Moment-free numerical approximation of highly oscillatory integrals with stationary points

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1 Moment-fee numeical appoximation of highly oscillatoy integals with stationay points Sheehan Olve Abstact We pesent a method fo the numeical quadatue of highly oscillatoy integals with stationay points. We begin with the deivation of a new asymptotic expansion, which has the popety that the accuacy impoves as the fequency of oscillations inceases. This asymptotic expansion is closely elated to the method of stationay phase, but pesented in a way that allows the deivation of an altenate appoximation method that has the same asymptotic behaviou, but with significantly geate accuacy. This appoximation method does not equie moments. 1. Intoduction We ae concened with numeically appoximating the highly oscillatoy integal whee ω is lage, I[f = 1 1 fx)e iωgx) dx, 0 = g0) = g 0) = = g 1) 0), g ) 0) > 0, and g x) 0 fo 0 < x 1. Note that the case whee g0) 0 can easily be handled by eplacing g with g g0), and multiplying the integal by e iωg0). The condition that g ) 0) > 0 implies that gx) > 0 fo 0 < x 1, and 1) gx) > 0 fo 1 x < 0. This condition, as well as the continuity of g and f can be elaxed, at the expense of complicating the poofs. Computing such integals using taditional techniques, such as Gaussian quadatue, is difficult. Accuacy depends on thee being seveal samples pe oscillation, which fo lage fequency is extemely inefficient. Fixing the numbe ) of quadatue points esults in an eo O1) as ω, whilst the integal itself decays like O ω 1 [9. The method of stationay phase [5 povides only an asymptotic esult; fo fixed fequency the accuacy of the appoximation is limited. It is possible to compute the integals by moving to the complex plane and integating along the path of steepest descent [3, howeve, this suffes fom the equiement that both f and g ae analytic, in ode to defom the integation path, and equies the knowledge o computation of the path of steepest descent. Filon-type methods allow efficient appoximation [4, howeve, they equie the knowledge of moments I [ x k, which in geneal ae unknown. In this pape we will pesent a new method such that the accuacy actually impoves as the fequency inceases, up to any chosen asymptotic ode. We begin by constucting a new asymptotic expansion that does not equie moments. Fom this asymptotic expansion we can find a basis fo a Filon-type method which can be integated explicitly in closed fom. This allows us to impove the accuacy of the appoximation fo fixed asymptotic odes. 2. Asymptotic Expansion Asymptotic expansions povide an invaluable tool fo high fequency integation. Fo the integal in question, thee exists two existing asymptotic expansions: the Iseles and Nøsett expansion found in [4 and Depatment of Applied Mathematics and Theoetical Physics, Cente fo Mathematical Sciences, Wilbefoce Rd, Cambidge CB3 0WA, UK, S.Olve@damtp.cam.ac.uk 1

