulate an uppe bound fo the exponents in (4.2) that avoids intege aithmetic. Coollay 4.4. Unde the same conditions as in Theoem 4.3, we have I[f; h x ] Q[f; h x ] = O(ω d+1 + d β ). (4.3) Poof. Fo the fist case of (4.2), note that k = d\β d/β. Fo the second case, assume that l = K + m 1 with < K <. Then k + l (m 1) = k + K < k + K+m 1 +m 1 = d β. Let us fist compae the esult of Theoem 4.3 to the esult based on using the exact path. One incus a loss of minimum k = d\β = d\( + m 1). In ode to achieve the full ode (d + 1)/, one should at least have k =, meaning d < β, m < d + 1. Full ode is then achieved if l m 1, which is always tue wheneve = 1, and moe likely to be violated fo lage. In the convese case, we have a maximum ode loss of one. Next, we compae to the esult based on using a tuncated asymptotic expansion. This is moe involved. An s-tem expansion has asymptotic eo O(ω (s+1)/ ) and equies the values g (j) (x), j =,..., + s 1 [15]. Using these same values, we can affod m = s. The asymptotic expansion also equies the s values f (j) (x), j =,...,s 1. The poposed method equies f(x k ) and g(x k ) fo k = 1,...,n. We choose n = s and continue by counting evaluations of f o any of its deivatives. Fo the asymptotic expansion, s values of f lead to ode s+1. A Gaussian quadatue ule with espect to the weight function e x yields d = 2s. By Coollay 4.4, the numeical saddle-point method then yields an ode geate than o equal to d + 1 d β = 2s + 1 2s + s 1 = s s [ ] s ( + 1) + s 1 Thus we ae guaanteed to do at least as good as the asymptotic expansion wheneve s + 1. Note that in the above, we ignoed the evaluations of g in the complex plane. This is justified in a setting whee many integals of the fom I[f] need to be evaluated fo the same oscillato g, fo example when computing moments fo late use in Filon-type quadatue [15]. Both these calculations show that the poposed method compaes well to both the method with exact paths and asymptotic expansions when is elatively small. In eal-life applications we do howeve not expect to encounte cases with being lage, we will typically have = 1 o = 2. 12

15 4.3 Suppoting lemmas We once again ewite the integal I[f; h x, P] in the following fom: I[f, h x, P] = e iωg(x) Q ψ(q)e iωr +m 1(q) e ωq dq, (4.4) whee 1 is the ode of the point x, R β (q) is a function of the fom and R β (q) = q β j q j, (4.5) j= ψ(q) = f x (q )q 1, is a smooth function independent of ω. This fomulation follows fom the constuction of the appoximate paths. In paticula, the fom of R +m 1 follows fom (3.11) with p = q. The following lemma is a genealization of Lemma 2.1 in [7]. That lemma chaacteized the asymptotic ode of a scaled quadatue ule applied to a steepest descent integal of the fom (1.2). Assume an n-point quadatue ule is given that satisfies the conditions (4.1). It was poved in [7] that, fo a function u(x) analytic in x =, the quadatue appoximation behaves as n u(x)e ωx dx ω 1/ w k u(x k ω 1/ ) = O(ω (d+1)/ ). k=1 Hee, we will allow the integand to depend on ω in a benign manne and show that the asymptotic ode changes in a way that eflects the possible gowth o decay of the integand as a function of ω. Lemma 4.5. Assume an n-point quadatue ule is given such that conditions (4.1) hold. Let u(x; ω) be analytic in x = with a positive adius of convegence R fo each ω ω, u(x; ω) = a j (ω)x j, x < R, (4.6) j= and such that a j = O(ω γ j ) with γ j R. If < P < R, then P u(x; ω)e ωx dx ω 1/ n k=1 w k u(x k ω 1/ ; ω) = O(ω max j d γ j j+1 ). 13

