A Numerical Approach To The Steepest Descent Method

Transcription

1 A Numerical Approach To The Steepest Descent Method K.U. Leuven, Division of Numerical and Applied Mathematics Isaac Newton Institute, Cambridge, Feb 15, 27

2 Outline 1 One-dimensional oscillatory integrals 2

3 Outline One-dimensional oscillatory integrals 1 One-dimensional oscillatory integrals 2

4 One-dimensional oscillatory integrals The model problem I := b f (x)e iωg(x) dx a with... ω: frequency parameter; f : amplitude; g: oscillator f, g: smooth real functions classical quadrature deteriorates rapidly as ω increases take fixed number of points per oscillation amount of operations scales linearly with ω new oscillatory quadrature methods asymptotic, Filon, Levin, numerical steepest descent

5 One-dimensional oscillatory integrals What determines the value of the integral? Example: (1 x 2 )e i ω x regions where the oscillations do not cancel: boundary points regions where the integrand is (locally) not oscillatory: stationary points: solutions to g (x) =

6 One-dimensional oscillatory integrals What is the size of the integral (asymptotically)? I = O(ω 1/(r+1) ) with r: the largest order of any stationary point ξ in [a, b] g (j) (ξ) =, j = 1,..., r; g (r+1) (ξ) Our goal: to find a decomposition of the integral b a f (x)eiωg(x) dx = F (a) + F (b) + i F (ξ i) with ξ i [a, b] the stationary points

7 Basic idea: select a new integration path assume f and g are analytic Cauchy s theorem: the value of I does not depend on the complex path taken C a b

8 The path of steepest descent (Cauchy(1827), Riemann (1863)) e iωg(x) = e iω(rg(x)+iig(x)) = e ωig(x) e iωrg(x) 1 The function e iωg(x) does not oscillate if Rg(x) is fixed 2 The function e iωg(x) decays exponentially fast if Ig(x) > new path at point a: h a (p) such that g(h a (p)) = g(a) + p i, p

9 Example 1: Fourier oscillator g(x) = x (f (x) = 1 x 2 ) g(h a (p)) = g(a) + pi h a (p) = a + pi 3 2 C a h a h b 1 1 b

10 Decomposition: I = b a f (x)eiωg(x) = F (a) F (b) F (a) = f (h a (p))e iωg(ha(p)) h a(p)dp = e iωg(a) f (h a (p))e ωp h a(p)dp = e iωg(a) f (a + pi)e ωp idp

11 One-dimensional oscillatory integrals A new integration path Example 2: Quadratic oscillator g(x) = x 2 (f (x) = 1 x 2 ) g(h a (p)) = g(a) + pi h a (p) = ± a 2 + pi

12 Decomposition: I = F 1 (a) F 1 (ξ) + F 2 (ξ) F 2 (b) 1 1 f (x)e iωx2 dx = e iω f (h 1,1 (p))e ωp h 1,1(p)dp + f (h,1 (p))e ωp h,1(p)dp f (h,2 (p))e ωp h,2(p)dp e iω f (h 1,2 (p))e ωp h 1,2(p)dp Numerical singularity of the path: h ξ (p) p 1/2, p

13 Example 3: Cubic oscillator g(x) = x 3 g(h a (p)) = g(a) + pi h a (p) = 3 a 3 + pi e i2π/3k, k =, 1, 2 C i I = F 1 (a) F 1 (ξ) + F 2 (ξ) F 2 (b) Numerical singularity of the path: h ξ (p) p 2/3, p

14 Example 3: Cubic oscillator g(x) = x 3 (and f (x) = 1 x 2 )

15 Implementation issue 1. How to evaluate F j? F j (x) = e iωg(x) f (h x (p))h x(p)e ωp dp = eiωg(x) ω f (h x ( q ω ))h x( q ω )e q dq if exponential decay: Gauss-Laguerre (w(q) = e q ) if singularity: generalized Gauss-Laguerre (w(q) = q α e q ) or, generalized Gauss-Hermite (w(q) = e qn )

16 F j (x) Q F [f, g, h x ] := eiωg(x) ω n w i f (h x (x i /ω))h x(x i /ω) i=1 F j (ξ) Q r F [f, g, h ξ,j] := eiωg(ξ) ω 1/(r+1) n i=1 absolute error: O(ω ( 2n 1)/(r+1) ) relative error: O(ω 2n/(r+1) ) w i f (h ξ,j (x r+1 i /ω)) h ξ,j r+1 (xi /ω) xi r Gaussian convergence rate 2n as a function of 1/ω (r = )