2 the well-known method of stationay phase [5. The fome of these equies knowledge of the moments I[1,...,I [ x 1 but leads us to the moe poweful numeical appoximation of Filon-type methods and Levin-type methods [6. Stationay phase does not equie moments, yet only povides an asymptotic esult, hence fo fixed fequency its usefulness as a numeical quadatue scheme is limited. In this section, we will deive an asymptotic expansion that bidges the gap between the two: it is oughly equivalent to the method of stationay phase, in that it does not equie moments, but pesented in a way analogous to the Iseles and Nøsett expansion, allowing us to deive a Filon-type method such that moments ae not necessay. We fist pesent the Iseles and Nøsett expansion, to motivate the methodology behind the new expansion. Fo simplicity, we take = 2, though the Iseles and Nøsett expansion exists fo highe ode stationay points as well, equiing the knowledge of additional moments. The standad technique of deiving asymptotic expansions, namely integation by pats, fails due to the intoduction of a singulaity at the stationay point. But we can make the singulaity emovable: I[f = I[f f0) + f0)i[1 = 1 iω [ f1) f0) = g e iωg1) 1) b a fx) f0) g x) f 1) f0) g e iωg 1) 1 1) iω I d dx eiωgx) dx + f0)i[1 [ [ d fx) f0) dx g x) + f0)i[1. Iteating this pocess esults in an asymptotic expansion. Howeve, this suffes fom equiing the fist moment, as well as highe ode moments when additional deivatives of g vanish at the stationay point. The idea behind the new expansion is to note that we do not necessaily need to subtact a constant, it is only necessay that the function we subtact is nonzeo at the stationay point. Hence we can eplace the moments I [ x k, which may not be computable in closed fom, with I[ψ k, whee ψ k is constucted in such a way that the integal is guaanteed to be computable. In ode to do this, we fist look at the canonical case of gx) = x. Suppose thee exists a function F such that d [ F x)e iωgx) = x k e iωgx). dx We can expand out the left-hand side to obtain the following equation: L[F = F + iωg F = x k. Replacing g with x 1 we obtain the equation F + iωx 1 F = x k. A solution to this equation is known: F x) = 1+k ω ) ) e iωx + 1+k 1 + k 1 + k 2 [Γ iπ, iωx Γ, 0, whee Γ is the incomplete gamma function [1. Incomplete gamma functions ae well-known, and can be computed efficiently [2. In fact, moden mathematical pogam languages, such as Maple, Mathematica and Matlab via the mfun function) have vey efficient built-in numeical implementations. Intuition suggests that if we eplace x with gx), then L[F will give us the ψ k we wee looking fo, hopefully independent of ω. The following lemma shows that ou intuition is indeed coect: Lemma 2.1 Let whee k+1 ω 1+k iωgx)+ φ,k x) = D,k sgn x) e 2 [Γ iπ Then φ,k C [ 1, 1 and, fo L[F = F + iωg F, 1 + k ) ) 1 + k, iωgx) Γ, 0, 1) k sgn x < 0 and even, D,k sgn x) = 1) k e 1+k iπ sgn x < 0 and odd, 1 othewise. L [ k+1 φ,k x) = sgnx) +k+1 gx) 1 g x). 2

3 Futhemoe, L [ φ,k C [ 1, 1. Finally, I [ L [ φ,k = φ,k 1)e iωg1) φ,k 1)e iωg 1). Poof : The fom of L [ φ,k away fom the stationay point follows immediately fom the equation fo the deivative of the incomplete Gamma function [1. The continuity of L [ φ,k follows fom the fact that and [ d sgnx) k k gx) = L [ φ dx 1 + k,k x) x 0, sgnx) k+1 gx) 1+k = sgnx) k+1 g 0) x + O 1+k x +1)! g 0) =! ) k+1 x k Ox)) k+1 is C [ 1, 1. Combining this with the fact that φ,k is continuous ensues that φ,k C. The final integal thus follows fom the fundamental theoem of calculus. Remak: The use of sgn and the case statement in the peceding lemma ae meely to choose the banch cut so that x ) 1/ = x fo both positive and negative x. We can also pove that { L [ φ,k } is a Chebyshev set [8, hence can intepolate at any given sequence of sample points. Lemma 2.2 The basis { L [ φ,k } is a Chebyshev set. Poof : Let u = sgnx) gx) 1/, whee u anges monotonically fom g 1) 1/ to g1) 1/. Let g+ 1 u) equal x 0 such that gx) = u, and g 1 u) similaly. When is odd then g± 1 = g 1. Note that sgn x = sgn u, hence x = gsgn 1 u u ). Thus we obtain ck L [ φ,k x) = sgnx) +1 g x) gx) 1 1 ck sgnx) k gx) k It follows that intepolating f by L [ φ,k is equivalent to intepolating = g x)u 1 ck u k. u 1 fx) g x) by the polynomial c k u k. This function is clealy well-defined fo u 0, hence we must show that it is also well-defined fo u = x = 0. But this follows since u 1 g x) = sgnx)+1 gx) 1 1/ g x) = = sgnx)gx) gx) 1/ g x) g x + O x +1)) x g + Ox)) 1/ )g + Ox )) = g + O x +1)) g + Ox)) 1/ )g + Ox )). set. 1 The limit of this as x goes to zeo, hence also as u goes to zeo, is g. Thus L [ φ,k is a Chebyshev Using L [ φ,k in place of x k, we can deive an altenative to the asymptotic expansion in [4, which does not depend on any moments: 3