16 Poof. We have P u(x; ω)e ωx dx = j= P a j (ω) x j e ωx dx. Using integation by pats, as in the poof of Coollay 4.2, we find that x j e ωx dx P Next, it is staightfowad to veify that x j e ωx dx = O(ω m ), m N. n x j e ωx dx ω 1/ w k (x k ω 1/ ) j = k=1 {, j < d, O(ω (j+1)/ ), j d The fist case follows fom exactness of the quadatue ule fo polynomials up to degee d 1. The second case follows because both tems in the left hand side have the given size: the integal can be computed explicitly, the summation contains the facto ω (j+1)/. Combining all of the above poves the esult. Note that u(x; ω) is evaluated in the points x k ω 1/ which, fo sufficiently lage ω, lie in the adius of convegence of u. Finally, we will examine the asymptotic size of functions of the fom e ωη(x) and thei deivatives. In ode to obtain the esult, we use a vesion of Faà di Buno s fomula expessed with intege patitions. A patition of an natual numbe n is a way of witing it as a sum of natual numbes. The numbe of diffeent ways to do this is the patition numbe of n, denoted a(n). We wite a patition p of the intege n as an aay p = (p 1, p 2,...,p n ), whee p j is the numbe of times the intege j occus in the sum, i.e., n j p j = n. (4.7) j=1 See, e.g., [4] fo a detailed teatment of patitions and [17] fo Faà di Buno s fomula, which we ecall in the following Lemma. Lemma 4.6 (Faà di Buno s Fomula). If g and f ae functions that ae sufficiently diffeentiable, then d n dx ng(f(x)) = n! p 1!p 2!...p n! g(k) (f(x)) ( f (x) 1! ) p1 ( f (x) 2! ) p2 ( f (n) (x) ) pn... n! whee the sum is ove all patitions p of n with enties p 1, p 2,...,p m, and k = p 1 + p p n. 14

17 Lemma 4.7. Let R β (q) be defined by (4.5) fo an intege β >. The deivatives of e ωr β(q), evaluated at q =, have an expansion of the fom d n β(q) dq neωr n\β = b j ω j, ω, q= j= whee \ denotes intege division. Poof. It is clea that R (j) β () =, j < β. (4.8) Using Faà di Buno s Fomula (Lemma 4.6), we have d n β(q) dq neωr = q= e ωr n! ( ωr β() β (q)) p1 ( ωr β (q) ) p2 ( (n) ωr β (q) ) pn... = p 1!p 2!...p n! 1! 2! n! n! ( (β) ωr β (q) ) pβ ( (β+1) ωr β (q)) pβ+1 ( (n) ωr β (q) ) pn,... p 1!p 2!...p n! β! (β + 1)! n! whee the sum is ove all patitions p of n. The last line follows fom equation (4.8). Clealy, each of the tems in this sum is popotional to ω P n j=β p j. It is also clea that the expansion consists of positive intege powes of ω. To find the dominating tem, we maximise the expession n j=β p j ove the set of all patitions of n. It emains only to pove that n p j n\β, j=β p patitions of n Assume a patition q of n exists such that n q j = n\β + M, j=β with M >. Fom q we can constuct anothe patition q as follows. We let q β = n\β + M and q j =, j > β. It follows fom ou constuction that n j q j j=β n j q j = β(n\β + M) > n. j=β No matte how we choose q j fo j < β, q can neve satisfy the summation popety (4.7) and neithe can q. This poves the esult eductio ad absudum. 15

18 The final lemma concens the maximal exponent of ω that may aise in the esult of Lemma 4.5. Lemma 4.8. Assume that and β ae integes such that β > and define the sequence s j = j\β j. Fo any positive intege d, let k = d\β and l = d mod β. The maximum of {s j } j=d is max j d s j = { k d, if l β, k + 1 (k+1)β, if l > β. Poof. Fo the intege division we have the identity j\β = j β 1 (j mod β). (4.9) β This means that s j = j( 1 β 1 ) 1 β (j mod β). The fist of these tems is deceasing monotonically. The second tem is non-inceasing, except when the intege pat of j/β changes. This implies that the lagest element in the sequence fo j d is eithe the fist element, s d, o s nβ fo some intege n. In the latte case, we have s nβ = n(1 β/), which again is deceasing. This means that a maximum must occu at the smallest admissible n. This is n = k, when d is a multiple of β, and n = k+1 othewise. This leads to s kβ = s d as above o s (k+1)β. Fom the identity (4.9), we find that the coesponding element is eithe o s d = d\β d = k d s (k+1)β = (k + 1)β\β (k + 1)β = k + 1 (k + 1)β. One easily veifies that the fome is lage than the latte if l < β. They ae equal if l = β. 16