17 Example: x eiωx dx I Q F [f, g, h a ] Q F [f, g, h b ] ω \ n E 3 3.1E 5 1.9E 6 1.7E 7 2.1E E 4 1.1E 6 2.3E 8 7.5E 1 3.2E E 5 3.9E 8 2.1E 1 2.E E E 6 1.2E 9 1.7E E E 17 rate 3.1(3) 5.(5) 6.9(7) 8.9(9) 1.8(11)

18 Example: x+2 eiωx3 dx I Q[f, g, h a ] Q 2 F [f, g, h ξ,1] + Q 2 F [f, g, h ξ,2] Q F [f, g, h b ] ω \ n E 4 2.4E 6 1.3E 8 6.9E E E 5 7.1E 7 2.3E 9 6.4E E E 5 2.1E 7 4.2E 1 6.1E E E 6 6.7E 8 8.1E E E 16 rate 1.3 (3/3) 1.7 (5/3) 2.4 (7/3) 3.4 (9/3) 3.9 (11/3)

19 Implementation issue 2. How to determine the path? if inverse of g is available: h a (p) = g 1 (g(a) + pi) otherwise...compute h a (x i /ω) and h a(x i /ω) numerically Apply Newton-Raphson iteration to g(h a (p)) g(a) pi = initial guess by truncated Taylor series of g only very few iterations necessary Derivative: g(h a (p)) g(a) pi = g (h a (p))h a(p) i =

20 Example: x eiω(x2 +x+1) 1/3 dx I Q F [f, g, h a ] Q F [f, g, h b ] with 2 nd order Taylor path ω \ n E 2 2.7E 3 7.4E 4 2.4E 4 8.9E E 3 2.6E 4 4.6E 5 1.E 5 2.5E E 4 1.8E 5 1.7E 6 2.E 7 2.9E E 5 1.1E 6 4.E 8 2.1E 9 1.5E E 6 6.8E 8 7.7E 1 1.6E E 13 rate

21 Example: x eiω(x2 +x+1) 1/3 dx I Q F [f, g, h a ] Q F [f, g, h b ] with 2 nd order Taylor path, followed by 1 to 4 Newton iteration steps ω \ n E 2 2.4E 3 7.4E 4 2.5E 4 7.5E E 3 2.4E 4 4.4E 5 1.E 5 2.4E E 4 1.5E 5 1.2E 6 1.5E 7 2.3E E 5 6.1E 7 1.8E 8 8.7E 1 6.2E E 6 2.1E 8 1.8E 1 2.7E E E 7 6.7E 1 1.5E E E 17 rate

22 Relation to Levin type methods Levin method: write I = F (b)e iωg(b) F (a)e iωg(a) with F (x) the non-oscillatory solution to F (x) + iωg (x)f (x) = f (x) It can be verified that the exact solution is given by F (x) = f (h x (p))h x(p)e ωp dp Numerical steepest descent: evaluate the analytical solution to the Levin ODE numerically!

23 The Generalized Filon s method (Iserles and Nørsett) Idea: approximate f globally on [a, b] by Hermite interpolation Assume: f (x) N i=1 c iφ i (x) Then: I N i=1 w ic i with w i = b a φ i(x)e iωg(x) dx Iserles and Nørsett: convergence O(ω s 1/(r+1) ) interpolate derivatives of order... s 1 at corner points interpolate derivatives of order... (s 1)(r + 1) at stationary points computation of moments, e.g., by numerical steepest descent

24 Localized Filon s method Idea: apply local Hermite (or Taylor) approximation of f for each integral contribution F j Define: From: We have: with: F j [f ](x) := e iωg(x) f (h x (p))h x(p)e ωp dp f (z) d j i= f (i) (x) (z x)i i! F j [f ](x) d j i= w i,jf (i) (x) w i,j := F j [ (z x)i i! ](x)

25 A classical quadrature rule Convergence result: I l dj j= i= w i,jf (i) (x j ) Let d j = s 1 at a non-stationary point, and let d j = s(r + 1) 1 at a stationary point then our rule has an absolute error of O(ω s 1/(r+1) ) and a relative error of O(ω s ), asymptotically for large ω.

26 Example 1: x 2 e iω(x 1/2)2 dx Filon type 3 weights localized Filon type numerical steepest descent Filon type 5 weights Filon (3 evals): O(ω 3/2 ) Local Filon (3 evals): O(ω 3/2 ) NSD (4 evals): O(ω 5/2 ) Filon (5 evals): O(ω 2 ) ω

27 Example 2: The Hankel oscillator I H [f ] := For large arguments H ν (1) (z) b a f (x)h ν (1) (ωg 1 (x))e iωg2(x) dx, 2 πz ei(z 1 2 νπ 1/4π), π < argz < 2π, z. Approximate oscillator: g(x) = g 1 (x) + g 2 (x) Quadrature rule: I H [f ] Q H [f ] := L l= dl j= w H l,j f (j) (x l ).