4 Theoem 2.3 Define µ[f = 2 k=0 c k φ,k so that Futhemoe, let Then I[f k=0 L[µ[f0) = f0),..., L[µ[f 2) 0) = f 2) 0). σ 0 x) = fx), σ k+1 x) = d dx σ k x) L[µ[σ k x) g. x) 1 { iω) k µ[σ k 1)e iωg1) µ[σ k 1)e iωg 1)} { 1 σk 1) L[µ[σ k 1) iω) k+1 g 1) e iωg1) σk 1) L[µ[σk g 1) k=0 1) e iωg 1) }. Poof : The poof is based on the poof of Theoem 3.2 in [4. Note that the existence of such a µ follows fom Lemma 2.2. We find that σ k C [0, 1, since is C [0, 1. Then σ k x) L[µ[σ k x) g x) = O x 1) g 0) 1)! x 1 + Ox ) = O1) g 0) 1)! + Ox) I[σ k = I[σ k L[µ[σ k + I[L[µ[σ k = 1 1 σ k L[µ[σ k d iω 1 g dx eiωg dx + {µ[σ k 1)e iωg1) µ[σ k 1)e iωg 1)} = 1 { σk 1) L[µ[σ k 1) iω g e iωg1) σ k 1) L[µ[σ } k 1) 1) g e iωg 1) 1) { + µ[σ k 1)e iωg1) µ[σ k 1)e iωg 1)} 1 iω I[ σ k+1. The theoem follows fom induction. The method of stationay phase can be deived as a consequence of Theoem 2.3. Conside the case of equal to two. Then µ[fx) = 2 g 0) f0)φ 2,0 x), since L[ φ 2,0 0) = g 0) 2. If we assume that gx) x 2 as x ±, then ± fe iωg dx = O ω 1) [5. Thus fomally we obtain ±1 I[f = = e iπ 4 2 ω = e iπ 4 fe iωg dx + O ω 1) = { [ 2 g 0) f0) lim x Γ f L[µ[f)e iωg dx + ) 1 2, iωgx) [ 1 lim Γ x 2π ωg 0) f0) + O ω 1). 2, iωgx) ) 1 Γ 2, 0 ) 1 Γ L[µ[f e iωg dx + O ω 1) ) } 2, 0 + O ω 1) We can demonstate this asymptotic expansion in action. Note that µ[σ k ±1) = O ω 1/), thus the patial sum up to s 1 of the asymptotic expansion has an asymptotic ode O ω s 1/). Conside the case whee fx) = cos x with the polynomial oscillato gx) = 4x 2 + x 3. The moments cannot be 4

5 Figue 1: The eo scaled by ω 3/2 of the one-tem asymptotic expansion left-hand figue), vesus the eo scaled by ω 5/2 of the two-tem asymptotic expansion ight-hand figue), fo 1 1 cos x +x 3) eiω4x Figue 2: The eo scaled by ω 4/3 of the one-tem asymptotic expansion left-hand figue), vesus the eo scaled by ω 7/3 of the two-tem asymptotic expansion ight-hand figue), fo x+2 eiω1 cos x 1 2 x2 +x 3) dx. integated in closed fom, hence the Iseles and Nøsett expansion is not applicable to this integal. On the othe hand, Figue 1 demonstates numeically that Theoem 2.3 does indeed give an asymptotic expansion. Fo a moe complicated example, conside the integal whee fx) = x + 2) 1 with the oscillato gx) = 1 cos x 1 2 x2 +x 3. Figue 2 demonstates that the expansion does indeed wok with highe ode stationay points, in this case is thee, and with nonpolynomial oscillatos. The following coollay, oiginally stated in [4, follows fom the asymptotic expansion. It is used in the poof of Filon-type methods. Coollay 2.4 Suppose that Then 0 = f 1) = = f s 1) 1), 0 = f0) = = f 2s 1) ) 0), 0 = f1) = = f s 1) 1). I[f O ω s 1/), ω. Poof : Note that σ k depends on f and its fist k deivatives, hence the equiement at the bounday points. We pove the equiement on the numbe of deivatives at the stationay point by induction. The case whee s = 1 is clea: we need f and its fist 2 deivatives to be zeo in ode fo µ[σ 0 = µ[f = 0. The coollay thus follows fom L Hôpital s ule, and the fact that g has a zeo of ode 1. 5