19 4.4 Poof of Theoem 4.3 We assembled enough esults in 4.3 to state a shot poof of Theoem 4.3. In the following, let β = + m 1. Poof. Leibniz fomula gives the deivatives of the integand of (4.4) as the sum d n [ ] n ( ) ψ(q)e iωr β (q) n [ d j dq n = j dq j eiωr β(q) dn j dq ψ(q) ] n j. j= Lemma 4.7 applied to each of the these tems gives something of the fom, d n [ ] n\β ψ(q)e iωr β (q) dq n = c j ω j. q= j= Hence, a Taylo seies aound q = has coefficients that ae O(ω n\β ). All conditions of Lemma 4.5 ae satisfied and we can conclude that the eo of the quadatue appoximation is I[f; h x, P] Q[f; h x ] = O(ω max j d j\β (j+1)/ ). The maximum in the exponent follows fom Lemma 4.8, since j\β (j + 1)/ = 1/ + [j\β j/] = 1/ + s j, whee s j is defined as in Lemma 4.8. This leads to the stated ode (4.2) of the quadatue appoximation. The case l β = m 1 follows immediately. Fo the second case, one can veify that k + 1 (k + 1)β 1 = d k + l (m 1). Fo = 1, the second case does not aise because then β = + m 1 = m and the condition l m 1 always holds, so the esult simplifies. Thus fa we poved the asymptotic eo in appoximating I[f; h x, P]. The final esult now follows fom Coollay Numeical expeiments In this section we will illustate the use of the method outlined in section 3 as well as the esults egading the asymptotic eo behaviou pedicted in Theoem

20 exact 2 tems 3 tems 4 tems 5 tems ω Figue 3: Log-plot of eo fo diffeent path appoximations. Case of egula endpoint. 5.1 Test of case with no stationay points Conside the highly oscillatoy integal I[f] = 1 1 sin(x)e iω/(x+2) dx The oscillato g(x) = 1/(x + 2) has no stationay points, meaning thee ae only contibutions fom the endpoints. The exact paths can be computed in this case. In Figue 3 we have plotted the eo of the two-point Gauss-Laguee quadatue applied to the esulting line integals with the given exact paths as well as appoximate paths with diffeent numbe of tems. Note that the appoximate paths ae constucted only with the knowledge of some deivatives of g. The loss of ode when using appoximate paths, which can clealy be obseved in Figue 3, is pedicted in Theoem 4.3. We shall test the conclusion of the theoem by using appoximate paths with diffeent numbe of tems and diffeent numbe of quadatue points, and then measuing the asymptotic ode by egession fo each combination. The esult of this test can be seen in Table 1 along with the pedicted ode, 2n + 1 2n\m. 18

21 n, m exact (3) 2.(2) 3.(3) 3.(3) 3.(3) 2 5.(5) 3.1(3) 4.(4) 4.1(4) 5.(5) 3 7.(7) 4.2(4) 5.1(5) 6.9(6) 6.2(6) 4 9.(9) 5.3(5) 7.1(7) 7.3(7) 8.(8) Table 1: Measued ode fo diffeent numbes n of Gauss-Laguee points with m tems in the Taylo expansion of the steepest descent path. Fist column is with the exact path. Numbes in paentheses ae odes pedicted in Theoem Case of stationay points Now conside the integal I = 1 cos(x)e iω(x3 +2x 2) dx, which has an ode one stationay point at the oigin. Even in this simple polynomial case the exact path oiginating fom the stationay point is cumbesome to compute. Instead we constuct the paths with the coefficients (3.9). The steepest descent integal coesponding to the path fom the stationay point at x = is computed with a scaled Gaussian quadatue. By using the exact path and a lage numbe of quadatue points, we can nealy eliminate the eo contibution fom the ight endpoint. Thus the eo will be dominated by the eo fom the x = -contibution. Running ove a ange of diffeent ω we estimate the ode by egession, and the esults fit with the pedictions fom Theoem 4.3 (see Table 2a). No attempt to use exact paths at the oigin was done, and the efeence solution was obtained with Matlab s standad quadatue package close to machine pecision. Fo completion, we include the esults fom paallel tests done on the integal I = 1 e iω(x4 +4x 3) dx, which has an ode 2 stationay point at the oigin(table 2b). Refeences [1] A. Asheim. A combined Filon/asymptotic quadatue method fo highly oscillatoy poblems. BIT, 48(3): , 28. [2] A. Asheim and D. Huybechs. Local solutions to high fequency 2D scatteing poblems. Technical epot, NTNU, Tondheim,