28 1 cos(x 1)H(1) (ωx)eiω(x2 +x 3 x) dx. Two quadrature points: x = : a singularity and a stationary point of order 1 x = 1: a regular endpoint ω \ (d, d 1 ) (, ) (1, ) (2, ) (3, 1) 1 1.2E 3 2.8E 5 1.3E 6 2.6E E 4 8.6E 6 2.9E 7 4.1E E 4 2.6E 6 6.4E 8 6.2E E E 7 1.4E 8 9.7E 11 rate 1.23 (1.25) 1.73 (1.75) 2.2 (2.25) 2.68 (2.75)

29 Outline 1 One-dimensional oscillatory integrals 2

30 Multivariate oscillatory integrals Model form I n := S f (x)e iωg(x) dx Contributing points? corner points critical points: g = resonance points: g S

31 Multivariate oscillatory integrals Main approach: repeated one-dimensional integration deform onto path of steepest descent for inner variable I 1 (x) := b(x) a(x) f (x, y)e iωg(x,y) dy = F (x, a(x)) F (x, b(x)) the function F is evaluated only in points on the boundary F (x, a(x)) = e iωg(x,a(x)) u(x, p) f (x, u(x, p)) e ωp dp p oscillator of F (x, a(x)) is exactly known: g(x, a(x))!

32 Example 1 Rectangular domain in two dimensions I := b d a c f (x, y)e iω(x+y) dy dx Step 1: deform onto path of steepest descent for y I := b a Smooth function G is given by G(x, c)e iω(x+c) G(x, d)e iω(x+d) dx G(x, y) = f (x, y + ip)ie ωp dp

33 Example 1 (continued) Rectangular domain in two dimensions b a G(x, c)e iω(x+c) = G(a, c)e iω(a+c) G(b, c)e iω(b+c) Total decomposition I := F (a, c) F (b, c) F (a, d) + F (b, d) Contributions are given by non-oscillatory double integrals with exponential decay in both variables F (x, y) = e iω(x+y) f (x + ip, y + iq)i 2 e ω(p+q) dq dp

34 Example 1 (continued) Rectangular domain in two dimensions I := b d a c f (x, y)e iω(x+y) dy dx (a,d) (b,d) (a,c) (b,c) I := F (a, c) F (b, c) F (a, d) + F (b, d)

35 Example 2: smooth boundaries 1. A two-dimensional simplex I := b x a a f (x, y)e iω(x+y) dy dx Step 1: deform onto path of steepest descent for y I := b a G(x, a)e iω(x+a) G(x, x)e iω(x+x) dx oscillators are different Result: contributions of the three corner points

36 Example 2: smooth boundaries (continued) 2. More general boundaries I := = b d(x) a b a c(x) f (x, y)e iω(x+y) dy dx G(x, c(x))e iω(x+c(x)) G(x, d(x))e iω(x+d(x)) dx new oscillator x + c(x) may have stationary points! resonance points: stationary point of oscillator g(x, c(x)) evaluated along the boundary happens when g S

37 Example 2: smooth boundaries (continued) Fourier integral on a circle 1 1 x 2 I := 1 f (x, y)e iω(x+y) dy dx 1 x ( 2 ) ( ) = F 2, F 2 2, 2

38 Example 3: critical points A rectangular domain with a critical point I := b d g(x, y) = x 2 xy y 2 g(, ) = g x (x, y) = x = y/2 g y (x, y) = y = x/2 a c f (x, y)e iω(x2 xy y 2) dy dx

39 Example 3: critical points (continued) A rectangular domain with a critical point (a, a/2) (a,d) (d/2,d) (b,d) (,) (a,c) (c/2,c) (b, b/2) (b,c)

40 Example 3: critical points (continued) A rectangular domain with a critical point Step 1: deform onto path of steepest descent for y I := b a G 1 (x, c)e iωg(x,c) G 1 (x, x/2)e iωg(x, x/2) + G 2 (x, x/2)e iωg(x, x/2) G 2 (x, d)e iωg(x,d) dx g 11 (x) := g(x, c) = x 2 cx c 2 g 12 (x) := g(x, x/2) = 5 4 x 2

41 Example 3: critical points (continued) A rectangular domain with a critical point Step 2: deform onto path of steepest descent for x I = F 111 (a, c) F 111 (c/2, c) + F 112 (c/2, c) F 112 (b, c) F 121 (a, a/2) + F 121 (, ) F 122 (, ) + F 122 (b, b/2) + F 211 (a, a/2) F 211 (, ) + F 212 (, ) F 212 (b, b/2) F 221 (a, d) + F 221 (d/2, d) F 222 (d/2, d) + F 222 (b, d).