6 3. Moment-fee Filon-type methods The majo failing of using an asymptotic expansion as a numeical appoximation is that fo fixed fequency the expansion in geneal does not convege. To combat this poblem Filon-type methods wee developed in [4. A Filon-type method is constucted by intepolating the function f, using the basis functions {ψ k }, at a sequence of nodes {x 1,..., x ν } and multiplicities {m 1,..., m ν }. The following theoem is fom [4. It states that we obtain the same asymptotic behaviou as the asymptotic expansion when we use the same numbe of deivatives at the endpoints of the inteval. Unlike an asymptotic expansion, we can add additional intepolation points within the inteval to educe the eo fo a fixed asymptotic ode. In fact, if the intepolant conveges unifomly to f, then it follows necessaily that the quadatue conveges to the exact value of the integal, fo fixed ω. Theoem 3.1 Let vx) = n k=1 c k ψ k, whee ψ k is independent of ω and n = ν x 1 = 1, x η = 0 and x ν = 1. If c k ae chosen so that k=1 m k vx k ) = fx k ),..., v m k 1) x k ) = f m k 1) x k ), k = 1, 2,..., ν, then, assuming this system is nonsingula, when m 1, m ν s, m η 2s 1) 1) and I[f Q F [f O ω s 1/) Q F [f = I[v = n c k I[ψ k. k=1. Assume that Poof : The theoem follows as a diect consequence of Coollay 2.4: I[f Q F [f = I[f v O ω s 1/). In pactice, ψ k is typically defined to be x k, i.e., we use standad polynomial intepolation. The eason is two-fold: polynomial intepolation is well undestood and guaanteed to intepolate at the given nodes and multiplicities, and the simplicity of the integand suggests that the moments I [ x k ae likely to be known. Howeve, when the moments ae unknown, Filon-type methods with the polynomial basis cannot povide an appoximation. In Lemma 2.1, we detemined a basis of functions such that the moments ae guaanteed to be known, hence it makes sense to choose ψ k = L [ φ,k in a Filon-type method. Moeove, it was poved in Lemma 2.2 that ψ k is a Chebyshev set, hence we know that it can intepolate at the given nodes and multiplicities. Thus using this basis we obtain the following theoem: Theoem 3.2 Let ψ k = L [ φ,k. Then I[f Q F [f O ω s 1/), whee n Q F [f = c k [φ,k 1)e iωg1) φ,k 1)e iωg 1). k=1 6

7 Figue 3: The eo scaled by ω 3/2 of a Filon-type method with endpoints and zeo fo nodes and multiplicities all one left figue, top) and a Filon-type method with nodes { 1, 1 2, 0, 1 2, 1} and multiplicities all one left figue, bottom), and the eo scaled by ω 5/2 of the two-tem asymptotic expansion ight figue, top) and a Filon-type method with nodes { 1, 0, 1} and multiplicities {2, 3, 2}, fo I[f = 1 1 cos x +x 3) eiω4x2 dx Figue 4: The eo scaled by ω 4/3 of the one-tem asymptotic expansion left figue, top), a Filon-type method with nodes { 1, 0, 1} and multiplicities {1, 2, 1} left figue, middle) and a Filon-type method with nodes { 1, 1 2, 0, 1 2, 1} and multiplicities {1, 1, 2, 1, 1} left figue, bottom), and the eo scaled by ω 7/3 of the two-tem asymptotic expansion ight figue, top) and a Filon-type method with nodes { 1, 0, 1} and multiplicities {2, 5, 2} ight figue, bottom), fo I[f = x+2 eiω1 cos x 1 2 x2 +x 3) dx. Unlike the asymptotic expansion, a Filon-type method allows inceasing the accuacy fo fixed ode. Figue 3 demonstates this with the same integal as in Figue 1. Note that the eos in the left figue ae of the same asymptotic ode as the left figue of Figue 1, howeve the eo is significantly less. This is despite the fact that we ae using exactly the same infomation abount f as we ae in the asymptotic expansion. Futhemoe, this figue demonstates how adding intepolation points can futhe educe the eo. The ight figue shows how adding sufficient multiplicities to a Filon-type method does indeed incease the asymptotic ode, and compaes the esulting quadatue with the equivalent asymptotic expansion. We obtain simila esults fo the integal with a highe-ode stationay point found in Figue 2, as seen in Figue 4. Remak: We puposely chose oscillatos such that g x) 0 fo 0 < x < 1. Without this, g x) would no longe be monotone and the basis L [ φ,k would diffe geatly in behaviou fom the polynomial basis. Though the theoems emain valid, numeical esults suggest that L [ φ,k becomes much less accuate fo intepolation, hence a lage amount of sample points would be equied. A simple wokaound is to choose a sufficiently small neighbouhood aound zeo such that this condition is satisfied, and use a Momentfee Filon-type method within this neighbohood. We could then appoximate the integal outside this 7