22 (a) (b) n, m (1) 1.5(3/2) 1.5(3/2) 2 1.5(3/2) 1.5(3/2) 2.(2) 3 1.6(3/2) 2.4(5/2) 2.5(5/2) 4 1.9(2) 2.4(5/2) 3.5(7/2) n, m (2/3) 1.(1) 1.(1) 2.6(2/3) 1.(1) 1.3(4/3) 3.9(1) 1.7(4/3) 1.3(4/3) 4 1.4(1) 1.6(5/3) 2.4(2) Table 2: Measued ode fo diffeent numbes n of Gauss points with m tems in the expansion of the steepest descent path. Numbes in paentheses ae odes pedicted in Theoem 4.3. a)-case of ode one stationay point, b)-case of ode two stationay point. [3] P. Bettess. Shot wave scatteing, poblems and techniques. Phil. Tans. R. Soc. Lond. A, 362: , 24. [4] M. Bona. A walk though combinatoics. Wold scientific co., 22. [5] M. Bon and E. Wolf. Pinciples of optics. Cambidge Univesity Pess, Cambidge, [6] P. J. Davis and P. Rabinowitz. Methods of numeical integation. Compute Science and Applied Mathematics. Academic Pess, New Yok, [7] A. Deaño and D. Huybechs. Complex Gaussian quadatue of oscillatoy integals. Nume. Math., 28. To appea. [8] A. Edélyi. Asymptotic expansions. Dove publications inc., New Yok, [9] P. Henici. Applied and computational complex analysis Volume I. Wiley & Sons, New Yok, [1] D. Huybechs and S. Olve. Supeintepolation in highly oscillatoy quadatue. Technical epot. In pepaation. [11] D. Huybechs and S. Olve. Highly Oscillatoy Poblems: Computation, Theoy and Applications, chapte 2: Highly oscillatoy quadatue. Cambidge Univ. Pess, 28. [12] D. Huybechs and S. Vandewalle. On the evaluation of highly oscillatoy integals by analytic continuation. SIAM J. Nume. Anal., 44(3): , 26. 2

23 [13] D. Huybechs and S. Vandewalle. A spase discetisation fo integal equation fomulations of high fequency scatteing poblems. SIAM J. Sci. Comput., 29(6): , 27. [14] A. Iseles and S. P. Nøsett. On quadatue methods fo highly oscillatoy integals and thei implementation. BIT Nume. Math., 44(4): , Decembe 24. [15] A. Iseles and S. P. Nøsett. Efficient quadatue of highly oscillatoy integals using deivatives. Poc. Roy. Soc. A., 461(257): , 25. [16] A. Iseles and S. P. Nøsett. Quadatue methods fo multivaiate highly oscillatoy integals using deivatives. Math. Comp., 75: , 26. [17] W. P. Johnson. The cuious histoy of Faà di Buno s fomula. The Ameican Mathematical Monthly, 19(3): , 22. [18] D. Levin. Fast integation of apidly oscillatoy functions. J. Comput. Appld. Maths., 67:95 11, [19] P. D. Mille. Applied Asymptotic Analysis. Ameican Mathematical Society, Povidence, 26. [2] F. W. J. Olve. Asymptotics and special functions. Academic Pess, Inc, New Yok, [21] S. Olve. Moment-fee numeical appoximation of highly oscillatoy integals with stationay points. Euo. J. Appl. Maths, 18: , 26. [22] S. Olve. Moment-fee numeical integation of highly oscillatoy functions. IMA J. of Nume. Anal., 26(2): , Ap. 26. [23] S. Olve. On the quadatue of multivaiate highly oscillatoy integals ove non-polytope domains. Nume. Math., 13(4): , 26. [24] J. Wojdylo. Computing the coefficients in Laplace s method. SIAM Rev., 44(1):76 96, 26. [25] R. Wong. Asymptotic Appoximations of Integals. SIAM Classics,