42 What can be proved Theorem 4.2 (Huybrechs and Vandewalle, 26) I n := f (x)e iωg(x) dx = s λ F λ (x λ ) + O(e ωd ), S size(λ)=2n Conditions are: analyticity of f, g in a complex neighbourhood of S piecewise analytic parameterisation of S stationary points may be degenerate two additional conditions exclude special cases

43 Additional conditions 1. Exclude curves of resonance points and critical points If for some λ we have g λ (x, y), y then the integral in y is not oscillatory. Solution: integration in y can be performed numerically on the real line

44 Additional conditions (continued) Example curve of resonance points on circular boundary Example: x 2 +y 2 <=1 f (x, y)eiω(x2 +y 2) dv

45 Additional conditions (continued) 2. Each lower-dimensional integral in x is either: an integral along a curve of stationary points (of the same order) in y an integral along a curve that has no stationary point in y In particular: this excludes critical points on the boundary Reason: existence of complex stationary points in y that may (or may not) be arbitrarily close to the real line

46 Additional conditions (continued) Example: integral on the part in the upper right halfplane (a, a/2) (a,d) (d/2,d) (b,d) (,) (a,c) (c/2,c) (b, b/2) (b,c)

47 Example 1: 3D balls and ellipsoids Sphere with Fourier oscillator No stationary points, two boundary points Z 1 Z q 1 x 2 Z q 1 x I 3 = x2 2 q 1 1 x 1 2 q 1 x 1 2 e x 1 +x2 2 x 3 (3x 3 + cos(x 2 ))e iω(x 1 +x 2 +x 3 ) dx 3 dx 2 dx 1. x2 2 Contributing points: g S ( 3/3, 3/3, 3/3) and ( 3/3, 3/3, 3/3)

48 Example 1: 3D balls and ellipsoids Cubature rule Convergence I 3 2 i=1 j k l w i,j,k,l j+k+l f x j 1 x k 2 x l 3 (x i ) ω \ d e 5 2.4e 6 1.2e e 6 3.6e 7 8.e e 7 5.2e 8 5.5e e 8 3.8e 9 2.7e e 9 5.2e 1 1.7e 12 rate 3. (2.5) 2.9 (3.) 4. (3.5)

49 Example 1: 3D balls and ellipsoids Generalisation to an ellipsoid 1 I 3 := 4π k2 (n(x) 2 1)e iω a x dx 3 dx 2 dx 1. E length scales R 1, R 2, R 3 along X, Y and Z axis oscillator: a x = a 1 x 1 + a 2 x 2 + a 3 x 3 two resonance points application: scattering of light due to propagation in an object with refractive index n(x) (Sigal Trattner)

50 Example 1: an ellipsoid (continued) Absolute errors R 1 = 1, R 2 = 2, R 3 = 1 tensor-product Gauss-Laguerre with m points per dimension ω \ m (4m 3 ) 1 (4) 2 (32) 3 (18) 4 (256) 1 2.2e 2 4.9e 3 3.8e 3 1.7e e 3 7.5e 5 7.8e 5 6.7e e 3 6.e 7 5.9e 7 5.9e e 6 3.8e 8 6.6e e e 6 2.4e 9 1.1e e 14 rate

51 Example 2: a degenerate stationary point A rectangular domain with a degenerate stationary point I 2 := x + y eiω(x3 +y 3) dy dx, (a,d) (b,d) (a,c) (b,c)

52 Example 2 (continued) Localised Filon-type quadrature rule nine quadrature points use d function values and derivatives at each point ω \ d e 3 2.5e 4 2.9e e 3 9.9e 5 9.e e 3 3.9e 5 2.8e e 4 1.6e 5 8.9e e 4 6.1e 6 2.9e 7 rate.98 (1.) 1.34 (1.33) 1.63 (1.66)

53 Concluding remarks 1 One-dimensional integrals two types of points: stationary points and endpoints integration along the path of steepest descent construction of Filon-type quadrature rules 2 Multi-dimensional integrals three types of points: corners, critical points, resonance points integration on a manifold of steepest descent construction of Filon-type cubature rules 3 Problems not treated here functions with complex poles or stationary points oscillatory integral equations