8 neighbouhood using a Levin-type method [6. Like Moment-fee Filon-type methods, Levin-type methods do not equie moments, howeve, they cannot be used in the pesence of stationay points. They ae not affected numeically by g vanishing. 4. Futue wok It might be possible to genealize these esults in the multivaiate setting, namely integating fx)e iωgx) dv, Ω whee Ω is piecewise smooth. In [7, the cuent autho deived a quadatue method by using the opeato L[v = v + iω g v to push the value of the highly oscillatoy integal ove a domain to a highly oscillatoy integal ove the bounday, assuming that thee ae no citical points within the domain: g 0. Initial numeical esults suggest that it should be possible to combine these esults with the esults fom this pape in ode to deive a numeical appoximation in the pesence of citical points. We would thus obtain the integal as an integal of incomplete gamma functions ove the bounday. Thee ae, howeve, seveal majo issues. The fist poblem is that the asymptotics of such integals is not known in all cases. Indeed, citical points need not be isolated: thee can be cuves of citical points within the domain. Futhemoe, geate cae must be taken so that ou constucted basis is C. Finally, once the integal is pushed to the bounday, thee is still the question of how one might integate the esulting incomplete gamma functions. Refeences [1 Abamowitz, M., and Stegun, I., Handbook of Mathematical Functions, National Bueau of Standads Appl. Math. Seies, #55, U.S. Govt. Pinting Office, Washington, D.C., [2 Cody, J., An Oveview of Softwae Development, Lectue Notes in Mathematics, 506, Numeical Analysis, Dundee G.A. Watson ed.), Spinge Velag, Belin, [3 Huybechs, D., and Vandewalle, S., On the evaluation of highly oscillatoy integals by analytic continuation, SIAM J. Num. Anal., to appea. [4 Iseles, A., and Nøsett, S.P., Efficient quadatue of highly oscillatoy integals using deivatives, Poceedings Royal Soc. A ), [5 Olve, F.W.J., Asymptotics and Special Functions, Academic Pess, New Yok, [6 Olve, S., Moment-fee numeical integation of highly oscillatoy functions, IMA J. Num. Anal ), [7 Olve, S., On the quadatue of multivaiate highly oscillatoy integals ove non-polytope domains, Nume. Math., to appea. [8 Powell, M.J.D., Appoximation Theoy and Methods, Cambidge Univesity Pess, Cambidge, [9 Stein, E., Hamonic Analysis: Real-Vaiable Methods, Othogonality, and Oscillatoy Integals, Pinceton Univesity Pess, Pinceton, NJ,

norges teknisk-naturvitenskapelige universitet Asymptotic analysis of numerical steepest descent with path approximations preprint numerics no.

norges teknisk-naturvitenskapelige universitet Asymptotic analysis of numerical steepest descent with path approximations preprint numerics no. noges teknisk-natuvitenskapelige univesitet Asymptotic analysis of numeical steepest descent with path appoximations by Andeas Asheim 1, Daan Huybechs 2 pepint numeics no. 2/29 nowegian univesity of science